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The
It is not necessarily
internal, and in fact is often taken to be countable. The finitary probability logic Lwp' or Lwp(I, IT) , with signature (I, IT) has the following symbols. variables equality symbol
"', v , 1
connectives
probability quantifiers
(Px
for each variable x and real number r constants
€
> €
rj , (Px
r)
~
[0, IJ ,
I
relation symbols An atomic formula of L
wp is an expression of one of the forms
where u
l"'"
ulT(i) are variables or constants.
The set of formulas of Lwp is the smallest set satisfying the following formation rules: (2.1)
All atomic formulas are formulas.
(2.2)
If 'P, ljJ are formulas, then 1'P, 'P ",ljJ , 'P v ljJ are formulas.
(2.3)
If 'P is a formula, and (Px
~
€
[0, 1J, then (Px > r)'P
r)'P are formulas.
The expressions (Px quantifiers.
x a variable, and r
> r) and
(Px ~ r) are called probability
Free and bound variables are defined as usual with respect to
H.KEISLER
18 these quantifiers.
An ntuple of vertables is written x == (Xl"'"
x >. n
m and an ntuple a
Given a hyperfinite model
E
An , the satisfaction
relation is defined as usual with the following quantifier clauses: (2.4)
mpo
(Py > r)
°fJ.{b
E
m j=
(2.5)
(Py;:::
°fJ.{b
2.6. finite model
Proof:
A:
E
m 1==
IFF
r)
>
r .
IFF
A: ~ j=
PROPOSITION.
For each formula
m, the relation on A defined by
By induction on the complexity of
At the quantifier
steps we use Theorem 1. II: If S
E
B(An+l) and r
E
[0,1] , then
{a: °fJ.{b: S(a,b)} > r}
E
a (An) ,
{a: °fJ.{b: S(a,b)};::: r }
E
B(An) .
The above proposition shows that the quantifier clauses (2.4) and (2.5) in the definition of satisfaction are always meaningful, that is, always applied to a measurable set. depends only on
0fJ. is
Moreover, the truth value of
mand the as signment a of X.
HYPERFINITE MODEL THEORY
19
If ~ is a hyperfinite model, the relation defined by cp(X) in
general be Borel but not internal. tifiers.
m will in
This is because of the probability quan
For example,
m F (px > O)S(x, y)(b])
{b , 00
U
m =1
1 {b:f1{a:S(a,b)}~iii}
and {c . m F (Px> r)(Py ~ s)S(x, y, z)[cJ)
Here is an interpretation of a hyperfinite model, which we shall call the opinion poll model.
It will be mentioned again in Section 5 as an
illustration of our analogue of the LClwenheimSkolem Theorem.
Let A be
a hyperfinite set and let f1 be the uniform internal probability measure on A.
Let S. (x, y), J J
€
W
,
be a list of internal relations on A. We think
of A as a large set of people,
S. as a statement about the tuple J
y,
and
sJ(a, b) is true iff the person a agrees with the statement Sj about the tuple of people
b.
Thus (px > tHPY > t)S(X, y)
means that a (noninfinitesimal) majority of xs agree with S about a (noninfinitesimal) majority of ys . The sentence (py
>
t )(Px > t jsrx, y)
means that for a majority of y' a , a majority of y.
XiS
agree with S about
Neither of the above sentences implies the other. In applications it is often natural to use two sorted models.
For
example, the Brownian motion model we shall discuss in Section 4 has two sorts, one for paths and one for time.
Everything in this paper can be
generalized in a routine way to two sorted models, but at the cost of a
H.KEISLER
20 more complicated notation.
We briefly define twosorted models and
thereafter confine our attention to the onesorted case. A twosorted hyperfin1te model is a structure
m=
(A, IJ. , B ,
where (A, lJ.)and (B, a.
1
E:
A,
b.
J
E:
a i ' b.J , Sk)'1
E:
I,J'
J k E:K
E:,
are' disjoint hyperfinite probability spaces,
II)
B, and Sk
II ,
C

AlT(k) X BT (k) are internal relations.
The
corresponding two sorted probability logic has two sorts of variables, one sort ranging over A and the other ranging over B.
The probability quan
tifiers are interpreted in the natural way, using the measure fJ. or
II
corresponding to the sort of the variable. We now return to the original onesorted logic. A formula 'P(X) of L
wp
is said to be valid, in symbols
finite model
a
m and ntuple
equivalent if 1=
E:
1= 'P , if
~
F 'P[a]
for every hyper
Two formulas 'P(X) and t!J(X) are
An.
t!J •

We shall see later that the set of valid formulas does not depend on the particular hyperreal universe
>~V(R)
in which we are working.
Here is
a simple result which reduces the problem of validity of formulas to the problem for sentences. 2. 7.
PROPOSITION.
occurring in
Then
Let c be an ntuple of constants not
1=
1= 'P(e'" •
The following more general fact is easily proved by induction
on complexity.
Let
c be an ntuple of constants interpreted by a in
~.
Then for any formula
u
1=
For many purposes in model theory the atomless models play the role in probability logic which infinite models play in classical logic. A model ~
is said to be atomless if every element a
fJ.{a)
r:::l
O.
E:
A has infinitesimal measure,
The class of atomless models is definable by a single sentence
HYPERFINITE MODEL THEORY
2.8.
PROPOSITION. ~
Proof:
~
A model ~ is atomless if and only if
j= l (PX > O)(Py > O)X =: y .
Consider an arbitrary m. °fJ.(a) > 0
Thus if
21
IFF
For each a
E
A ,
~ j= (Py> O)x =: y[a]
is atomless, {a: m j= (Py > 0)
X
=: y[aJ}
¢,
whence
On the other hand if
m j=
l (Px > O)(Py > O)x =: y .
m is
not atomless, say °fJ.(b) > 0 , then
so mj=(Px>O)(Py>O)x=:y. In atomless models one can eliminate quantifiers from pure identity formulas, that is, formulas without relation symbols (but with the identity and constants).
More precisely, for each pure identity formula rp(x) there
is a pure identity formula <jJ(X) without quantifiers such that j= l (Px > O)(Py > O)x =: y  ('I' 
<jJ) •
In fact, for each pure identity sentence 'I' without constants, either 'I' holds in all atomless models or (lrp) holds in all atomless models.
The
proofs are left as exercises. The probability quantifier
(Px
~
r) can be defined from the quantifier
(Px > 1  r) , because it is easy to see that (2.9)
j= (Px
~
r) 'I' 
l (Px
> 1  r) l
'I'
22
H.KEISLER
Thus the quantifiers (Px?: r) could have been introduced as abbreviations instead of included in the list of symbols of the logic.
It is convenient to
have both types of quantifiers because of the following useful prenex normal form theorem. A prenex formula of L is a formula of the form wp
where
°
1,°2 " " , Om are probability quantifiers of either type (Px > r) , (Px?: r) and lIJ is a formula without quantifiers. 2. 10.
Every formula cp(x) of
PRENEX NORMAL FORM THEOREM.
L is equivalent to a prenex formula cp' (x) wp
.
Like the classical proof, the argument is by induction on the
Proof:
complexity of cp(X) and uses the fact that certain schemes of formulas are valid.
In each case it is a routine matter to check that the formulas are
valid. We shall skip the details and simply list the required facts about validity as a hint. (1)
If cP is obtained from a tautology in propositional logic by
substitution, then (2)
If
(3)
and (4)
(5)
F
cp 
cp'
F cp
•
then
F (Px>
r) cp
(Px > r) cp' ,
and
F (Px ?:
r) cp
(Px ?: r) cp'
F l(Px> r) cp
(Px ?: 1  r)l cp ,
F 1 (Px?: r) cp
(Px > 1  r] 1
If cp(x, y) is prenex and z does not occur in cp(x,
F (Px > r)
(pz
F (Px?: r) cp(x, y)
(pz?:
>
y) ,
y) , r) cp(z, y) . r) cp(z,
If x does not occur in cp , then
F
r)lIJ
(Px > r)(
r) lIJ
(Px?: r)(cpvllJ)
F CPA(Px >
r)lIJ 
(Px> r)(CPAlIJ)
r
< 1 .
then
HYPERFINITE MODEL THEORY
F 'P1\(Px
== r)tjJ 
(Px
23
== r)('Pl\tjJ) ,
>
r
0
.
1= l (Px > 1) 'I' ,
(6)
1= (Px == 0) cP •
The completeness problem for Lw p is the problem of characterizing the valid sentences by means of valid axioms and rules of inference.
The
reader can easily discover schemes of valid formulas in addition to those listed in the above proof.
Here are some examples which depend on
properties of the probability measure. If
F (Px
r > s ,
> r) cP 
(Px > s) 'I' .
F [(Px > r)(cpl\tjJ)I\(Px > s)(cpl\ltjJ)] 
F (Px>
r + s) 'P 
If rs
(Px> r+ s) cP ,
[(Px> r)(cpl\tjJ)v(Px > s)('P1\ ltjJ)
+ r' a' == 1, then
F l~PX > r)(Py > s) cP
1\
(Px > r')(Py > S')l 'P]
We shall now tum to the infinitary logic L P'
wI
L
wl
has all the
formation rules of L plus an additional infinitary rule: wp If
(2.11)
are formulas, then
x is an ntuple of variables and
1\ cP (x) and
m m
'l'm(X), m
E: W ,
V'P (X) are formulas. m m
Notice that the probability quantifiers may be applied to formulas of L P as well as L P wI w
In general the formulas of L P belong to V("'R)
wI
but if the signature (I, ()')
r
belongs to V(R) , then the formulas of
belong to V(R) • Each formula of L has at most finitely many free variables. wlP
Since
countable intersections and unions of measurable sets are measurable, there is no difficulty in defining the satisfaction relation
for formulas of L
wl
We state the analogue of Proposition 2.6.
24
H.KEISLER 2.12.
PROPOSITION.
For each formula ;a(x) of L P and
wI
hyperfinite model m, the relation {a: m1=
on A is a Borel set.
We next prove a normal form theorem which is a uniform version of the monotone class theorem from measure theory.
Like the Prenex Normal Form
Theorem, it will be useful later on. By a monotone conjunction we shall mean an infinite conjunction
1\
wl
such that for each m ,
Bya monotone disjunction we mean a disjunction
Vt\Jm(X) such that
m
for each m ,
Thus monotone conjunctions correspond to intersections of decreasing chains, and monotone disjunctions to unions of increasing chains. The set of monotone formulas of L is the least set such that: "lIP (2.13) Every prenex formula of L
wp
is monotone.
(2.14)
A monotone conjunction of monotone formulas is monotone.
(2.15)
A monotone disjunction of monotone formulas is monotone.
The monotone complexity of a monotone formula is defined as follows. Prenex formulas have monotone complexity zero. of
I\
The monotone complexity
(x) is the least ordinal greater than the complexities of
m m
m < co .
The monotone complexity of V 2.16.
of L
wl
(x)
m m
Proof:
(x),
is defined similarly.
MONOTONE FORMUlA THEOREM.
is equivalent to a monotone formula
m
Every formula
(X)
Let M be the set of all formulas of L P which are equivaWI
lent to monotone formulas with the same free variables. formula is prenex and hence belongs to M.
Every atomic
Any infinite conjunction
HYPERFINITE MODEL THEORY
25
is equivalent to the monotone conjunction
/\ 'I'm
m
/\l\J m, m
Therefore to prove the theorem it suffices to prove that M is closed under negations, finite conjunctions, monotone conjunctions, and the quantifiers (Px> r] . It is easily seen by induction on monotone complexity that if 'I' is
monotone then (lq» Suppose 'I' m(X) conjunction.
If follows that if 'I'
E:
M.
E:
M for each m < wand
E:
M then (lq»
E:
M .
/\ 'I' (X) is a monotone
m m
For each m , choose a monotone formula l\J (X) equivalent m
'I'm(X) .
Then ~l\Jm(X) is a monotone formula equivalent to ~q>m(X), so
/\'1' (X)
E:
m m
M .
To show that M is closed under the quantifiers [Px > r) we first observe that for a monotone disjunction V cp (X) or conjunction /\ 'I' (X), m m m m (1)
1= (px> r)Vq>
(2)
1= (Px> r)/\q>
V (Px
m m m m
m

V
O
>
r) 'I'
m
,
1 /\m(Px> r+ r)({Jm
Since 1= 'I'  l\J
implies
the disjunction V (Px > m
1= (px> r)q> 
(Px> r)ljJ ,
on the right of (1) is monotone.
conjunction /\(Px > r + m
on the right of (2) is monotone.
Also, each Moreover,
since r > s
implies
1=
(Px > r) ({J

(Px > s) cp
r
the disjunction
on the right side of (2) is monotone.
Obviously, if 'I' is a prenex formula
then (Px > r) 'I' is prenex and hence monotone.
Using (1) and (2), we see
H.KEISLER
26
by induction on monotone complexity that if ep is monotone then (px > r) ep EM.
Therefore if ep E M then (Px > r) ep EM.
To prove that M is closed under finite conjunction, we borrow a trick from classical measure theory (see Halmos [12]).
Fix the ntuple x .
For each ep(X) in L p let wI
Then K(ep) is closed under equivalence. II'
E K(ljJ).
tions,
By symmetry, if tjJ E K(cp) then
Since M is closed under monotone conjunctions and disjunc
K(ep) is closed under monotone conjunctions and disjunctions.
By
the Prenex Normal Form Theorem every formula of Lwp belongs to M. Suppose epo(X) is in Lwp '
Then ljJ E K(epO) for all tjJ(X) in Lwp '
It
follows that tjJ(X) E K(cpO) for all monotone ljJ(X) , and hence tjJ(X) E K(epO) for all tjJ(X) EM.
Now suppose ep(X) EM.
For any tjJ(X) in Lwp ' we
have ep E K(ljJ) and hence ljJ E K(ep). As before it follows that whenever ljJ(X) EM, tjJ E K(ep).
Therefore if ep(X) , ljJ(X) EM, then ep(X) 1\ ljJ(X) EM.
This completes our proof. By leaving the quantifiers out of the above proof we obtain an analogous result. 2.17.
MONOTONE FORMUIA THEOREM WITHOUT QUANTIFIERS.
Every formula ep(X) of L P without quantifiers is equivalent to a monowI tone formula ep' (X) of L p without· quantifiers. wI We remark that every formula of Lwp without quantifiers is prenex and hence monotone. We shall now introduce an infinitary logic which is richer than L p' WI the Suslin logic L)O(p'
The analogous Suslin logic
~w
' with universal
and existential quantifiers instead of probability quantifiers, was studied by Ellentuck [8]. In addition tothe symbols of L P' WI
L"p has a new infinitary connec
xx.
HYPERFINITE MODEL THEORY tive XX, called the Suslin connective. sequences of elements of w.
Let
w$
27
be the set of all finite
The logic Lxxp has all the formation rules
(2.1), (2.2), (2.3), (2.11) of L P plus the following new rule: wI (2.18)
x is an
If
ntuple of variables and 'Ps(X) , s
€
w,S;;, are
formulas, then XX '" (X) is a formula. s
wl '
Thus LXXpiS an extension of L
Notice that probability quantifiers
'
and connectives can be applied to formulas of L XXp as well as L
wl
and each formula of L XXp has at most finitely many free variables. The satisfaction relation
mF 'P[a] for L XXp is defined by induction
on the complexity of 'P(X) as usual.
The clause of the definition giving
the interpretation of XX is: (2.19)
mF XX"'s[a]
if and only if there exists f
Intuitively the finite sequences s
€
€ WW
such that
w,S;; should be thought of as nodes
~
in a countable tree, and XX 'Ps(X) means that for some branch f
€
w w
of
the tree, the conjunction ~ 'l'f tm(X) of the nodes in the branch holds. prove that the satisfaction definition
mF 'P[a]
'P(X) of L XXP , we must show that the relation
To
is meaningful for formulas
{a:
m F 'P[a]}
is always
Loeb measurable so that the probability quantifiers can be interpreted. We need a classical lemma (for a proof see Kuratowski and Mostowski [15J, pp. 430431). 2.20. space.
If X
s
LEMMA.
Let (B, 1') be a complete probability measure
is measurable for each s
€
JS ,
then
is
measurable. 2. 21. PROPOSITION.
For every formula ",(x) of L XXp and
H.KEISLER
28
and every hyperfinite model ~, the relation {a: \U 1=
defined by
By induction on the complexity of
Suslin connective
)Q(
uses Lemma 2. 20.
that the relation {(a, b) : \U
1=
<jJ[a, bj}
The step for the
For the quantifier step we assume
is Loeb measurable.
It follows
from the Corollary 1. 13 that the relations
and
are Loeb measurable. and (Py
2;
These are the relations defined by (Py > r)<jJ(x, y)
r) <jJ(X, y) respectively.
The other steps of the induction are
easy. There are several reasonable variants of probability logic which we have not investigated.
One possibility, which was suggested by P.
Bankston, is to use the more general quantifiers (Px Borel subset of [0. lJ.
€
B) where B is a
Another possibility is to introduce probability
quantifiers on rituples of variables, so that
(Pxy > r)
that the relation
{ (x, y)
iH
has measure greater than r , The results in this section hold for a much wider class of models, namely for all models with a family of ITadditive probability measures v n satisfying the Fubini Theorem.
We postpone our discussion of such models
until Section G, where we show that any sentence of L P which has a
WI
model of the general type has a hyperfinite model.
In the next two sections
we obtain results which apply only to hyperfinite models.
HYPERFINITE MODEL THEORY III.
29
THE MANYVALUED LOGIC
In this section we shall introduce a second logic,
5L,
for the study
of hyperfinite models.
SL
is a manyvalued logic with truth values in the hyperreal line '~R .
For this reason it is convenient to identify a hyperfinite model
with the twovalued model
where F : AtT(j) _ j
{O, l} is the characteristic function of Sj'
Our
results will apply to a broader class of manyvalued hyperfinite models. A manyvalued hyperfinite model is a structure
where (A, fJ.)
is a hyperfinite probability space, each a
€ A, and each i F. is an internal function mapping AtT{j) into *[1,1] . The F. are
J
J
called the functions, or random variables of lU. The signature (I, c ) is defined as before. finite model.
In this section, model will mean manyvalued hyper
These models are of interest in their own right since they can
often be used to represent physical processes. ple of Brownian motion in Section 5.
The logic
We shall discuss the exam
5L
has the following
symbols: individual variables equality symbol an nary connective C for each continuous real function C:Rn_R,
n2::0
three quantifiers
(max xj , (min xj ,
constants
€
I
5... dx
30
H.KEISLER
function symbols We use the notation is the quantifier Because formula.
SL
SL
for this logic because its most interesting feature
S... dx . is manyvalued, we shall use the word term instead of
By an atomic term we mean an expression of one of the forms
where "i: ... , ulT(j) are constants or individual variables. ~ of
SL
The set of
is the smallest set which satisfies the following formation
rules. (3.1)
Every atomic term is a term.
(3.2)
If C is an nary connective and T
C (Tl, ... ,Tn) is a term.
... , Tn are terms, then l, (In the case n = 0, C is a real
number and C is a term. ) If T is a term and x is a variable,
(3.3)
(max x)T,
(min x)T,
STdx are terms.
Free and bound variables are defined as usual, with the quanttfiers binding the variables.
x.
T(X) denotes a term with at most the free variables
A term without free variables is called a constant term.
Notice that
a variable or constant alone is not a term, since terms are built up from atomic terms as defined above. The truth value Tlll[a]
of a term T(x) in a model III at an assignment
a is defined by induction on the complexity of terms. (3.4)
(3.5)
eq(x,y)Q![a,b] = {lo if a=b if a f b  ~Q!~ Fj (X) [a] = Fj(a)
Atomic terms with constants are interpreted in the natural way.
HYPERFINITE MODEL THEORY (3.7)
(3.8)
31
(max y)T(X,y)'!l[a]
maX{T'!l[a,b]: b
(min Y)T(X, y)'!l [a]
min {T'!l[a,b] : b lOA}
(jT(x'Y)dy)'!l[a] =
10 A}
L: T'!l[a,b] f1 (b) bEA
It can be shown that the truth value of T(x) depends only on
the assignment a of X.
'!l and
ST(X, y)dy gives the expected
The quantifier
value, or average value of T(x, y) as y varies. We see by induction on the complexity of T(X) that for each model '!l, the function f: An _ each term T(i) of say even more.
SL
= T'!l[ a]
*R defined by f(a)
Suppose the signature of L is internal.
Sl, and for each T the function f: SlX
>I'
n
N

We can
Then the set of
*N is an internal set,
~
'~R defined by
'!l
~
~
f('!l, a) = T [a]
Recall that, on the other hand, the interpretation of a formula
'!l is in general Borel but not internal.
3.9.
LEMMA.
For each term T(X) of
real number IITII such that for every model '!l and a
Proof:
Thus
is interpreted in '!l by an internal function.
all hyperfinite models '!l with universe sets A
is internal.
is internal.
S1 10
there is a least
An ,
By induction on the complexity of T .
Linear combinations of terms are terms because linear functions are connectives.
The function IITII
is a serninorm on the terms of
SL,
that
is, liT + UII ::s IITII + IIUII,
lIaTIi
A set D of terms is said to be dense in
SL
and every real
E
=
S1
> 0 there is a term TO (X)
lailiTH if for every term T(x) of
10
D such that
liT  Toll < e .
Two terms T(i) and U(i) are said to be equivalent if liT  UII = 0 . This holds if and only if for all '!land a,
T'!l[a]"" U'!l[a]
.
32
H.KEISLER n For each n and each real closed interval [r, s] , the set C (R ) of
continuous functions C: Rn __ R is a linear space with the seminorm
The classical StoneWeierstrass theorem shows that there is countable set Co of continuous real functions which is dense in the following sense: For each n , [r, s], C Co
Co
€
n C(Rn)
€
n) C(R , and real
such that IIC  Coil <
E
E
> 0 there is a
on [r, s t . In particular, the
Set of polynomials with rational coefficients is dense. continuous real functions, let SL(e
o)
Given a set Co of
be the set of all terms T(x) such
that every connective in T(x) belongs to CO' 3.10.
STONEWEIERSTRASS THEOREM FOR TERMS.
If Co is
a dense set of continuous real functions, then SL( CO) is dense in SL. Proof:
We show by induction on the complexity of terms T(x)
that for all real e > 0 there exists To(X)
SL(C
€
Every atomic term already belongs to SL(C ) ' O
o)
such that liT  Toll <
The quantifier steps in the
induction are easy, because
and
Assume the result holds for terms Tl(X), ... , Tm(X) , and let
Let
E
> 0 be real. Let
Then we always have
of SL E.
HYPERFlNlTE MODEL THEORY !If..~ ~ (TlLa], ... ,Tm[a])€
*[r,r] m
There is a real Ii > 0 such that whenever max{ls£  t£
I:
£ < m choose Co € Co
n
£ s m} < Ii, we have T£O € SL(C O)
m) C(R
such that
33
~
~
s,t € [r,r]
IC(s)  C(t) I < IIT£  T.eo ll <
such that for all s € [_r,r]m,
m
and
e/2.
s .
For each
Take
IC(S)  Co(s)1 <
e/2,
and let
Then for all
~
and a,
< e/2 + e/2
=
e .
Therefore liT  TO II < e . This complete s the proof. Our use of the StoneWeierstrass Theorem will depend on the fact that if L has a countable, signature then SL(C
o)
is a countable set of terms.
We shall frequently use the following consequence of the StoneWeierstrass Theorem. Two models
~, ~
are said to be infinitely close in symbols
""
~
~
,
W
if T "" T~ for every constant term T . 3.11. tions.
If
LEMMA.
Let Co be a dense set of continuous func
T~ "" T'.J> for all constant terms T € SL(C o)' then
Proof:
Consider an arbitrary constant term U.
W ""
~.
For each real
e > 0 , choose a term T € SL(C o) such that liT  UII < e. Then
34
H.KEISLER
Therefore U!ll "" U~ . The hyperreal universe (*[V(R)], E) is said to be ICsaturated, where IC
> w,
if for every set X of fewer than IC internal sets such that every
finite subset of X has a nonempty intersection,
nX f e ,
By the Internal
Definition Principle, this is equivalent to the property that for each n , every set of fewer than IC bounded formulas rp{x) with constants from "'(V(R)] which is finitely satisfiable in *[Vn(R» is satisfiable in ~'[Vn(R)] .
f is a near isomorphism from !lI to
~,
in symbols f:!lI:::'
is a bijection of A onto B and for every term T(X) of SL and
Equivalently,
('U, ala E A "" (~. fa)a EA'
~
, if f
a E An
r
Obviously, if !lI and 'J3 are
nearly isomorphic then they are infinitely close. 3. 12.
cardinal of *N.
ISOMORPHISM THEOREM.
Let IC be the (standard)
Suppose (*[ V(R)) ,E) is ICsaturated and the signature
of L is of cardinality less than IC. If !lI "" 'J3 then 'U and 'J3 are nearly isomorphic. Proof: Let
SLa
Assume 'U ""
~.
We use a back and forth construction.
be SL with a extra constants, and let A = {a
a
: a < IC}
r
B
= [ba :
a < x} .
We construct c E A and dEB such that a a A
= {ca :
a < Ie} ,
B
= {d0' :
a < z} ,
and for all constant terms T E SL/C(C ) ' T!lI "" T~. O T
(!lI,c/3)/3
0'
(We abbreviate
!lI by T .) Suppose c/3' ~ have been chosen for /3 < a.
is even, 1. e.,
0'
= X.
+ 2n where
Consider finitely many terms
X.
is a limit ordinal, put
CO'
If
= a1l. + n .
HYPERFINITE MODEL THEORY
35
where Co is a countable dense set of connectives. Let 0 < n < Choose functions C l, ... , C p
x
= T£~[C)
•
such that C£ (x) is near 1 at
Co
€
w
T£~[CJ . Precisely,
and near zero when x is away from
Then in the model (~, 93)13 < a ' the term
has value ('13.
~)f3
(1  1..)P. n
2:
< O!
.
Therefore this term has value
It follows that there is an element d
€
2:
(1  1..)p in n
B such that for
£=l, ... ,p, 1
<: n
There are fewer than
IC
terms in SLa(C ) ' O
of *[ VIR)] , there is an element d
O!
For odd
0'
= X. + 2n + 1
and then construct ca'
€
Therefore by xsaturatron
B such that for all terms
the argument is similar but we take dO' Then
= bx.+ n
T~ I'::! T'13 for all constant terms T
€
SLIC( Co)
By the StoneWeierstrass Lemma 3.11 we have
Let f: A 
B be defined by f(c ) 0'
so dO' = df3 . Bya dual argument,
= d0'
,
0'
<
IC •
f is a function because
f is one to one.
f maps A onto B
•
H.KEISLER
36
because the ci s and d's exhaust A and B.
It follows that f is a near
isomorphism. The following analogue of the Isomorphism Theorem follows from the results in the Continuous Model Theory book [7].
We shall discuss the
relationship further in Section 6. 3.12.
ISOMORPHISM THEOREM WITHOUT INTEGRAL QUANTIFIERS.
Assume the hypothesis of 3.12.
Let M be the set of all terms T(X) of
S ... cIx
in which the quantifier
does not occur.
If
SL
T~ '" T'P for all constant
terms T € M, then there is a bijection f of A onto B such that for all T(X) € M and
a € An,
3.13.
T~[a] '" T'P[fa]
HYFERRATIONAL APPROXIMATION THEOREM.
model m there is a model 'P such that A and all the values of v Proof:
Let a
O
= B,
For every
(m, ala € A '" ('P, ala e A '
and G. are hyperrational numbers. J
be the element of A with the smallest non zero
measure fJ.(a ) • Let H be an infinite hyperinteger large enough so that O
H
1
€
o(fJ.(ao»
For a € A let mea)
=
[HfL(a)]
.
Define v
by
veal = m(a)/K
Thus v(a)/fL(a) '" 1 whenever fJ.(a)
fo
0 , and veal
=0
whenever fJ.(a)
'P has the universe A, the measure v , and the same constants a
i
=0
as m.
The relations of 'P are G. where J
By a straightforward induction on the complexity of terms one can show that for all T(x) and a € An, T(X) =
SU(X, y)dy
Tm[a] '" T'P[a] . The key step from U(X', y) to
is as follows.
T~ [a] = ~ u~ [a, b] fJ.(b) '" ~ U'P[a, b] fJ.(b) b€A
b€A
.
HYPERFINITE MODEL THEORY "" LU'J>[a,b]v(b)
=
37
T'J>[a)
bEA Our next theorem is an analogue of the Ehrenfeucht Mostowski theorem for classical logic.
The classical theorem may be stated as follows (see
[ 6]). EHRENFEUCHrMOSTOWSKI THEOREM.
Every infinite w l saturated model 21 for a countable first order logic L has an infinite ww set (X, <) of indiscernibles. That is, X ~ A and for any increasing ntuples xl<· .. <xn' Yl<"'
Our analogous result will use Ramsey's theorem for finite sets. 3.14.
FINITE RAMSEY THEOREM.
For all positive integers
b , c , d there exists a positive integer a such that d
a ...... (b)c . That is, if A has power a, then for every partition of the set [A]d of unordered dtuples from A into c parts, there is a subset B
~
A of
power b such that [B]d is included in one partition class. Note: If d a ...... (b)c ,the relation remains true if a is increased and b. c , dare decreased. Consider an internal model 21. mean an internal set X
~
By a set of indiscernibles in m we
A and an internal linear ordering < of X such
that for all n < wand any two increasing ntuples a, (21 ,a) "" (m
,b). m is said to be infinite
3.15. signature.
from (X, <) ,
if A is infinite (but hyperfinite).
INDISCERNIBILlTY THEOREM.
Then every infinite model
b
Let SL have a countable
mcontains an infinite set of indis
cemibles. proof:
Let Co be a countable dense set of connectives, and let <
38
H.KEISLER
be an internal linear ordering of A.
of terms in SL(C numbers. not T,e ~
o)
Let p(
raj '"
and a finite sequence q
T, q)
q,e , where
the Finite Ramsey Theorem,
IAI ....
d
(b)c
=
(ql"'" qm)
of rational
be the partition of [A]n determined by whether or
partition has Zm classes.
Therefore
Consider a finite sequence
a is increasing with respect to Let JAI
<. This
be the internal cardinality of A. d
By
for all finite b , c , and d.
[AI .... (b)c
for some infinite hyperintegers b , c , d.
Therefore
for each p( T, q) there is an internal set X ~ A of cardinal b such that [X]n is included in one partition class of p( T, many pairs (T, q).
q).
There are only countably
Thus by w 1 saturation there is a single internal set
X ~ A of cardinal b such that for all (T, q), [X]n is included in one class of peT,
(Tz , qz)
q) .
(Here we need the fact that two pairs (T
can be combined.)
l,
It follows that for any term T(X)
and any two increasing ntuples
a, b
qll and €
SL(C
o)
from (X, <), T IU [a] ~ T~ [b]
Finally, by the StoneWeierstrass Lemma 3.11,
for all finite increasing
a, b
from (X,
c ) .
Given positive integers b, c , dIet r(b, c , d) be the least positive d
integer a such that a .... (b)c 3.16.
. The letter r stands for Ramsey.
INDISCERNIBILITY THEOREM WITH EXTRA CONSTANTS.
Let SL have a countable signature.
Let ~ be an infinite model and S an
internal subset of A such that for all positive b , m , d m r(b,Z ISl ,d):s Then the expanded model (~, ala Proof:
€
€
W
,
IAI .
S has an infinite set of indiscemibles.
There exist infinite b , m , d
€
~'N such that
HYPERFINITE MODEL THEORY m r(b,zISI ,d) :s JA[
39
.
Consider ktuples of terms
of fL(Co) and rational numbers q = (ql"'" qk)'
Let q have length P.
This time peT ,q) is the partition of [A]n such that
"ii, b are in the same
class if and only if for all ;
Zk.JSJP
There are at most
E
sP and i:s k ,
< zlS[m partition classes. The proof of 3.15
can now be carried through to get an infinite set of indiscernibles for
Here is another indiscemibility result. 3. 17.
m'"
(A, Il.' Fj ) j
internal set.
E
Let
INDISCERNIBILITY THEOREM FOR FUNCTIONS.
J
be a model whose functions (F j ) j
E
Let
J form an infinite
< be an internal linear ordering of J. Then there is an
infinite internal subset K C J such that for any two increasing finite sequences
from (K, <) ,
Proof:
We may assume without loss of generality that all the F. 's J
have the same number m of argument places. dense.
Let Co be countable
By a constant term scheme T (8) we mean an expression which
becomes a constant term when symbols.
G is
replaced by a sequence of function
There are only countably many constant term schemes from
40
H.KEISLER
SL(C ) . O
Consider pairs (T(G),
q)
where T(G) is a ktuple of constant
term schemes with G of length n , and p( T(G).
q)
q
is a k tuple of rationals.
Let
be the partition of [JJn determined by whether or not the
value of T,E (Fi ' ...• Fi ) in I n
~ is ~
q,E' ,E
= 1,
... , k.
k There are 2
partition classes. As in the preceding proof. there is an infinite internal
q) ,
set K ~ J such that for all (f(G). partition class for peT (G),
q).
[K]n is included in a single
Using Lemma 3.11 again, we see that K
has the required property.
j
We conclude this section with two examples of preservation theorems in SL.
The first example is quite elementary, while the second example
is more difficult and uses an analogue of the compactness theorem. By a homomorphism of
~
onto '.Il, in symbols f:
~
 '.Il , we mean
an internal function of f of A onto B such that: (3.18)
For al I b
(3.19)
For all i
(3.20)
For all j
The image '.Il
B. v (b )
=
€
I • f(a
= bi
€
J and b
€
= f( ~)
i)
E
fl.(f
1
(b)
.
Bcr(j) ,
may be thought of as the average of
~
under f .
A constant term T is said to be preserved under homomorphic images if whenever f I
~

'.Il , T~ "" T'.Il.
morphic images if whenever f:
~ 
(Recall that '.Il is atomless if each b
T is preserved under atomles s homo'.Il and '.Il is atomless, E
T~
"" T'.Il
B has infinitesimal measure
v(b) "" 0 .)
Given a function symbol F with m places. we define constant term
Sf
to be the
HYPERF1N1TE MODEL THEORY
Let T be a constant term.
3. 21. THEOREM. (i)
41
T is preserved under homomorphic images if and only if there is a
term U of the form
such that Tlll
!':l
U~ for all ~ .
T is preserved under atomless homomorphic images if and only if
(ii)
there is a term U of the above form such that T
mRJ
U
lll
for all atomless
m.
Every term of the form U is preserved under homomorphic
Proof:
images because each
SFj
is.
We prove (ii). Assume T is preserved under atomless homomorphic images.
Let
FI , ..• , Fn
be the function symbols occurring in T.
To
simplify notation we assume that these are all the function symbols of L. Consider two atomless models
~
, 'Jl such that
Let 1ll x 'Jl be the model
m X 'Jl
= (AXB,fJ.Xv,(ai,bi),H.). I '< JIE,Jn
where
Let "ITA: A X a  A, "ITa: A x B  B be the projection mappings. "ITA:
mx 'Jl 
each a
1ll
and
"IT
mx 'Jl  'Jl
B: such that f.l(a) i 0 ,
are homomorphisms, because for
~ Hj(a, b)(f.l x vXa, b)/IJo(a) b
Then
H.KEISLER
42
[Fj(a) 
(j"fj )~] ~ v(b) + ~Gj(b)v(b) b
b
and similarly whenever v (b) of o ,
~Hj(a,b)(flXV)(a,b)/V(b) "" Gj(b) a Therefore
T~ "" T~ x '+l "" T'jl . Hence we may define a real function C(st( where
~
models set, and
c:
[0. l]n _
JFl) ~ , ... , st( JFn) ~ ) = stT ~ .
ranges over all atomless models. ~
such that for all a ~
R by
E
A , flea)
is atomless if and only if
Let M(a) be the set of all
< a . Then
~ E
M(a) is an internal
M(a) for some
a "" 0 . For
each real e > 0 and infinitesimal a > 0 , we have:
Therefore for each real
E
> 0 there is a real
It follows that C is continuous.
for all atomless
a> 0
such that (1) holds.
Thus
~.
The proof of (i) is similar but without the restriction to atomless models. There is also a simpler proof of (i) using the oneelement homomorphic image of
~
instead of
~ X
'+l •
43
HYPERFINITE MODEL THEORY 3.22.
PROPOSITION.
The sentence
JJeq( x, y)dxdy
is preserved under atomless homomorphic images but not arbitrary homomorphic images.
In fact,
~
is atomless if and only if
[JJeq(x, y)dXdY]!U """ 0
.
Our next result will characterize terms which can be approximated by terms without equality. We shall need the following analogue of the compactness theorem.
Bv a statement for
S1
we mean a relation of the
form T E X where T is a constant term and X E ':'P(R).
A set g of
statements is satisfied in ~ if T!U E X for each (T E X) E g . 3. 23. is
1C saturated
COMPACTNESS THEOREM.
Suppose
51
and g is a set of statements for
(*[ VCR)] ,
E )
of power less than
IC
If every finite subset of g is satisfiable, then g is satisfiable. Proof: (
'~[V(R)],
There is a bounded formula W( x, y) in the language
E)
a statement for
which holds if and only if x is a hyperfinite model,
5L,
and y is satisfied in x.
y is
For all Yl"'" YnEg ,
By lCsaturation there is a model x such that
(*[ VCR)] , E) t= A {
is said to be a refinement of 'j3 if there is a homomorphism f:
which preserves function values, that is, for all
A special type of refinement is a duplication. finite probability space U
='j3 • tJ
~
= (D, rr)
at do is defined by
a in
~
 'j3
A and j E I .
Consider an atomless hyper
, and let dO ED.
The duplication
H.KEISLER
44 A
Then
~
B
x D •
f!(b,
= b.
is a refinement of 'j:\ with feb, d)
by splitting each b
€
B into a copy of
= v(b) IT(d)
d)
~
Intuitively,
~
is formed
.
m is
A constant term T is preserved under refinements if whenever
T~ "" T'll .
a refinement of 'll. 3.24.
PROPOSITION.
If T does not contain the equality
symbol then T is preserved under refinements. Proof:
Show by induction on the complexity of T{x) that if
refinement of 'll with f: ~ 
3. 25.
EXAMPLE.
'll , then for all
a
€
~
is a
An •
The term
JJe{ x, y)dxdy is not preserved under refinements. We shall prove a converse of 3.24. 3.26.
LEMMA.
First we need a lemma.
For every constant term T there is a
constant term U without the equality symbol such that for every duplication
~ = 'll. ~, T~ "" U~. Proof:
The intuitive idea is that since
IT
is an atomless measure
on D. each a e A has a set of duplicates of measure large compared to f!{a).
Formally the construction of U is a complicated recursion on terms
T{X) . For each ntuple of variables x, consider partitions p of x .
For
each partition class c of p, introduce a new variable vc ' forming a new sequence o~ variables partition p of
x we
x., = (v
C : C E
p).
shall define a term T
p
For each term T{X) and
(Xp ). First. given a partition
HYPERFINITE MODEL THEORY
45
q on y =: (x, y), q  y is the induced partition on x and Tq  y is the term obtained from Tq by replacing vc where y
E
c by vc {y} • Here
are the details. [Fj (X)]p is obtained by replacing each
'1c
by vc where
'1c
E
c .
[eq(x,y)]p =: 1 if x p y , 0 otherwise. 1 m  1 m [C(T , ... , T )]p =: C(T , ... , T )
P
[(maxy)T(X,y)]
=:
p
P
max({(maxv)(T y):p=:qy}) ¢
q
[(min Y)T(X, y)]p is similar.
The idea is that for integrals we ignore the points where y equals something in X, while for max and min we take each case into account. It can be shown by induction that
of
x determined by a.
T
p
T~ [a) ~ Tp~[a)
if p is the partition
does not contain the eq symbol, so for a
constant term T, U =: T¢ has the required property.
3. 27.
THEOREM.
The following are equivalent for a
constant term T. (i)
T is preserved under refinements.
(ii)
For every real that liT Proof:
UII <
E
> 0 there is a constant term U without eq such E
•
(ii) ~(i) by Proposition 3. 24.
Assume (i).
Let
SLo
be the
sublanguage of SL with the constants and relation symbols occurring in T . Let Co be the set of polynomials with rational coefficients, and let u be the set of all constant terms U in U.
E
SLo(C
u is not empty because 0, 1
For each U
€
E
such that eq does not occur o) 1.1. Moreover, u is countable.
1.1 and rational number r , we have r  U
We first prove:
€
'tl and U  r
€
1.1 •
46 (1)
H.KEISLER For all rational r < sand
8
> 0 there is a term U
r,s,
rational Ii (r, s , 8) > 0 such that for all
oS
E
II and a
~,
To prove (l), let
By compactness, for each U (\I~)T
~
E
II there is a rational r.s
E
[r, s]
~U
~
2:
o(U) > 0 such that
o(U) .
Let '13 be such that (\lU E lr,s l ) st U'13 > 0 ,
and let ,'} = {U Ell: st U'13 :s o} .
Let g be the countable set of statements g= {U:S.!..:O
The set
s
U{T
E
"r r, s]}
is finitely satisfiable, because if
is not satisfiable and C( r
l,
... , r
m)
E
Co approximates
within 1/Zn, then we have the contradiction
By compactness the countable set
max] r
l,
... , r
m)
HYPERFINITE MODEL THEORY g U{T
is satisfied in some model For each U
€
47
':'[r, s ] }
~.
U, we have
€
st U'tl "'" 0 => st U~ s 0 , and it follows that for all U
SL
Lemma 3.11 (for
without eq ), every U without the eq symbol
satisfies U~ "'" u'tl. respectively.
U, U~ "'" U'tl . By the StoneWeierstrass
€
Let ~, , tI' be duplications of ~, tI
Taking U as in Lemma 3.26, we have
so T m"", T'tl . Therefore st T'tl
[r , s]
€
Thus any tI which satisfies g
satisfies st Ttl satisfies g real
E:
r, s
€
{U
r, s
[r, s]
2::
5(U)/2: U
€
[r 
> 0 there is a term Ur, s ,
minimum of finitely many U
liT  UII <
E:
as follows.
E:
intervals [rk, rk + 1]
€
€
g
E:
> 0 , every 'tl which
E:, S
+ E:]
•
E: €
U
formed by approximating the
r
By compactness, for each
r , s ,and a real 5 > 0 such that (1) holds.
> 0 we may construct a term, U Take IITII < M
of length
U
Pick C (u) k
ur , s}
Hence for each real
satisfies T
For each rational
€
Co such that
E:
U with
wand partition [M, M]
0 "'" k "'" K .
E: ,
rk , rk + i:
€
€
Let
, 5 k = 5( r:k , r k + l' E:) •
into
48
H.KEISLER
Finally, let
where C E Co and C(u O " ' " uk) is within jukl s M+ E • Then UE e 1..1
and
E
of max(u O " ' " Uk) for
liT  UE!I < 3 E
A constant term is preserved under refinements
:E atomless
for every atomless '.Jl and every refinement !II of '.Jl, T!II "" T'.Jl.
models if Here is
an atomless version of Theorem 3.27. 3. 28.
THEOREM.
The following are equivalent for a
constant term T. (i)
T is preserved under refinements of atomless models.
(ii)
For every real
E
> 0 there is a constant term U without eq such
that for all atomless '.jl, Proof:
IIT'Jl  u'Jl lI <
E
•
The argument is like the proof of 3. 27 but restricted to
atomless models.
The compactness theorem holds for atomless models
because '.jl is atomless if and only if '.jl satisfies the countable set of statements
{f f eqj x, y)dxdy < n : 0 < n< w} 3.29.
EXAMPLE.
•
The term
(max x)(max y)[ F (x, y)  eq( x, y)1 is not preserved under refinements of atomless models. A model !II is uniform if fLea)
= l/IAI
for all a EA.
Our last result
of this section shows that for every model !II there is a uniform model
~
HYPERFINITE MODEL THEORY such that T~
TIS: for every constant term T without the eq symbol.
""l
3. 30.
refinement '13 of ('J5.b)bEB
""l
49
PROPOSITION. ~
For every model
and a uniform model
(~,b)bEB
. If
~
~
there is a
~
such that B
=C
and
is twovalued. we may take 'J5 and IS:
to be twovalued. Let a
Proof:
measure f1(a ) ' O
O
be the element of
m with the
smallest nonzero
Let H be a hyperinteger large enough so that to each
a E A we may assign a hyperinteger 0 < mea) such that Lm(a)
aEA and
L {rnra) : f1(a)
= o} /m(a o)
""l
0 .
Let the universe B have internal cardinality H.
Form 'J5 by letting f
1
split each a E A into rma) elements of measure floral/mea) , with any compatible interpretation for the constants, and let
~
has the same universe, constants, and relations as 'J5, but each b E B
has measure l/H in ~.
IV.
ELEMENTARY SUBMODELS
This section contains a. downward LgwenheimSkolem theorem for SL which is closely related to the weak law of large numbers.
It also contains
a theorem about elementary chains which is related to the strong law of large numbers. Bya model we shall again mean a manyvalued hyperfinite model. shall concentrate on infinite uniform models (where flora)
= l//AI
""l
We
0 for
50
H.KEISLER
all a
E
A ) and atomless models (where fL(a)
Rl
0 for all a
E
Our notion of a submodel differs from the classical notion.
A ). The idea
is to keep the universe A and the functions and constants the same, but restrict the hyperfinite measure fL to an internal subset of
~.
We shall work with the manyvalued logic 5L. We ~ throughout this section that L has
~
countable signature.
Consider a model !lI and an internal subset B S A of positive (but possibly infinitesimal) internal measure fL(B) > O.
The conditional
measure fLlB is the measure on A defined by (4.1)
(fLIB)(D)
=
jJ.(Bf1D)jjJ.(B)
for internal subsets DCA.
Thus for an element a
E
A,
if a
E
B
By the sub model \\lIB of !lI restricted to B we mean the hyperfinite model !lIIB
= (A'fLIB,ai,F.). J 1EI,
JE J
If \\l is a twovalued model, then so is !lIIB. We say that
~
IB is an elementary submodel of \\lIB
< \\l
~
, in symbols
,
if for every term T(x) of 5L and every ntuple
a
E
An ,
In other words,
Given B SA, we use the notation '15
= ~ lB.
Notice that the parameters
"a" range over the universe A rather than the subset B. We shall prove a series of LgwenheimSkolem theorems of the following
HYPERFINITE MODEL THEORY type.
For certain internal sets rl
for a large proportion of the B
of subsets of A , we shall show that
rl ,
€
51
mIB
is an elementary submode1 of
m.
To state our results precisely we introduce the notions "almost all" and "nearly all". Consider a hyperfinite probability space (rl, v). Loeb measure on rl S
~
rl
generated by v. An event in (rl, v ) is a set
which is measurable with respect to
a statement s(w) with the event {w
sure, or holds for nearly all w
€
o
v.
Sometimes we identify
An event S is
rl : s(w)} .
€
almost sure, or holds for almost all w
such that v(So)
Let °v be the
€
rl • if °V(S) = 1.
S is nearly
rl , if there is an internal set So ~ S
1. Thus "nearly sure" implies "almost sure".
"oJ
Our
LgwenheimSko1em results will be strong "nearly sure" assertions. 4.2.
The intersection of countab1y many
PROPOSITION.
nearly sure events is nearly sure. Proof:
Let SO' Sl' . ..
event such that T
m
~
be nearly sure and let T be an internal m Sm and v(Tm) "oJ 1 .
By w1 saturation there is an internal event T such that for each O<m<w, v(T)
2:,1 
I m'
Then v(T) "" 1 and Ten S m<w m Example ,
Let T1
sets such that v(T
m)
"oJ
~
1
T2
T ~ TO so
~
_.!.. . m
n... n Tm
.
n S is nearly sure. m<w m
."
be a countable chain of internal
Then
U T is almost sure but not m<w m
nearly sure.
Because if T is internal and v(T) "" 1 , then by wIsaturation
there is a w
€
T such that w
1/
T for all m < w , hence T is not a m
subset of U T m<w m By an internal random variable on a hyperfinite probability space
(n , v} we mean an internal function X: n  *R. If H
€
~'N
, an
52
H.KEISLER 0 :0; m < H) of (necessarily internal) random m: is independent if for all internal r E RH ,
internal sequence (X variables X m
n
rn
v { W: X (w)::=: r } m m
.
The mean of X is the hyperreal number E(X)
=
L.
X(w) yew) ,
WEn
and the variance of X is defined by
Since n is a hyperfinite set, the mean and variance always exist. X is said to be finite if for all
WEn,
X(w) is finite.
Our results will
depend on the following basic inequality (cf. [3]). 4. 3.
(X
m
: 0
:0;
Let H
BERNSTEIN'S INEQUALITY.
E
'~N
and let
m < H) be an internal sequence of independent finite random
variables on a hyperfinite probability space (rz • v) . Assume that for each m < H, E(X )
m
=0
and V(X )
m
a < 1 and every hyperreal y
v {w r]
L:
E
:0;
Then for every real number
(1"2 •
o(erft) ,
Xm(w) I::=: y(l"y'H }
_.!... a y 2
:0;
2e 2
m
p[ (I
L:
X I ::=: yeryH)] m
:0;
2e
I 2 ay 2
m
We use Chebyshev's inequality in the form P(Y::=: a)
:0;
for Y a random variable and a > 0..
E(y)/a
Since each Xn is finite there is a
HYPERFINITE MODEL THEORY finite bound B
€
N such that for all n < H, w
€
53 rl,
IX n (w) I s B . Let
X (w) , and define n
us write X( n, w)
Yew)
L X(£, w)
=
£< n Consider a positive hyperreal 0 . Since
e
x
122 22 = (l+x+ZX [l+TIx+4TX + ... ]:s
12x l+x+zx e
we have e OX( n, w)
Using E(X(n, w))
:S
1+
=0
sxr n, w) +
i (oX( n, w))2 e oB
,
The X I s are independent, so n E(eoY(w))
=
E( TTeoX(n,w)) n
Let
01
be a positive hyperreal number.
Replacing
01
by e 01
P(Y(w):2:
01/
,
5)
By the same argument
By Chebyshev's inequality,
we have :S
1 2 2 sn exp(ZH5 (J" e 
01)
•
54
H.KEISLER
so
Now let y
P( I Yew) [a:
a/oj
vB).
Put
€ 0(0
s 2exp(
21
2 20B
HI) 0 e
 a )
Since a < 1 is real and 0:0 y/oft ~ 0 , we have
21
H£2 2 oB
uoe
O!
Thus I 2 P( I Yew) I a: yoft ) os 2exp( 2 ay )
We shall use the following consequence of Bernstein's inequality here. LEMMA.
4.4. M s
(.£L) log A
1/
4
Suppose 1 s 10gA
Let (X
m
: m :s H )
€
o(H) , ana let
be an internal sequence of
independent finite random variables on a hyperfinite probability space
(n, v) with E(Xm) :0 0 . Then P( I B
1
\' X I a: .!..) w m M
:s
AM .
m
Proof:
Since each Xm is finite, each V(Xm) is finite.
Let
55
HYPERFINITE MODEL THEORY 2 Then (l" is finite.
Let cry
M.jlogA (l"2,/H
y(l",/H
P(I H
logAl/ 4
s
L:
m
m I 2:
X
L:
. We have
:s
(l"2 Hl/4
l
P(I H
= M/logA
, so y
o«l"fi)
By 5.3,
I 2 ay
Y (l"vB ) :s 2e 2
1
xmt
E
M(l"2
2:
~
2:
~)
)
:s 2e
2
"2 aM 10gA/(l"
2
m
P( I H
L:
X I m m
In the above proof we could have taken M es (H/IogA)b for any real b < 1/3 .
Our first L1:\wenheimSkolem theorem concerns an internal partition 'T1" of the universe set A, that is, an internal set 'T1" of nonempty disjoint internal sets with union A. By a choice set for 'T1" we mean an internal set B C A which contains exactly one element of each partition class D
E
'T1".
Let A'T1" be the set of
all choice sets for 'T1" . We shall use the following convention: Given a hyperfinite set A, the cardinality of A is also denoted by A. 4.5. model.
Let
PARTITION THEOREM.
~
be an infinite uniform
Let 'T1" be an internal partition of A with H partition classes,
such that (i)
log A
(il)
For each partition class D
€
o(H) .
Then for nearly all choice sets B measure,
~
E
E
'T1" , HD/A'" I .
A'T1" with respect to the uniform
I B is an elementary submodel of
~.
H.KEISLER
56
Before proving the Partition Theorem we shall discuss its relation to the weak law of large numbers.
In our present context the weak law of
large numbers may be stated as follows. Let (X
m
: 0 :s m < H) be an internal sequence of independent random
variables on a hyperfinite probability space (11. v). infinite,
E(Xm)
=0
, and V(Xm)
H
l
o(H).
€
~
Suppose H is
Then for nearly all w
€
11 ,
Xm(w) "" 0 .
m
When the X
are finite, Lemma 4.4 gives a much stronger conclusion. m The weak law of large numbers implies the following analogue of the Partition Theorem. Assume the hypothesis of 4.5 but with "log A "H is infinite".
and each a
€
€
o(H)" weakened to
Then for each term T(x) with at most one quantifier
An , the event T~ I B [a] se T~ [a] is nearly sure.
not follow that the event
(Va E An) T~ I B [a] "" T~ [a]
The weak law is used for the case U(x) no quantifiers. We let 'IT
= {D m:
be the unique element of B
n Dm
consider the random variable
and let
Then for nearly all B E A'IT
r
is nearly sure. )
= ST(X,y)dy
0 :s m < H} , and for B '
~
n
Fix a EA.
(It does
where T has €
'IT we let b
For each m we
m
HYPERFINITE MODEL THEORY
u'P [a]
H
l[
H
L;
Ym(B) m
L; Xm(B) m
H l
l
57
+
L;
E(Y )]
m
m
T~[a,b] D  l
m
m
R<
AI
L;
T~[a,b]
bEA
We now give the proof of the Partition Theorem 4. 5. Proof:
SL (1)
We show by induction on the complexity of terms T(x) of
that there is an infinite hyperinteger K(T) such that for all 'P~]
1T
P { BEA :IT [a
~ ~ I} T [a]l2: K(T)
:0;
a E An ,
K(T) A .
Observe that making K(T) smaller preserves the truth of (1). For atomic terms T(X), we always have T'P [a]
= T!ll [a] ,
holds for any K(T) . Let C be an £ary connective and assume (1) holds for Tl(X), ... , T£(X) .
and for all
t .S
Choose an infinite K such that
E TI [  IIT II, IIT II] i i i:o;£ '
K exists because C is continuous.
Then for all
a E An ,
so (1)
58
H.KEISLER :s;
L AK(Ti)
:s; 2A K :s; AlK
i:s;2
Therefore (1) holds with K(T)
= K 1
.
Assume (1) holds for T(X, y) and let U(x) ~
for all a
€
n
A
(max y) T(x, y).
Then
,
'+l ~ ~ ~ 1 p{B:IU [a]  U [a]I"" K(T)} :s; p{ B: (3b
'tI ~ ~ ~ 1 A) IT [a, b]  T [a, b] I "" K(T)}
€
Therefore (1) holds with K(U)
=
K(T)  1 .
The same argument works for (min Y)T(X, y) . Finally, assume (1) holds for T(x, y) and let U(X) ~
Let a
€
n
A
.
Let (D
For each choice set B
m: €
0 s m 1T
A
r
= ST(x, y)dy
< H) be the partition classes of
let b m be the unique element of B
.
1T •
n D m,
so that B
= {bm : 0
:s; m < H}
Consider the random variables Y on (A1T, v) , where v is the uniform m internal measure on A1T , defined by
Then (Y m : 0 s m < H) is an internal sequence of independent finite . random variables.
We shall estimate the differences ~
~
U [a]  H
1 \' LJ Ym '
m
U'tI[a]  H l
L;
Y m m
HYPERFINITE MODEL THEORY
59
Let
=0
Then the X are still independent and E(Xm) m m and B,
IXm(B)1
211UII
:5
.
Moreover, for each
so Xm(B) is finite.
By the Bernstein Inequality Lemma 4.4, there is an infinite M with P[HIl
~
Xml
~]
2:
A
:5
M
r
m
l
~
Y  E(Hm
m
The expected value E(H
l
~
l
~
Y )I m
2:
~]
:5
AM .
m
Y ) is infinitely close to uQl[aJ
m
m
whence
~
~ TQl[a, bJ (A/DmH)
m
~
.~
m
m T
"'" 1 .
Ql
[a, b]
m
Therefore there is an infinite M' such that for all a € An, P[lH
l
~ m
Y  Um[a] m
I
:5
~,]
:5
A
M'
' because
60
H.KEISLER
For each ':ll, each b
B has measure H
€
L:
H l
T':ll[a, b
m
l,
so
m]
An ,
€
':ll 1 '\' LJ Y m  U [a]
1 I 2: K(T)]
m
l
L:
T~[a,bm]
 H
m
:s;
P[ (3b
€
l
L:
T':ll[a,b m] m
 b]  T':ll [a,  b] A) IT ~ [a,
12: K~T)]
1 ] I 2: K(T)
Therefore if K = min(M' • K(T)  1) ,
Hence (1) holds with K(U) zs K/2 . This completes the proof of (1). Let Co be a countable dense set of connectives. countably many symbols,
SL(C ) is countable.
conclude from (1) that
Then for each T(x) we have
O
Since SL has
For each term T(X), we
HYPERFlNlTE MODEL THEORY for nearly all B
€
A'IT .
for nearly all B
€
A'IT.
61
By Proposition 4.2,
By the StoneWeierstrass Lemma 3.11, the above
event is the same as the event
'tI
The above theorem shows that
<~ ,
so 'tI
<~
for nearly all B
A'IT .
€
has a large collection of elementary
~
submode1s 'tI of each cardinal H where 10gA Since 10gA
o(H),
€
H:o; A/2
o(A) , we may choose H so that H
€
€
o(A).
There is a more
natural LowenheimSkolem theorem corresponding to the set A[H}
of all
internal subsets B S. A of internal cardinality H with the uniform measure. We shall prove later on that if log A submodel of
replacement. B
o(H) then 'tI is an elementary
~ for nearly all B € A[H] . The difficulty in proving this
result is that the subsets B
order,
€
€
A[H}
correspond to sampling without
If we order A and write the elements of B in increasing
= {bm:
because the b
0
:0;
m < H} ,
b
m I s must be distinct.
is not independent of {be: £ < m }
We first prove an auxiliary result m corresponding to sampling with replacement.
H
Let A
~
be the set of all internal sequences b
= (b m:
0
:0;
m < H)
of elements of A. By the frequency measure fl b
corresponding to a sequence b
€
AH
we mean the internal probability measure on A given by {m: b
m
€
D}
H
Thus
a
€
fl~(D)
A,
b
is the relative frequency of b
m
's which belong to D.
For
62
H. KEISLER
J.Lb(a) In particular,
J.L ~(a) b
=:
0 if a
By the subsequence
b
frequency of a in H
=:
r/ range of
~~
b
(b) .
m we mean the model
We say that m~ is an elementary subsequence of b
(m~ ,
b
4.6.
a) a
A "" (m. ala
E
E
o(H).
respect 'to the uniform measure,
a
E
b
< m,
if
A .
Let
Then for nearly all
bE
m be a
AH with
mI:) is an elementary subsequence of m.
The proof is similar to the proof of 4.5.
occurs in the induction step from T(x,y) to U(x) ~
m~
ELEMEmARY SUBSEQUENCE THEOREM.
uniform model and let logA
Proof:
E
m,
H
n
A , let Y be the random variable on A m
=:
The only difference
ST(x,y)dY .
Given
defined by
Then E(Y ) m
= ~ T~ [a, c] AI cEA
and hence by the Bernstein Inequality Lemma 4.4, P{bE AH:IH
I
~ Ym(b)  U~[a] /0: ~}:S AM . m
The remaining computations are exactly as in 4.5, and we show that for
HYPERFINITE MODEL THEORY each T(X) there is an infinite K(T) such that for all
63
a E An , t
4.7.
LEMMA. ~
Let G:5 A . Then
G ~ ,
P{ b E A : b
1S
oneone
}
2:
2 1 1 G A
Proof: For each m < G , the probability that b E {b n : 0 :5 n < m} m x, l. is at most GATherefore the probability that b is not one to one is at Z most G AI . t Recall that A[H]
is the set of all internal B c A of internal
cardinality H. 4. 8.
ELEMENTARY SUBMODEL THEOREM.
Let 21 be a
uniform model and let H be a hyperinteger such that log A E o(H) , Then for nearly all B E A[B]
H:5 A .
with respect to the uniform measure,
'13 " 211 B is an elementary submodel of 21 .
A(H) " {b E AH : For each BE A[HJ B" range
cbl .
b
is one to one}
there are exactly HI
Whenever B
sequences bE A(H) with
= range (b),
bE A(H) , we have ~b
"
211 B .
By Lemma 4.7, A(H) has internal measure at least 1 HZA l "" 1 in AH . By Theorem 4.6, the event A(H) is nearly sure.
n {b
EA H:
u., b
< !ll}
It follows that for nearly all B E A[H]
r
21 I '13
<
21.
t
The proof of Theorem 4. 8 in the general case is more technical and and is postponed to the Appendix. Our L~wenheim Skolem theorems can be interpreted in the opinion poll
H.KEISLER
64
model
m is an infinite uniform model with
introduced in Section 2.
jjl
y).
countably many internal relations Sj(X, hyperfinite set of people and Sj (a,
b)
is true if the person a agrees with
the statement Sj about the ntuple of people procedures for taking an opinion poll.
The set A represents a
b.
We consider various
Every elementary submodel
<m
'.J3
is a sample which is good in the sense that for any statement Sj and any
b€
ntuple
An, the percentage of a
B who agree with S. about
€
J
b
is
infinitely close to the percentage of a in the total population A who agree
b. The Partition Theorem shows that we are nearly sure to
with S. about J
get a good sample by taking an internal partition of A into H equal classes where logA class.
€
o(H) , and picking one person from each partition
The Elementary Submodel Theorem shows that we are nearly sure
to get a good sample by picking an arbitrary set of H people from A if logA
€
o(H) . 4.9.
Then nearly all submodels of
* peA) ,
jjl ,
Let log A
be an infinite uniform model.
~
with respect to the uniform measure on
are elementary submodels of
Proof: all B €
Let
COROLLARY.
~
.
O(H , H € o(A). Then HO:S B for nearly O) O * P(A). For HO :s H s A we have '.J3 < jjl for nearly all B € A[H] be infinite. For T(X) € SL and Ho:S H :s A let K(T, H) be the €
Let K O greatest hyperinteger K :s K such that O
Then K(T, H) is always infinite.
Let
and put E
Suppose
= max] fL(b) :
,::j
0
is an internal partition of A with H partition classes such
IT
that (i)
b € A}
E
AlogA
€
o(H) .
HYPERFINITE MODEL THEORY For each partition class D
(ii)
Then for nearly all choice sets B mea sure on A1T, 4.6'.
~ ~
mI B
1 and
HD /A ~ I
€ 1T ,
€
65
A1T with respect to the uniform is an elementary submodel of
ELEMENTARY SUBSEQUENCE THEOREM.
hyperfinite model and let logA
€
o(H) .
m be
Let
Then for nearly all
b€
m. a
AH with
respect to the measure v (b)
=n
m
H, on A of
fJ.(b ) m
m~ with the frequency measure is an elementary subsequence b
a,
4. 8'.
ELEMENTARY SUBMODEL THEOREM.
Let
m be
an
atomless internal model with
e
= max] ....(a):
a
E
A} ~ 0 .
Suppose f;AlogA K(T)
E
o(H) ,
= min{K(T, H):
H
H:o; A .
O
s H sA}
Let X(T) be the set of all internal B C A such that H
a
all
E
~
o s B
,
Then X(T) is an internal set whose measure in each A[H] , H is at least I  K(T)1. {B
€
and for
O
s H sA,
It follows that the measure of X(T) in
'~r(A): HO :0; B} is at least 1 K(T)l , and therefore X(T) has
measure
~
I in "r(A).
Let Co be a countable dense set of connectives.
Then
H.KEISLER
66
is nearly sure, and by the StoneWeierstrass Lemma 3.11, 'll nearly all B
EO
for
X.
4.10.
j
Let
COROLIARY.
internal subset of A, and log A the set of all B
EO
€
m be a
o(H) , D
€
uniform model,
D an
o(H) , H s A.
Let
such that D ~ B. Then for nearly all B
A[H]
with respect to the uniform measure on model of
<~
Q
,
~
be
Q
€ Q
I B is an elementary sub
m.
The above result is analogous to the downward L~enheimSkolem result of Tarski and Vaught (see [6 [). The next three results concern models which are atomles s but not necessarily uniform. 4. 5'.
PARTITION THEOREM.
Then for nearly all B
~
"'!
1 and
mI B
EO
A[H]
Let l!I be an atomless model
with respect to the uniform measure on A[H] ,
is an elementary submodel of ~ .
We tum now to the subject of elementary chains. An elementary chain in an internal model ~ is an ordered set ('llm: m
€
S) ,
s C {o, 1, ... , H} , of submodels of l!I such that whenever m , n
EO
S
and m < n ,
That is,
B m
~
Bn
~
A and
The set of all internal linear orderings < of A may be identified with the set A(A) of one to one sequences
b= with range A.
For each
b
EO
(b
m : 0 s m < A)
A(A) , let
HYPERFINITE MODEL THEORY
67
(':j>m:o~m
be the chain of submodels given by
Thus ':j>Al is W itself. 4. ll. model.
ELEMENTARY CHAIN THEOREM.
Then for nearly all internal linear orderings
Let
b
~
be a uniform
of A with respect
to the uniform measure on A(A) , ( ':j>m: logA is an elementary chain in .
Proof:
nearly all
~
€
o(m), m < A)
.
At this point we shall only prove the weaker result that for
b€
A(A) , (':j> : logA m
is an elementary chain in
ojrn) , m
€
€
O(vA))
The full result is proved in the appendix.
~.
It follows from the proof of the Elementary Submodel Theorem that for each
fixed
m
€
O(vA) with logA
€
ojm) and each term
rex)
of
SL,
there
is an infinite K(T, rn) such that
Thus for all infinite H and finite k > 0 , we see that for all hyperintegers m between H log A and vA
IH ,
It follows that for all finite k > 0 and all sufficiently large finite h , (1) holds for all m between h log A and h is sufficiently large, the event
vA/h
. Thus whenever k > 2 and
68
H.KEISLER ('vIm)[h log A :s m s vA" Ih _ ('via
has probability
2::
€
An) IT 't5m[ a]  TlU [a]
lk 1 A and hence is nearly sure.
I<
t]
Let D be the
internal set D
{rn
€
*N: 10gA
€
oim) and m
O(.!A")}
€
.
Then the event ('vim
€
~ n 't5m ~ lU ~ D)( 'via € A ) I T [ a]  T [a]
1
I< k
is nearly sure.
Recall that a countable intersection of nearly sure events
is nearly sure.
Thus for each term T(x) (Vm
is nearly sure.
€
€
SL(C o) , the event
~ n 'llm ~ 'lI ~ D)( 'via € A ) T [a] "" T [a]
Since SL(C ) is countable, we see by the StoneWeierstrass O
Lemma 3.11 that the event
t
is nearly sure.
The Elementary Chain Theorem is analogous to the strong law of large numbers.
In our context the strong law may be stated as follows.
Let (X : 0 :s m < H) be an internal sequence of independent random m
variables on a hyperfinite probability space (n , v). infinite,
E(Xm)
=0 ('vi
, and V(Xm) is finite.
infinite K
:S
H)K
l
Then for nearly all w
L:
Xm(W) "" 0 • m
This can be proved from Kolmogorov's inequality
L:
P[(3K:s H) I Xml m
Suppose H is
2:: tKlT] :S
t
2
€
n ,
HYPERFINITE MODEL THEORY where each V(Xm)
S
0"2.
69
The strong law of large numbers implies the
following analogue of the Elementary Chain Theorem. Let ~ be a uniform model. quantifier and each
a c An,
Then for each term T(X) with at most one
the event
(for all infinite m < A) T':pm [ a] "" T~ [a] is nearly sure with respect to the uniform measure on A(A) . Again, the proof is easier for m
E
o(fi) .
We state without proof an Elementary Chain Theorem for atomless models. 4.11'.
ELEMENTARY CHAIN THEOREM.
Let
~
be an atomless
model with s = max] f.L(a): a Then for nearly all linear orderings
b
E
A} "" 0 .
of A with respect to the uniform
measure on A(A) , (':p
m
is an elementary chain in
: eA log A ~
E
o(m) , m < A)
.
V. PROBABILITY QUANTIFIERS REVISITED
In our first look at probability quantifiers we could not go very far because the interpretation of the quantifiers depends not only on the relations of
~
but also on the internal probability measure f.L.
Our
study can now be carried further with the aid of the manyvalued logic 5L, which codes up the connection between the relations of ~ and the measure
f.L . Before proceeding further we need to study a slightly
more general class of models for L.
H.KEISLER
70
Bya Borel model for L we mean a twovalued structure
where (A, .... ) is a hyperfinite probability space, each a
i
€
A. and each
Sj in a Borel relation on A, Sj E B(Ali(j». We now have three types of models.
Remember that:
(a)
Hyperfinite models for L are twovalued.
(b)
Borel models for L are twovalued.
(c)
Hyperfinite models for
SL
are manyvalued.
Each hyperfinite model for L is a Borel model. model
SL
m for
induces a Borel model
universe, measure, and constants as
u,
Each hyperfinite
mo for L. mO has the same and the relations Sj of
mo
are the Borel relations S.
1
When
~
= {aE Ali(j):
is twovalued,
mo
st(F.(a)) > o} 1
u .
5. 1. PROPOSITION,
All the results of Section 2 hold for
Borel models. We wish to show that every formula of
L P which is satisfiable
WI
in a Borel model is satisfiable in a hyperfinite model. Consider a pair of Borel models
m=
(A ...... a.,S.). I. l'E J' I lIE
m'
with the same universe A, measure .... ' and constants ai' that
~
and
We say
m' are equal almost everywhere if for every atomic formula
cp(x) the sets
differ by a set of Loeb measure zero, and for every atomic sentence cp
HYPERFINITE MODEL THEORY wehave
~
iff m'
~
Two Borel models
~ ~
m' are said to be L pequivalent if they "1.
and
satisfy exactly the same sentences of L
WI
Let ill and
THEOREM.
5.2.
71
which are equal almost everywhere.
p'
m' be Borel models for L
Then ill and ill' are L pequivalent.
WI
Moreover, for every formula ,!,(X) of L P' the sets
WI
differ by a set of measure zero. We argue by induction on the complexity of rp(X).
Proof:
result holds for atomic /\ are routine. (Px > r)
by definition, and the steps for 1,
Assuming the result for
Y).
y) ,
The 1\,
and
we shall prove it for
Let 8
{b:ill ~(Px> r)
8'
{b:~'~(px>r)
8uppose 8 A 8' has measure
> 0 in 0lJ.n. Then either 8  8' or 8'  8
has measure > 0 , say 8  8'.
Then there exists s > r and a set
T c 8  8' of measure m > 0 such that for all b
€
T,
ill j: (Px ~ s)
r)
{ (a , b) : has measure
~
b€
T,
m(s  r) > O.
m 1= ",[ a ,b] ,
el'
l= 1 ",[ a . b]}
This contradicts the hypothesis that the sets
{(a,b): ill ~
m'
~rp[a,b]}
This completes the induction.
H.KEISLER
72 5. 3.
THEOREM.
Let
countably many constants, L e., hyperfinite model Proof:
~'
~
be a Borel model for L with at most
I is finite or countable. Then there is a
for L which is equal to 21 almost everywhere.
8ince each atomic formula contains at most one relation
symbol, we may assume without loss of generality that
~
has only one
relation symbol, sayan nplaced symbol 8. Assume first that m has only finitely many constant symbols, say c l"'" c We form 8' by m' piecing together internal sets along various diagonals. Partition An
a
into finitely many classes B ... , B such that and b belong to the l, k same partition class if and only if they satisfy the same equations involving c l"'" c m ' xl"'" x n' For each Be' let ~ be a maximal subsequence of such that for b € Be ' the assignments of are
x
Yi
distinct from each other and from the constants.
The length of y£ is a
number f(£) :s n representing the number of degrees of freedom of B£ (In case B£
£ :s k
is a single ntuple of constants,
and b
€
f(£)
=0
.) For each
B£ ' let b£ be the subsequence of b corresponding to
y£ ' and let T£
= {~ : b €
f(£) Then T£ is Loeb measurable in A f(£) set T£ A which differs from T£
.s
8
n
B£} .
Therefore there is an internal by a set of measure zero in the
Loeb measure on Af(£) . Let
Then 8' is an internal relation in An.
is equal to
m
almost everywhere.
The hyperfinite model
HYPERFINITE MODEL THEORY For the case of w constants c
~
c ' ... , l' 2 with only the first m constants, and let
let
73 ~
m
be a hyperfinite model equal to mm almost everywhere.
be the reduct of
By wIsaturation
the sequence (S~.: m
€
N)
may be extended to an internal sequence of relations (S'
m
: m
€
~'N)
.
For all finite m < p < wand each atomic formula
{b: ;
j=
has internal measure less than p
1
j=
. Therefore for some infinite p, the
model
has the property that for every m < wand atomic formula
{b: ~~
j=
has internal measure less than p
1
. Since p
j= 1
~
is almost everywhere equal to ~'p
5.4.
COROllARY.
If a set T of sentences of L p with wI
at most countably many constants has a Borel model, it has a hyperfinite model. We now use the manyvalued logic
SL
to expand L wp.
The logic
74
H.KEISLER
Lwp has a canonical expansion to the twovalued logic L(5)wp described as follows.
The constant symbols of L(S) are the same as the constant
symbols of L.
The relation symbols of L(5) are the expressions of the
form [T(X) > 0] where T(x) is a term of the logic 5L. L(S)wp has the same connectives A, v,
(Px;:: r) as L
wp
The logic
and quantifiers (Px > r) ,
l
' It is understood that 5L has the same signature as
L, and we identify each relation symbol symbol [Pj (x) > 0]
of L(S).
8 j (X) of L with the relation
Thus
Lwp
S
L(S)wp
We define the infinitary logic L(S) p by adding the countable conjunction WI 1\ and disjunction V .
Each hyperfinite model
m(S) for
L(S) .
m for
5L induces a unique twovalued model
However, in general ~ (S) is not a hyperfinite model but
a Borel model.
m(S) has the same universe, measure, and constants as relation symbol [T(X) > 0]
~.
Each
of L(5) is interpreted in ~(5) by the
relation
{ a~
E
n
~
~
A : steT [a]) > o} .
Thus
if and only if
Since all the results of Section 2 hold for Borel models, they hold for the logic L(5) p and the Borel models ~(S). In particular, the satis . WI faction relation
HYPERFINITE MODEL THEORY makes sense for formulas
WI
L(S)l<Xp'
75
P and even of the Suslin logic
From the definitions we see that the reduct of ~l(S) to L is
.
the Borel model ~O. If ~ is twovalued, then ~o" ~ Bya
Smodel of a 
model ~ for
SL
sentence
such that ~ (S)
F
L(S)WI P
of
Thus
we mean a hyperfinite
Smodels of
are in
general manyvalued.
is
Svalid if every hyperfinite model
for
SL
is a
Two formulas '1'(5<), tjJ(x) of L(S) p are said to be
the sentence
'P(c" 
WI
tjJ(c" is
Consider a sentence a
~
Smodel of
WI
A hyperfinite model ~ for SL is
if and only if ~o is a Borel model of
is the reduct of ~(S) to L.
Sequivalent if
Svalrd,
of L p'
Smodel of
Thus if rp has a
'1', because ~o
Smodel it has a
Borel
model. 5.5.
COROLLARY.
A sentence rp of L
wl
has a
Srnodel if
and only if it has a hyperfinite model. Proof:
If ~ 1s a
and by Corollary 5.4,
Smodel of
rp
r
then ~O is a Borel model of v ,
'I' has a twovalued hyperfinite model
is a hyperfinite model of
then ~ is a
Smodel of
~.
If
~
because
j
~ " ~O'
5.6.
COROLLARY.
If a sentence
(in all hyperfinite models for L(S) ) then it is Proof:
of L(S)
Svalid.
WI
p is valid
If 'P is valid, then by 5.4 'P holds in all Borel models for
L(S) , and hence 'P holds in ~(S) for all hyperfinite models ~ for SL. j The logic L(S) p is adequate for expressing most of the classical
WI
properties of stochastic processes.
To illustrate the expressive power of
this logic we describe a hyperfinite model representing Brownian motion, based on the paper of Anderson [1].
It is convenient to use a manyvalued
hyperfinite model with two sorts of elements.
76
H.KEISLER Let H be a positive infinite hypertntecer, let at
*[ 0 , 1]
S be the hyperfinite sub set of
= (HI) 1 ,
and let
defined by
S = {a t , za t , ..• , H I at}
Then each standard rational number q
€
Q
n [0, 1]
the standard part function maps S onto [0. 1] .
belongs to S, and Let n be the set of
all internal functions vr : S ..... { .i .rj . Our model will have two uniform hyperfinite probability spaces (o , fJ.)
and (S. v ) •
We think of elements of S as points of time and elements w
€
n as
representing the paths
L;
B(w, t) =
w(S)v!.TI
s:st B(w, t) is an "infinitesimal random walk". function
B ; n X [0, 1] ..... R
Anderson proved that the
defined by
O
iff
(Yt"" r)B(w,t) "" s
is a Brownian motion on the Loeb space (n , 0 fJ.) in the classical sense.
m with the function
We would like to form a model
B(w, t) : n X S ..... *R, but this is not permitted because the values of B(w, t) are not always in *[ 1,1] , or even finite. truncated version of B(w, t).
For
fl €
wand r
€
*R let
n_niiff r > n ri es r es n r
r xn { We now describe the model
To get around this we use a
~.
if r < n
m has
the form
m= (n , fJ., S, v r an' F, Bn )0< n < w We let x, y be variables of the sort ranging over over S.
The constants an of
n , and s, t, u ranging
m all belong to S, and
HYPERFINITE MODEL THEORY { an : 0 < n < w} Thefunctionsare F: 8*[0,1]
=
Q
77
n [0, 1]
and Bn:nXS 2 ...... *[0,1] where
F(s) = s , Bn(X, s, t)
1 = Ii" [Brx,
s)  B(x, t)] /\ 1 .
We shall express some basic properties of Brownian motion in the language L(S)
WI
T(X):S r means l T6~)  r > 0 , and other inequalities
p'
are used in a similar way.
For 0 < n <
[B(x, s)  B(x, t)] /\ n
00 ,
= n.Bn(x, s,
t)
The inequality I Bpc, s)  Btx, t)1 s r where r is real and [r] < n is expres sed by [Bpc, s)  B(x, t)] An s r .
The fact that for s, t
E
8 , B(x, s)  B(x, t) has expected value zero is
expres sed by 1\ (max s)(maxt)IS[B(X,s)  B(x,t)]l\ndxl:s 0 . n
The fact that B(x, s)
= B(x, s)
 B(x, 0) is almost surely finite is
expressed by (Px 2:: 1) V (max sj] B(x, s)A(n+ 1) I :s n . n
The fact that Brownian motion is almost surely continuous is expressed by (Px2::l) 1\ V (max s)(max t)![B(X, s)B(x,t)]"lloC (F(s)F(t»:s m' m n
n
where C (r) is the polygonal function which is zero for n for [r] :S 2~
1\
1\
n CEC O DECO
2:
~ and one
The fact that B(x, t)  B(x, s) is independent of
B(x, u)  B(x, t) when s < t 1\
Ir]
< u is expressed by
(max s)(max t)(max u)(F(t)  F(s)). (F(u)  F(t)) 0
H.KEISLER
78
The law of the iterated logarithm is expressed by (PX == 1) 1\ V m n
(Px
[B(X, q)"l] s jZqloglog.!.. q
1\
qEQ
(1+1.. ) ,
m
1 O
== 1) 1\ 1\
qEQ 1 0< q
m n
[B(x,q)"l] ==
j
Zq log log~
1 (1  m)
The fact that B(x,l) is normally distributed with mean zero and variance 1 is expressed by
1\
1\
(Px
qEQ
n
== 4> (q»)[B(x, l)"n] < q ,
Iql < n 1\
1 (Px > 4> (q»[ B(x, 1)1\ n] < q ,
1\
qEQ
n
Iql < n where 4> (r) is the normal distribution function. We now return to our general discussion. A formula
WI
P
is said to be quantifierfree if it contains no probability quantifiers (Px > rj , (Px
== r) . The next theorem allows us to use L(S) P in the
WI
same manner as a Skolem expansion is used in classical model theory. 5.7.
1(S)
WI
L(S)
WI
QUANTIFIER ELIMINATION THEOREM.
p is
Every formula
Sequivalent to a quantifierfree formula
.p(x) of
P .
Proof:
It suffices to prove that for every quantifierfree formula
.p(X,y) of L(S)
WI
P' the formulas (Py> r).p(x,y) • r E [0,1], are
HYPERFINITE MODEL THEORY
79
Sequivalent to
quantifierfree formulas cp(X) of Ld) p Let M be wI the set of all quantifierfree monotone formulas of L(S) p' and let Q
WI
be the set of quantifierfree formulas of L(S)
WI
P'
By the Monotone
Formula Theorem Without Quantifiers, every formula in Q is equivalent
Sequivalent
to a formula in M.
By Corollary 5.6, every formula in Q is
to a formula in M.
We show by induction on monotone complexity that
for every formula IJ!(X,y)€ M and r€[O,l],
Sequivalent to
(Py>rjlJ!(x,y) is
some cp(X) € Q .
Consider first the case where IJ!( X, y) is a single relation [T( X, y) > 0] Then (Py > r)4J(x, y) is satisfied by
a in
meS)
if and only if for some
positive m , n < w ,
Let C
n
be the polygonal function with Cn(s) = 0 for s :s zln '
Cn(S) = 1 for s
2:
~
Then for each n ,
Therefore (Py > r)4J(x,y) is Sequivalent to
v
V
O<m<w O
JC n (T(x, y»dy > r +1..m ,
or V V [ O<m<w O
JC
n
(T (
x, y»dy 
(r + 1..)] > 0 m
The last formula belongs to Q . Now suppose 4J(X, y) € M has monotone complexity zero, that is, IJ!(X, y) is a quantifierfree formula of L(S)wp' 4J is S equivalent to a
80
H.KEISLER
formula in disjunctive normal form,
.v /\ ((Tij(X,y) > O]Al(Uij(X,y)> 0]) 1=1 J=1 p
This in tum is
q
~
Sequivalent to
p q ~ ~ 1 V I\«T'j(X,y»O]A 1\ (Ui,(X,y)<]) i=l j=l 1 O
p
~ I ~ q 1\ 1\ ((Tij(X,y»O],,[u .. (X,y»O]), i =1 j = 1 0 < n <w n 1J
V
p V
1\
i=lO
0< n<w
1
~
~
((Ti(x,y) > O]A[n:  U,(x.y) > 0]) , 1 ~ 1 ~ «(T(x, y) > 0],,(  U(x, y) > 0] , n
1\
O
(V (x, y) > 0] n
Here
T(;;, y)
U(;;, v)
The terms Vn(X, y) monotonically decrease in value as n increases, Therefore (Px > r) ljJ (x, y) is V
Sequivalent to 1\
0< m<w 0< n<w
(Px
> r + 1.)( V m
n
We have already seen that the inside formulas are
(x, y)
> 0]
Sequivalent to formulas
81
KYPERFINITE MODEL THEORY of Q , so (Px > r)ljJ(x, y) is
Sequivalent to a
formula of Q .
Now assume the result for monotone complexity less than ~ 'P (x, y) in M have monotone complexity
01.
n monotone,
F (Py > r)
I\cP
n
Hence (Py > r) 'riCPn is
n
0',
and let
Since the conjunction is
1 ~ V 1\ (Py>r+)cpn(x,y). O<m<wn m
(x, y)
Sequivalent to a
V'P (x) in M have monotone complexity
n n
formula in Q. 0'.
Finally, let
Then since the disjunction
is monotone,
F (Py > r) V cP (x, y) n n Thus (Py > r)
X'Pn
V (py
n
>
r) 'P (x, y)
n
is Sequivalent to a formula in Q.
This completes
our induction. The proof of the Quantifier Elimination Theorem gives an algorithm which, given a formula of L
wp
formula of L(S)
WI
p'
Sequivalent quantifierfree
The exact sense in which the algorithm is effective
will be discussed later. be quite complicated.
, produces a
The translation of a simple formula of Lwp may
For example, given the sentence (Px> r)(Py
~
of Lwp ' the algorithm produces the
s) F(x, y) > 0
Sequivalent
sentence
VI\VVI\I\VI\VV abcdefghij
1 1 1 1  (r++++)] > 0 d g i a
5.8.
SL,
COROLlARY.
For all hyperfinite models
the following are equivalent.
~
and
'fI for
H.KEISLER
82 (i)
'13(S) are L(S) pequivalent, 1. e., they satisfy the WI
(11)
same sentences of L(S)
WI
Proof:
If m,.,
P .
13 then m(S) and 13 (S) satisfy the same atomic
sentences, and hence the same quantifierfree sentences of L(S) p
WI
By the Quantifier Elimination Theorem they are L(S) P equivalent.
WI
Thus
(i) implies (11). If (11), then for every constant term T and real r we have st(Tm) > riff iff '13(S) and hence m,., 5.9.
r] > 0 iff st(T'13) > r ,
13 .
WI
m and
13 are the reducts of ~(Sl and 13 (S) to L p' WI
eq symbol does not occur. ~.
13 
and
Let If
tp
tp
f
be a sentence L(S) P in which the
WI
has asmodel m then it has a uniform
If ~ is twovalued we may take It to be twovalued.
Let ~ (J)
Proof: f:
13 are two valued hyperfinite
13 , then m and 13 are L p equivalent.
5.10. THEOREM.
Smodel
If m,
COROU.ARY.
models and m,., Proof:
F [T 
m(S) l= [T  r] > 0
F
tp •
By Proposition 3.30 there is a refinement
=C m,., By 3.24, T T'13 for all constant terms T without the eq
~ and a uniform hyperfinite model It for SL such that B
13'" ~.
symbol. Therefore m(S) and '13(S) satisfy the same quantifierfree sentences of L(S) that
tp
WI
P without eq.
From the proof of Theorem 5.7 we see
is Sequivalent to a quantifierfree sentence IjJ of LcS) P
without eq , Therefore UI is a
Smodel of
Smodel of
WI
tp
and hence of 1jJ,
IjJ and hence of tp , and finally It is a uniform
13 is a
Smodel of tp .f
HYPERF1N1TE MODEL THEORY
83
We now consider the effect of changing the hyperreal universe '~[V(R)]
.
If the signature (I, cr) of L belongs to V(R) , then the set of
P belong to V(R). Let us write ~ E ':'[V(R)] if every reduct wI of m to a finite sublanguage of L belongs to *[ VCR)] . formulas of L(S)
s.u, Let
~<[V(R)]
THEOREM.
Suppose SL has a countable signature in V(R).
and #[V(R)] be two (wIsaturated) hyperreal universes.
every hyperfinite model 'J3 € #[V(R)] for Proof:
SL
Let
m€
"~[VCR)] for
such that
For
SL there is a hyperfinite model
m~ 'J3 .
Co the set of polynomials with rational coefficients,
plus the minimum function mirur , s).
Let g be the set of constant terms
T
€
SL(C such that Tm 2: O. Then g is a countable set. For each o)
T
€
g, the statement (3 '.jl)['.jl is finite and T'.jl 2: 0]
is expressible by a bounded sentence in (V(R), 8).
This sentence holds
in (,~[ VCR)] ,8) because T~ 2: 0 , and therefore it holds in (V(R), 8) . E s then min(Tl, m if and only if Tl2: 0 for /, = 1,
If Tl, ... ,T
,T
m)
E
, m.
g.and [min(T
l,
... ,T
m)]132: Thus for all T .. , T E g m l,·
0
there is a finite model 13 such that
It follows from the Compactness Theorem 3.23 that there is a hyperfinite
model '.jl € #[V(R)] such 'that T'tl2: 0 for all T
E
g.
Using the Stone
Weierstrass Theorem we see that for any constant term U so
~ ~
E
SL,
U~ ~ U'J3 ,
'.jl . 5.12.
ABSOLUTENESS THEOREM.
Let '<[VCR)]
and #[V(R)]
be two hyperreal universes, and let the signature of L belong to VCR) • A sentence rf[V(R)] .
cp
(!)f L P is valid in
WI
~'[V(R)]
if and only if it is valid in
H.KEISLER
84 Proof:
Smodel in
We show that every sentence *[V(R)] has a
many symbols occur in signature.
WI
#[V(R)].
Since only countably
Smodel of
'I' in *[V(R)] . By 5.11 there is a
'13 E # [V(R)] for SL such that '13 "" Ul.
Therefore
'13(S) satisfy the same quantifierfree sentences of
P WI The proof of the Quantifier Elimination Theorem uniformly gives us a Ul(S)
and
of L(S) P which has a
we may assume that L has a countable
q>,
Let Ul be a
hyperfinite model
Smodel in
q>
sentence ljJ of q>
L(S, max)
WI
P which is quantifierfree and
in every hyperreal universe.
~ is a
Smodel of
L(S)
Sequivalent to
Since Ul is asmodel of 'I' in *[V(R)],
ljJ in *[V(R)], hence '13 is asmodel of ljJ in
#[V(R)], and '13 is a Smodel of
q>
in #[V(R)] .
j
Our next result is related to the completeness problem for Lwp' We assume the reader is familiar with the notion of an admissible Barwise [21 or Keisler [13]), Theorem: If
a
~
a
(see
We shall use the Barwise Completeness
is an admissible set with all a
a C H C), then the set of valid arecursively enumerable.
E
a
countable (that is,
sentences in the infinitary logic L
a
is
By Lap we mean the set of all formulas of L P which belong to the
WI
admissible set (Px
> r) , (Px
~
a.
Notice that formulas in Lap contain only quantifiers
r) such that r e
n.
5. 13. ABSTRACT COMPLETENESS THEOREM. admissible set with
W
E
aC
Let
a
be an
HC , and let L have an afinite signature.
Then the set of all valid sentences of Lap is arecursively enumerable. Proof:
We give the proof in the case that L has a finite signature.
The afinite case is similar.
Let
Co be the set of connectives of SL
consisting of the linear functions with rational coefficients, the rational polygonal functions in one variable, and the max and min functions. Then the set of terms of SL(C ) is afinite. O
In order to obtain the result for arbitrary
a
with
W
E
a~
HC we
HYPERFINITE MODEL THEORY
85
need a weakening of the notion of a monotone formula.
A conjunction
/\ q, is said to be directed if for all 'PI' 'P2 "q, there exists 'P" q, such
that 1= 'P 
'PI" 'P2 • Directed disjunctions are defined dually.
formula 'P of
L(C O'
S)wrP
A
is directed if it is built up from finite quantifier
free formulas, using only directed conjunctions and disjunctions and probability quantifiers.
Let J
a. Ja
which belong to
be the set of sentences of
a
is an arecursive set.
L(C
o ' S)w 1p
By effectivizing the
proof of the Monotone Formula Theorem it can be shown that there is an
a recursive function fl: Ja 
Ja such that for each sentence
fl'P is a directed sentence which is equivalent to 'P.
'P"
Ja '
The details of this
effectivization, which require some care in the inductive definition of f
are given in Keisler [21]. Let K be the set of quantifierfree a l, sentences in J a' The proof of the Quantifier Elimination Theorem gives
an arecursive function
f 2: J
a

K
a
such that whenever ljJ is a
directed sentence of Ja' f 2ljJ is a sentence lent to ljJ •
e" K a which is
Sequiva
Let f be the composition
Then f: Ja lent to 'Po
K
a is arecursive and for each
'P"
Ja' f'P is
Sequiva
(This is the precise sense in which the algorithm of Theorem
5. 7 is effective. ) Call a model ~ for SL hyperrational if all the values of are hyperrational numbers.
If
f1 and
Fj
~ is a finite model for SL and the values
of f1 and F. are all rational, we call J
m rational.
The rational models
are elements of HF, the set of hereditarily finite sets. For each constant term T " SL(C in the language of (HF, 8)
o) ,
such that
there is a finite formula eT(u, v)
H.KEISLER
86 if and only if
~
function T 
aT is arecursive.
is a rational model for
(HF, e)!= if and only if ~ (S) 1= [T > 0]. K
a
SL
and
T~ ~
k '
and the
Moreover, V a ( ~ , m) 0< m < w T
Let g be the
arecursive function from
into La such that g[T> 0]
V
0< m < w
aT(u, m)
Then g(l1J) has one variable u and is built from finite formulas using only finite and infinite connectives (no quantifiers).
Let
r be the set of all
L which hold in (HF. e). Since w € a, r ww is arecursively enumerable. We claim that for each sentence tjJ EO K finite sentences y
€
a'
the following are equivalent. (1)
(gtjJ)(u) is consistent with
(Z)
tjJ has a
r.
Smodel.
(1) holds if and only if (g tjJ)(u) is satisfiable in some model of
r.
Since (gtjJ) is built from finite formulas without quantifiers, (1) holds if and only if (g
r. For each constant term T
EO
fL(eo) and 0 < m < w , we have
if and only if ~ is a hyperrational model in ~'HF and T~[ follows that for hyperrational
~ EO
a]
*HF ,
Therefore (1) holds iff tjJ has a hyperrational
Smodel.
By the
~
k.
It
87
HYPERFINITE MODEL THEORY Hyperrattonal Approximation Theorem 3.13 and by 5.8, (1) is equivalent to (2).
Thus for sentences 'P of Lap' 'P has a hyperfinite model if and
r.
only if g(f('P)) is consistent with
r
~
g(f(rp)).
By the Barwise Completeness Theorem, the set of conse
r
quences of
Hence 'P is valid if and only if
in La is arecursively enumerable.
Since f and g are
arecursive, the set of valid formulas of Lap is arecursively j
enumerable.
5.14. signature and let (Px
> q),
COROLIARY.
where q is rational.
of valid sentences of Proof:
be the set of formulas of Lwp with only quantifiers
L~p
(Px 2: q)
a
Let
Suppose L has a hyperarithmetical
L~p
is a
1Ti
Then the set of Qjdel numbers
set of natural numbers.
be the smallest admissible set such that w
set X C w is hyperarithmetical iff it is afinite, and
1Ti
€
G.
A
iff it is
arecursively enumerable (see Barwise (2 J).
j
We conclude with a result about the expanded Sus lin logic 1(S)\O( p .
5.15. has <
IC
THEOREM.
Suppose
'~[V(R))
constant and function symbols, where
cardinality of '
Let
m, '+'
is ICsaturated and L IC
is the standard
be hyperfinite models for SL with
Then W(S) and ,+,(S) are L(S)XXpeqUivalent.
If Wand
m""
'j) •
tI are two
valued, they are !xxpequivalent. Proof: f:
By the Isomorphism Theorem there is a near isomorphism
m.::: '+'. Form K by adding a constant 0a to L for each a
E: A
.
We have
and hence (m(S), a)
a
E:
A and (tid). fa)
a
E:
A are KcS) p equivalent. WI
For each n , let D(An) be the set of all subsets of An defined by n) n) formulas 'P(x) of K(S) r : and define D(B analogously. Then D(A WI n) n) and D(Bn) are rr algebras, and X E: D(A if and only if fX E: D(B .
88
H.KEISLER
Let fl be the completion of the restriction of °fl to D(An ) , and n n similarly for Y Then X is fl measurable iff fX is "n measurable. n n' Also flnex) = Yn(fX) , because the probability quantifiers insure that each section of X has the same measure in 0fJ. as its image in °Y.
It
follows from Lemma 2.20 that for each formula
is fJonmeasurab1e.
Then by induction on complexity we see that for each formula
S
~
K( )xxp and each a
E
n
A ,
t
VI.
STANDARD PROBABILITY MODELS It is natural to consider models for the logic L
w P 1
u
= (A,
1',
of the form
ai'S.J ).IE I ,JE . J
called standard probability models, where
l'
is a standard ITadditive
probability measure on A and each S. is a relation on AlT(j) which is J
measurable in the product measure with the hyperfinite models.
1'lT
O) . We shall compare such models
It turns out that for standard probability
models, the logic L P and the logic L(S) P without the (max x] and w1 w1 (min x] quantifiers can be interpreted in the natural way, but we do not have an interpretation for the full expansion L(S) p' w1
The difficulty is
2 that given a measurable bounded realvalued function F( x, v) : A  R,
the new function (sup x] F( x, y): A  R is not necessarily measurable, so that S(sup ~F{x, y)dy may be undefined.
The logic L(S)
P without
wI
the (max x) and (min x) quantifiers is denoted by L(r)
P .
wI
The main result of this section is that every standard probability
HYPERFlNlTE MODEL THEORY model is L(S)
wI
89
p equivalent to some hyperfinite model.
This gives a
precise mathematical form to the intuitive idea that standard probability structures can be replaced by hyperfinite structures. Bernstein's Inequality.
The proof uses
The advantage of hyperfinite models over standard
probability models is that in the former we can "count" elements and submodels, and we also have the full language L<5) p available. wI
We have
seen that the almost sure continuity of Brownian motion can be expres sed in L(S)
wI
p using the (max x] quantifier.
This is related to the fact that
the almost sure continuity of Brownian motion is obtained more easily in Anderson's hyperfinite model than in the standard (Wiener) model. We now introduce a general class of probability models which includes both the hyperfinite models and the standard probability models. If
JJ
is a ITadditive (hence realvalued) probability measure on A,
we let In(A, v) be the set of v measurable sets. sequence of ITadditive probability measures v
n
Given a set A, a on An is said to have
the rubini property if : For all m, n , v +n is an extension of the product m
(6.1) .
measure v m
vn
X
(6.2).
JJ
n
that is, for all
S
€
is, preserved under permutations
1T
of {L, ... ,n},
n
In(A , fL n ) , n  lIl(A ,vn)
(6.3). S
€
(i)
lIl(A
m+n
,v
For all
The analogue of Theorem l.12 holds.
+ ) we have: m n
b€
An , { a : S( a, b)}
~
(ii)
For all r
€
R,
~
~
€
m
In(A . v
m) .
That is, for all
90
H.KEISLER
o
(iii)
then
v {b: v {a: S(a,b)} > o} n m
=0
.
Bya twovalued probability model we mean a structure 21=(A,v n ,ai'S.) i EI , E j J J n<W ,
where v
n
a
i
£
is a sequence of probability measures with the Fubini property,
A, and each Sj is a
vIT(j) measurable relation on AIT(i)
The twovalued hyperfinite models, the Borel models, and the standard probability models are all probability models in the above sense. For hyperfinite models we take v n to be the restriction of the Loeb measure o(fln) to the Borel sets
n
a(A ) .
The results of Section 2, suitably formulated, hold for twovalued probability models 21.
21 po ",[ a]
The satisfaction relation
for '" (x) in
L p is defined exactly as before. We state the main facts from Section
WI
2 in the general setting.
A realvalued probability model is a structure 21=(A,v n,ai,F.) · J n < W , l.£ I , j
where v
n
has the Fubini property,
£
J
IT(i) a E A, and each Fj: A _ [1,1] i
is a fl measurable function. n Each manyvalued hyperfinite model 21 induces a realvalued probability model
st 21
= (A, 0 (fln) • ai' stF.) < i J n w, £
called the standard part of 21, where
I , J. £ J
HYPERFINITE MODEL THEORY
91
The functions (st F ) are Loeb measurable by Theorem 1. 8. j
Let SL be the set of terms of the manyvalued logic SL in which the quantifiers (max X), (min x] do not occur. realvalued probability model defined in the natural way.
~,
Let T(~) E SL. ~
In a
~
the interpretation T [a] can be
The quantifier clause is
( T( x~ , y)dy) ~~ [a]
J
=
f~~
T [a, b ] d
lJ
(b) .
For each formula q'(X) of L and wlP twovalued probability model ~, the relation 6. 4.
PROPOSITION.
is IJ. n  measurable. 6.5.
q'(X) 
q"
6.6.
For every formula
PRENEX NORMAL FORM THEOREM.
wp such that
(x) is valid in all twovalued probability models.
MONOTONE FORMULA THEOREM.
of L P there is a formula
wI
q"
For every formula
(X) of L p such that
wI
and equivalent to
q"
If q'(X) has
(X) .
We do not have a natural interpretation of terms of SL containing the quantifiers (max xj , (min x) within the scope of an integral quantifier. Let L(r) be the sublanguage of L(S) with the function symbols [T(X) > 0] for T(X)
E
SL , and form L(r)
realvalued probability model
WI
~
p in the usual way.
Each
determines a twovalued probability
model~(S) for L(S ) , and formulas of L(S ) P can be interpreted in
WI
the natural way in ~(S). We do not have a natural expansion to a model for the full language L,(S).
92
H.KEISLER The standard part st ~ of a hyperfinite model is an especially well
behaved realvalued model, in which the full logic 6.7.
SL,
tenus T(x) of
Tst~[a]
the inductive definition of T
st~
SL.
For all
~
[a] makes sense and
= st(TU[a]) .
We argue by induction on the complexity of T(x).
Proof:
result holds for atomic tenus. U(x)
can be interpreted.
Let ~ be a hyperfinite model for
THEOREM.
~
SL
= ST(x,y)dy.
Assume the result for T(X, y).
Let a € An. fey) = T
st a
The
Consider
The function
~
~
~
[a,y] = steT [a,y])
is the standard part of an internal function and hence is measurable.
For
internal g(y) ,
J( st g (y» dOf.1(y)
st( ~b € A g(b) f.1(b» .
Therefore
U
st U
~ [a] =
f T st U[a,y]d ~
0
f.1(y) =
\' ~~ = st(Lib€AT [a,b]f.1(b» =
Now consider W(X)
= (max
v) T(
has a maximum value r , the set value st r.
x, y).
JsteT U[a,y])d f.1(y) ~
~~
st(U [a]) . Since the set {T U [a, b ] : b € A} ~
~
{st T [a, b] : b € A} has maximum
Therefore st r
~
~
st(IN [a]) .
The term (min Y)T(X, v) is similar. Assume the result for Tl(X), ... , Tm(X) and let
Then since C is continuous,
0
HYPERFINITE MODEL THEORY
C(stT
l
~
, ... , stT
~
m
93
)
t 6.8.
COROLLARY.
Let ~ and 'j.\ be hyperfinite models for
Then ~ "" 'j.\ if and only if for every constant term T of
SL.
SL,
Two hyperfinite models for SL have the same standard part, st m " st'.jl, if they have the same universe A" B, the internal measures f.L
and
II
generate the same Loeb measures
they have the same
0f.L" 01',
constants a " b ' and the functions are infinitely close, i i
By 6. 8, if st m " st '.jl then
~
R!
'.jl.
Here is an easy characterization of models which are nearly twovalued. 6.9.
PROPOSITION.
Let m be a hyperfinite model for
following are equivalent. (i)
st ~ is a twovalued model.
(ii)
Fj(a)
(iii)
There is a twovalued hyperfinite model 'j.\ R!
(iv)
For all j
and
R!
0 or Fj(a) "" I for all j
€
J,
€
J, iii € AlT(j) • ~
•
SL.
The
H.KEISLER
94
We digress briefly to discuss the effect of removing the integral quantifier from 5L while retaining the (max xj and (min x) quanttffers. Let M be the set of all terms T(X) of 5L in which the integral quantifier 5 .. dx does not occur.
Each T(X)
€
M has a natural interpretation
T m[a] in a realvalued probability model m which is independent of the measures
jJ
measures.
•
n
We are thus led to consider realvalued models without
The model theory "for such models with the manyvalued logic
M has been studied in [7] (with a compact Hausdorff space in place of the space of values [1, 1]).
If m is a hyperfinite model and
xsaturated, the realvalued model st(m) turns out to be the sense of [7].
*[V(R)]
IC_ saturated
is in
The Isomorphism Theorem Without Integral Quantifiers
3.12', which we stated in this papeljfollows from the uniqueness theorem for saturated models in [7] . We now return to our study of the logic L(r)
WI
the proof that every probability model is L(r)
WI
p'
We work towards
pequivalent to a hyper
finite model. It is easy to check that each term T(x) liTII
a
€
which is the supremum of An.
Tm[a]
€
5L
has a finite norm
~ and
for all probability models
Moreover, the StoneWeierstrass Theorem holds as before.
The
Quantifier Elimination Theorem and Bernstein's Inequality can be proved in the following formulations using the arguments from Section 5. 6.10.
QUANTIFIER ELIMINATION THEOREM.
WI
For every formula
p there is a quantifierfree formula tJ!(X) of L(S)
such that
WI
p
tJ!(X) is valid in all realvalued probability models.
Let X , n < W be a sequence n of independent random variables on a probability space (0, jJ). Assume 6.11.
BERNSTEIN'S INEQUALITY.
that the X are uniformly bounded by K, that is, n sup{ I X (w) I : n n
< cc .
W €
n} s K
<
00
HYPERFINITE MODEL THEORY Suppose that for all n < w , E(Xn) a < 1, every
HEW,
95 2
=0
and V(Xn) :S iJ . Then for every 1(J".jHI09(2 and every y:s K a) ,
p(1 L: Xnl2:: YiJVR)
:S
1 2 ay 2e 2
n
COROLIARY.
dent random variables on
Let X n < W be a sequence of indepenn, (0, v) . Assume the X are uniformly n
bounded by K 2:: 1, and suppose each E(Xn) let
2
3K
:s (H flog HF 4 .
If
P(IH
1L:X n
! 2:: ~):S
Suppose L has a finite signature.
Let
m for
b E AW
SL ,
Let 2:s H <
H
M
.
Consider a realvalued probability
For each m <
W
,
let ':fI(m) be
the finite model ( B(m), fJ.(m) , a. ,Fj~B(m»)i 1
E
1 .
,J E
J
where B(m)
= {be:
£
:s;
m} U A o
and fJ.(m) is the frequency measure fJ.(m)(a) = We write ':fI(m) 
and
and let A be the finite set of constants o
be a countable sequence.
':fI(m)
W ,
M:s (H flog H)1/4 ,
n
model
= O.
U
:S
m . b£ = a}/m.
m if for each constant term T in m_a>
SL ,
96
H.KEISLER Given a probability model ~, the probability measure "won AW is
generated in the usual way be the measures "n on An . 6.13. ~
=:
(A,,,
'j3(m) 
LEMMA.
Suppose L has a finite signature.
Let
, •.. ) be a realvalued probability model for SL. w for almost all sequences b € A •
m
~
"w
Proof:
For T(x) in
SL,
a€A
n,
b€A w ,and H<w, we may
define T'j3(H)[a), even when a iB(H)n , as follows.
T~[a]. If T(X)
T']>(H)[a]
=:
If T(x)
JU(X, y)dy,
=:
m
iii" €
If T(X) is atomic,
C(Tl(X), ... ,Tk(X)),
H l ~ u']>(H)[a, b
T']>(H)[a]
Thus when
Then
~
m]
B(H)n, T']>(H)[a] coincides with our previous definition.
We prove several facts by induction on the complexity of terms T(X) in
SL.
In each case the steps for atomic terms and for T
=:
C(T
l,
" ., T
k)
are routine. Claim 1.
For every T(x) and G < w there exists K (T, G) < w l
such that whenever Kl(T, G)
S
H < w, a € An , and b,
C€
AW differ
on at most one place,
Assume Claim lfor U( X, y) and let T(x)
=:
SU(X, y)dy.
If b.
differ on at most one place and H == Kl(U, G) , and H == 1, then
c
HYPERFINITE MODEL THEORY
97
Therefore Claim 1 holds for T with KI(T, G) = max(K1(U, ZG), 4GIIUII) . Claim Z.
For every
rex)
and G < w there exists Kz(T, G) < w
such that whenever Kz(T, G) s H < wand a
€
An ,
The proof is similar to the argument for the Elementary Subsequence Theorem 4.6 and uses
6.IZ. Assume the claim for U(X, v) , let
T(X) = SU(x,y)dY and let KZ(U, G) s H.
a
€
An, G < eo , Kl(U,G) s H < co , and
Let Y be the random variable m
on the space (AW
,
vw ) . and let
x m (b) =
Y
m
(b) 
E(Y
)
m
Then
and the X are independent. By Claim I, for each b m conditional probability inequality
€
A we have the
H.KEISLER
98 Therefore, putting r" ZiG,
We now obtain a bound for the probability that using Corollary 6.12. G :s (H Ilog H)1/
4
Assume H is large enough so that
 Tm[a]
I ~ 3/G
3(211T 11)2 ,
. We have
P(I
:s P(
I T~(H)[a]
L:
HluNH)[a, b  Tm[a] I <: ~) m] m
L: I H 1 U'+l(H)[8,b m]
 HIYml
m
+ I L HIYm 
Tm[a]
I~ ~)
m
:s
peL:
IH I u '+l (H)[ 8 , b m ] Yml~&)
m
+
P(I
L HIXm I ~ ~)
:s 2H G :s H G/ 3
.
m
Therefore Claim Z holds for T(x). Then for all constant terms T in G
~
2,
SL
and all H ~ K (T, G) where 2
99
HYPERFINITE MODEL THEORY
Since
2 ~ LJ H is finite, it follows (by the BorelCantelli lemma) that for
H=l
each 2:$ G < co , the probability that there are infinitely many H < w such that
is zero.
Therefore for each T, the event lim T~(m) m_oo
has probability one.
call this event
set of continuous real functions.
= T'!I ~.
Then
Let
Se(Co)
Co be a countable dense
is countable.
Thus
has probability one. By the StoneWeierstrass Theorem, if T
€
b€
SL , W
A
whence ~ (rn)  '!I.
b€ e
Therefore '.Jl (m) 
then b
€
eT
for all
'!l for almost all
f
•
The above argument still holds under the weaker assumption that L has a countable signature and finitely many constants. 6.14. THEOREM.
Suppose L has a countable signature.
every realvalued probability model '!I there is a hyperfinite model
SL
such that '!I and ~ are L(f)
WI
Proof:
P equivalent.
Write L as the union of a chain of sublanguages with
finite signatures,
L
U Ln' Let '!In be the reduct of ill to Ln . n<w
For ~
for
100
H. KEISLER
b E AW ,
For
let '+> n (m) be the reduct of '.fl(m) to 1 n .
'J3 n (rn) 
we have, by 1emma 6.13 W
~
for almost all bEl\.
mn
for almost all
For each n < W
b E AW.
Therefore
we have
(1)
Choose
b E AW
such that (1) holds, and consider the sequence of finite
models '.flm(m) for 1
By WI saturation this sequence can be extended
m,
to a sequence of hyperfinite models '+>H(H) , H E >'N.
For infinite H let
be the reduct of 'J3 (H ) to 1. As usual let Co be a countable H dense set of connectives. Using WIsaturation again, there is an infinite ~(H)
H such that for all constant terms T E S1 (CO) and all 1
I TIE (H)
m _ T I <
fz.
<
k
<W
,
By the StoneWeierstrass Theorem, this holds for
all constant terms T E Sc , and therefore TIE (H) "" Tm for all T E S1~(H)(S)
It follows that rp of 1(S)
WI
p'
WI
r
'P iff m(S)
l=
'P for all quantifierfree sentences
Finally, by the Quantifier Elimination Theorem,
Ii(H) are 1(S)
6.15.
l=
m and
pequivalent. Let 1 have a countable signature, and let
COR01lARY.
be a set of sentences of
1(S)
WI
model it has a hyperfinite model.
p' If If
r
has a realvalued probability
r has a twovalued probability
model it has a twovalued hyperfinite model. Proof:
By 6. 14 and 6. 9.
VII. APPENDDC
This appendix contains the proofs of the Elementary Submodel Theorem 4.8 and the Elementary Chain Theorem 4.11 in the general case. throughout that 1 has a countable signature. hyperfinite model.
We assume
"Model" means manyvalued
HYPERFINITE MODEL THEORY
101
Let Co be the set of polynomials with rational coefficients. hyperintegers A, H and all terms T(x)
E
SL(C
o)'
For all
define K(T, A, H)
inductively as follows. (7. 1)
For T(X) atomic,
(7.2)
If T(x)
=
K(T, A, H)
=:
(H Ilog A)1/4
(maxy)U(X,y) or (miny)U(X,y),
K(T,A, H)
=:
(7.3)
If T(x)
=:
(7.4)
If C
Co and
K(U, A, H)  1
jT(x, y)dy , 1 K(T, A, H) = "2 (K(U,A, H)  1) E
C(TI(X), ... , T,e (x»)
T(X) then K(T,A,H)
The point of the above definition is that K(T, A, H) does not depend on a model
but only on the hyperintegers A, H.
~
are fixed, we shall write K(T)
7.5.
=:
When A and H
K(T,A, H) .
If logA E o(H) then for every term T(x) of
LEMMA.
SL( CO), K(T, A, H) is infinite. By induction on the complexity of T(x) .
Proof:
7.6.
Let !ll be an infinite uniform model and let
LEMMA.
H be a hyperinteger with logA E o(H).
Then for every term T(x)
and a E An , ....
H
~

!ll 
1
P{b EA :IT [a]  T [a]l2: K(T)} ~ A
(i)
Proof:
E
SL(C
o)
K(T)
This is a sharp form of the Elementary Subsequence Theorem
4.6, and has a similar proof.
We only mention the case
H.KEISLER
102
in the induction.
Let t be the maximum of
I1T II , i :=; £. Since C i is a polynomial, its partial derivatives are bounded in [t, t]£ . Therefore there is a finite k such that for all r,
K(T) :=; ~K(Ti) < K(Ti)
By definition, whenever
I ri  si I
S
Ik
I
Co
[t, t]£ ,
€
for all i :=;
I/K(T ) for all i s: i
I C(r)  CIs)
S
€
s.
It follows that
s,
s k/min(K(T.): i :=; 1
n
:=; I/K(T)
Therefore if (i) holds for each T then (i) holds for T. i 7. 7.
and let
~
LEM:MA.
Let log A
ot H) , let !II be a uniform model,
€
be a submodel of !II such that D
=A 
B
I ¢
and for every
(i)
(ii)
Proof:
We argue by induction on the complexity of T(i).
The
result holds trivially for atomic T(i) . The steps for the quantifiers (max y) and (min y) are easy. T(i)
= C(TI(i), ... , Tt (i)),
Tl(i) , ... , T (i) . t
€
Co' and assume (i) holds for
For any positive
K(T)
•
and any r,
Iri  si l :=; K(T.) 1
,we have
c; c;
Iril , Isil :=; I1T i li and because by 7.4,
C
€
Let
o(K(T for i:=; i))
S
with ~
c;
~
I C(r)  C(s)1
S
K(T)
t. Therefore, taking a = 3B ID
we see that (ii) holds for T(i) . Assume (i) holds for uri, y) and let T(i) =
SUfi, y)dy.
To simplify
,
HYPERFINITE MODEL THEORY
103
notation we consider the case where x is the empty sequence.
IT~ T~I
s
U
t
L:
U~[b] I
bED
l \' UU[b] / _ D
I TU
IT
IT~
=
6
+
3B 1 D"K(U)
+
~~
bED
~ ~

L:
Then
U~ [b]
bEA
L
UU[b] [ + bEB
~" K(lU)
U B 'j) B 4B 1 U 'j) B 4B 1 IT (i)+T "j)[+D"K(U)s!T T 1"j)+DK(U) 3B
1
D" K(T) Therefore (ii) holds for T , This completes the induction. 7.8.
LEMMA.
Let U be a uniform model and let D be an
internal subset of U such that D /A
RJ
1.
Then :tl = U I D is an
elementary submodel of U. Proof:
for all a EA.
By induction on the complexity of terms T(X) we show that
The only nontrivial step in the induction is the step from
U(X,y) to T(X)
= SU(x,y)dY,
Assuming the result for U(X,y), we
have [TU[a] 
T~[a]1 =
Ik L:
UU[a,b] bEA
t
L:
U:tl[a,b]/ bED
H.KEISLER
104
os
Ii ~
U~[a,b] i~ ~ u~[a,b]l+
b€A
It ~ U~[a,b]1
b€A
b€AD
Therefore the result holds for T(X) Let
ELEMENTARY SUBMODEL THEOREM 4.8.
~
be a uniform
model and let H be a hyperinteger such that log A € o(H), Then for nearly all B € A[H] , Proof:
'13
H os A
= ~ IB
is an elementary submodel of
Since log A € o(A), A is infinite.
~.
Choose a hyperinteger
G such that logA € o(G),
G € O(.jA),
G € o(H) .
To simplify notation we shall assume that H /G is a hyperinteger, although this assumption is not necessary. We shall work with the set A(H) of one to one sequences in A of length H.
o os
For
b € A(H) and
m < H / G, let
Thus the sets Am depend only on Dm €
~( G ]
and
\n
b restricted to mG. For each m ,
has internal cardinality A  mG.
Let Co be the set of rational polynomials, and consider a term T € SL(C ) ' To avoid difficulties arising when Am is too small, we shall o choose a hyperinteger M < H /G such that
HYPERFlNlTE MODEL THEORY
105
Such an M may be found by choosing e "" 0 large enough so that
e~€ ~'N,
G€O(!AH(le)),VAsAH(le),
H and putting M = Ci( I  e) •
AM
= A
MG
K~T)€O(E)
For then
= AHOe),
G € O(~)
and
Ei  ell
MG 1 AM· K(T)
H(lK(THA  H(
c

K(T)e
From the definition of K(T, A, G) we see that K(T,Am,G)
2:
K(T,A,G).
7.6 that for all m
S
M,
S K(T)e 1
1!..::..£)
J\n S
b
€ A(mG), and a € An,
for m .s M, so
vA
s AM' in the space A (MG) the event
~ n ~ ~m ~ (3m< M)(3a €A) IT mea]  T [a]1 i': has probability at most
.
A implies
Thus putting K(T) = K(T,A,G), we see from
By Lemma 4. 7,
Since
"" 0
1
if(Tj
H.KEISLER
106
L:
2(AmfK(T) An :s 2MAn
(AMK(T)
:s 2M(A )2n  K(T) "" 0 . M
m<M
Thus for nearly all bE: A(MG) with respect to the uniform Loeb measure, we have (VT
(1)
10
~ n ~m ~ ~m ~ 1 jL(CO))(Vm< M)(Va lOA) IT [aJ  T [aJI < K(T)
We shall show that (4) implies
where D
=A 
AM
=
range(b).
Assume that bE: A(MG) T(X)
10
SL(C
o) ,
satisifes (1).
and m < M
By Lemma 7.7, for all
we have
1 e_
K(T)
3G
1
e
Amt 1
K(T)
:s
3G
Arvr e
1
K(T)
.
Therefore for m :s M we have
3MG
AM
By assumption,
MG
1
AM • K(T) ~
Hence for all a and m es M ,
Using (1) again,
""
o .
1 e
K(T)
a
10
An ,
107
HYPERFINITE MODEL THEORY T
!i)m ill [a] "'T[a]
forallm<M.
It follows by induction on the complexity of T(x) € !i),.~
St(e o) that
~ ~
T La] '" T [a]
(3)
The only nontrivial step of the induction is from Vex, y) to T(X)
=
Assuming (3) holds for V(X,y) , we have
SV(x,y)dy.
!i)
=
T [a]
1 LJ V [ a , b ] • MG bED
1"
'" M
,,!i)
LJ
~
V
!i).e 
1
[a,b]·G
.e<M b €D.e
b Thus (2) holds for Now suppose D
z
T!i)£[a] '" £<M
S. S
b L:
T~[a]
£<M
Since (1) is nearly sure, so is (2).
= range( S) , D IB = MG IH '"
€ A(H) and let B
=A 
AM = {b k <: MG}. Then k: !i) is an elementary submodel of 'jl.
7 . 8,
I, so by Lemma
We conclude that the event (4)
holds for nearly all
b
€ A(H) and hence for nearly all B € A[H] .
By
the StoneWeierstrass Lemma 3.11, every 'jl satisfying (4) is an elementary sub model of
~.
ELEMENTARY CHAIN THEOREM 4.11. model.
Then for nearly all linear orderings
b
Let € A(A) ,
~
be a uniform
108
H.KEISLER
( 'l\: log A € is an elementary chain in Proof: H
=A
otk) , k
< A)
~.
We use the proof of the Elementary Submodel Theorem.
and let G and M be as in that proof.
Choose G
We have already shown in Section 4 that for nearly all
b€
l
Let
such that
A(A) ,
(1)
is an elementary chain in aI.
The proof of the Elementary Submodel
b e A(MG)
Theorem shows that for nearly all
( 'tlmG: 0 ~ m
(2)
is an elementary chain in ~.
Let
b€
,
< M)
A(A) be any ordering of A such
that both (1) and (2) are elementary chains in
~.
For any k where
G s k < MG, we have l mG
~
k
< (m+l)G
for some m < M.
Since G c O(G , we have G € o(k) and hence l) mG/k "" 1. Therefore by Lemma 7. 8, 'tl is an elementary submodel of mG Similarly, if MG ~ k
< A 'fle have MG/k "" 1 and hence 'tlMG is an
elementary submodel of \ . (3)
( 'l\: logA €
is an elementary chain in aI. elementary chain in aI.
It follows that otk) , k < A)
Therefore for nearly all
b€
A(A) , (3) is an
I
HYPERFINITE MODEL THEORY
109
REFERENCES
(1]
Anderson, R. A non standard approach to Brownian motion and its integration. To appear, Israel J. Math.
[2 J
Barwfse, J. Admissible sets and structures. 394 pages. SpringerVerlag, 1975.
(3 J
Bernstein, S. N.
[4]
Bretrnan, L. Probability. 421 pages. AddisonWesley, 1968.
(5]
Brown, D. and Robinson, A.
Probability Theory (Russian). Moscow, 1946.
Nonstandard exchange economies.
Econometrica 43 (1974), pp. 4155.
[6 J
Chang, C. C. and Keisler, H. J, Model Theory. 550 pages. NorthHolland Pub!.
cc.,
1973.
Chang, C. C. and Keisler, H. J. Continuous Model Theory.
[7]
165 pages. Princeton Univ. Press, 1966.
Ellentuck, E. The foundations of Suslin logic. J. Symb. Logic 40
[8J
(1975), PP. 567575. [9 J
Fagin, R. Probabilities on finite models. J. Symb. Logic 41 (1976), pp. 5058.
[10
J
Fenstad, J. E.
Representations of probabilities defined on first
order languages. Pages 156172 in Sets, Models, and Recursion Theory, edited by J. Crossley, NorthHolland Publ. Co., 1967.
[11]
Ga1fman, H.
Concerning measures on first order calculi.
Israel J.
Math 2 (1964j, pp. 118. [12 J
Halmos, P.
Measure Theory. 304 pages. D. Van Nostrand Inc.,
1950.
[13J
Keisler, H. J. Model Theory for Infinitary Logic. 208 pages. NorthHolland Publ,
ce.,
1971.
110 [ 14]
H.KEISLER Keisler, H. J. Foundations of Infinitesimal Calculus. 214 pages. Prindle, Weber, and Schmidt Inc., 1976.
[15]
Kuratowski, K. and Mostowski, A.
Set Theory. 514 pages. North
Holland Publ , Co., 1976. [16]
Loeb, P.
Conversion from nonstandard to standard measure space
and applications to probability theory. Trans. Amer. Math. Soc. 211 (1975), pp. 113122. [17]
Los, J. Remarks on the foundations of probability.
Proc. Int.
Conq, for Math. of Stockholm, 1962, pp, 225229. Almquist and Wiksells, 1973. [ 18]
Robinson, A. Pub!.
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co.,
Non standard analysis. 293 pages. North Holland
1966.
Scott, D. and Krauss, P. Assigning probabilities to logical formulas. Pages 219264 in Aspects of Inductive Logic, NorthHolland Publ. Co., 1966.
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Stroyan, K. and Luxemburg, W. A. J. Introduction to the Theory of Infinitesimals. Academic Press, 1976.
[21]
Keisler, H. J. The Monotone Class Theorem in Infinitary Logic. To appear,
R. Gandy, M. Hyland (Eds.), LOGIC COLLOQUIUM 76 © NorthHolland Publishing Company (1977) ON THE KIND OF DATA NEEDED FOR A THEORY OF PROOFS
G. Kreisel
Part I I of this article responds to the question impl icit in the title: there are reasonably adequate data for a natural history of proofs, but not for a systematic science. The distinction between natural history and systematic or fundamental science is elaborated in the Introduction to Part II. Part I (§§13) prepares for Part I I by 1isting some striking successes of early proof theory and the diminishing returns of later elaborations. The present article complements recent publ ications (Kreisel (1976) and (in press» which stress negative aspects of current proof theory. PART I:
THE PAST
INTRODUCTION Over the last half century proof theory has made a good deal of progress of permanent interest. One need only compare what WP. know now with the impressions current in the twenties concerning such general issues as Hilbert's programme; cf. §l below for two extreme examples. Also quite specific questions raised by mathematicians about the content of thp.ir own rroofs; have been answered with the help of early work in proof theory; cf. §2. This was considerably elaborated in the sixties, both w.r.t. the analysis of various informal notions of proof in the foundational I iterature and w.r.t. the mathematical techniques used. Some of the elaborations show unquestionable logical or mathematical wit, but none has excited broad (active) interest in the si lent majority of logicians; or, perhaps, a ) ittle more pointedly, what authors regarded as interesting about their elaborations, that is, the results stated, left the silent majority unimpressed. Of course, every science produc.es ,ome duJ I stuff; the point here is that the harder results in proof theory are so to speak systematically dull  with reason (cf. §3and below). The commonplace view of the (sociological) facts just described jy that the silent majority lacks the philosophical sensibility needed to appreciate the full inwardness of those elaborations. It can hardly be expected that the silent majority expresses its sensibility very well. But  and this is perhaps the main point of Pa4t I of this article  there is also another side to the matter. The elaborate results are stated in language proper to (philosophical) aims which themselves make dubious aeeumptione, For example, the aim of Hilbert's programme was to el iminate abstract methods because it assumed that these methods are unrel iable or otherwise 'unjustified' (or, at least, more so than elementary methods). Now, expectations derived from this assumption have been refuted explicitly by work on Hilbert's programme, and impl icitly by general mathematical experience. This has corrected false first impressions, and thus constitutes philosophical progress. Recognition of such progress (by the silent majority) would show philosophical acumen, not insensibility. Put differently, the concepts which existing foundational schemes (suggested by first impressions) consider to be basic are simply off the mark. To get some perspective on future work, it is natural to look at the past of some successful sciences, and to match up the early stages in their development with those of proof theory over the last 50 years. Two old branches of physics,
111
112
G.KREISEL
kinematics and the mechanics of continuous media, illustrate the kind of thing that has happened in the history of several sciences. Initially only qual itative impressions of the world are available, but this is often sufficient to speculate on the true nature of the phenomena involved. In the branches of physics mentioned, those early ideas were expressed by phrases I ike perfect shape or ideal fluid, and made more expl icit by use of the corresponding branches of mathematics (Euclidean geometry, Laplace's partial differential equations). As time went on, these ideas or conceptions of the world were elaborated by more advanced mathematics  or else radically revised. Sometimes a conception can be rejected on purely mathematical grounds simply by developing it to a point where it confl icts with very general qualitative or otherwise familiar experience. In such cases, the 1 iterature speaks of imagined (Gedanken) experiments: they need only be mentioned, not carried out since their outcome is not in  genuine  doubt. In the theory of ideal fluids the standard example is the result that a steady current exerts no drag (on objects of arbitrary shape). For similar negative examples in logic see §l; for positive ones, see §2, and successful appl ications of elementary geometry or mechanics. As to the elaborations of §3 they may perhaps be compared to the notorious appl ied mathematics of the Cambridge Tripos at the turn of this century; cf. Littlewood (1953). This comparison is of course immensely optimistic: presentday geometry and mechanics have gone well beyond the level of the Cambridge Tripos. The para] lel above will be continued in the introduction to Pant II.  In Pant I the main use of the parallel concerns the passage from (informal) proofs to adequate formal izations; for example, it can no more be assumed that mathematical texts provide exactly those data which are significant for proofs than that descriptions by sailors of waves in the sea provide the data which are hydrodynamically significant. EARLY SUCCESSES:
FORMAL SYSTEMS AND FORMALIZATION
1. About 50 years ago there was widespread interest in the need for analytic methods in number theory; cf. Ingham (1932). Opinions varied. In accordance with Hilbert's programme some thought that references to reals, sets of reals, etc., were mere shorthand for appropriate approximations, and straightforwardly el iminable. Others thought that the opposite was true. And, above all, both sides thought that it was a matter of 'opinion' that would never be settled. Work in proof theory showed that they were especially wrong where they agreed. (a) Codel's incompleteness theorem, and especially his interpretation of it in footnote 4Sa of Godel (1931), corrected the assumption about the general innocence, that is, el iminabil ity of set theoretic methods, even for proving number theoretic results. Incidentally the assumption was widespread even among those who did not know its precise formulation in the form of Hilbert's programme.  NB. The correction was discovered in connection with Hilbert's proof theoretic programme. Today it is best to use different results, spl itting the notion of formal system into 2 parts: (definabil ity in a) formal language and formal rules (for the consequence relation). The results for the restriction to formal operations are corollaries of general results in recursion theory; (invariant) definability and the consequence relation are best studied in model theory without this restriction. Also the implications of axioms of infinity asserting the existence of sets of high type On Godel 's footnote 4Sa ) are val id without any restriction to formal systems; cf. p. 182. t.10 to t.6 of Kreisel and Krivine (1971) . (b) Actual mathematical practice, in contrast to the possibilities pointed out by Godel, turned out not to use analytic methods in an essential way. As eariy as 40 years ago, Gentzen pointed out that number theoretic practice did not use the full force of first order arithmetic (and so his consistency proof was not needed to 'justify' actual, only possible numbertheoretic reasoning).
THEORY OF PROOFS
113
cf. p. 136, 170 and 200 of Gentzen (1969). In the fifties I reformulated his point more generally as fol lows: one uses the language of set theory or analysis, but 'weak' (existential) axioms> conservative over arithmetic. In fact, imagination was needed to find any specific result of analytic practice which cannot be proved by quite elementary methods: I had to turn to such curiosities as the theorem of CantorBendixson (which will come up again at the end of §4 below). In short, the pious business about the collapse of mathematics if set theoretic or even all nonconstructive methods are excluded, turned out to be hollow. Historical Remark. The logical status of set theoretic principles actually used in mathematical practice is not widely known even now; cf. Bishop (1967) and Stolzenberg's review, which quote the pious business mentioned above and other impressions current in the twenties, that is, before the work of Gentzen and GOdel. By implication, Bishop's exposition is presented as being the first refutation of those impressions!  NB. Of course, as Gentzen certainly real ized, these facts about (the limitations of) current practice are not relevant to Hilbert's original programme, which was concerned with the nature of all (possible) mathematics, not with perpetuating current defects. At most, those facts added to the plausibil ity of the programme (provided one started off with the relevant general assumptions of formal ist or nominal ist epistemology). EARLY SUCCESSES: UNWINDING INDIRECT PROOFS 2. Though the incompleteness and conservation results in §l (a), resp. §l (b) were discovered in proof theoretic contexts, they do not refer to the proofs themselves but only to the set of provable theorems. As mentioned already, many such results are better treated by use of recursion theory (in the case of formal systems) or model theory (for general izations to arbitrary sets of axioms) . We now turn to results that involve operations on proofs. As is to be expected from experience in mathematics, especially category theory, close attention to the choice of data used to represent proofs is needed here, for example, to ensure efficiency of the operations involved. More generally, as mentioned at the end of the Introduction, the details of the passage from informal proofs to t~eir formal izations are more significant here than in §l. Historical Remark. The methods used below were discovered in connection with doubts about the 'legitimacy' of nonconstructive proofs, sometimes called 'indirect' in the case of existential theorems 3 x F. The purpose was to show (for example, for Ej O formulae) that for some t, F[xlt] can be proved by 'legitimate' methods. This purpose is out of date as soon as the 'legitimacy' of the indirect methods is recognized. But the question remains whether or not a given (indirect) proof n of 3 x F provides an instance t:F[x/t]; and, if, it does, to determine t mechanically> that is, to find mechanical or, equivalently, formal rules: rr ~ trr' In short, the original question of justifying proofs (that is, principles of proof) is replaced by questions of handling proofs (mechanically). (a) Two examples of mechanical unwinding. They need only proof theory of first order predicate logic. They span a period of nearly 30 years. They answer genuine questions, (actually) asked by excel lent mathematicians about their own work.
(i) A theorem of Littlewood (proved in 1914, refuting a guess of Riemann): rr(x)  £i (x) changes sign infinitely often where n(x) is the number of primes < x and £i (x) = fX dx/log x . e
Of course, this means that there is some (recursive) method of determining a, in fact the least natural number n such that n In) > £i (n); since £i (n) is not l ntecr a l , we can compute n In)  £i (n) to sufficient accuracy to determine whether n(n) ;;; ~i (n), and then take the least N:rr(N) > 1i (N). Littlewood's proof consists of two
G.KREISEL
114
parts, one assuming the (still open) Riemann hypothesis, the other its negation. As Littlewood (1953), p. 113 puts it, 'it appeared later that this proof is a pure existence theorem and does not lead to any explicit numerical value [of N].' Unquestionably, the proof is indirect, since it appl ies the law of the excluded middle to an undecided proposition. Nevertheless, an essentially routine appl ication of proof theory (Etheorems or cut el imination) appl ied to Littlewood's original proof, extracts a bound for N; c f , Kreisel (195Z). Of course, this use cannot be expected to give optimal bounds for which further ideas are needed, and it doesn't: it gives a bound of about 10
10
10 34
compared to (1.65)10
1165
found in Lehman (1966). Actually, as Littlewood real ized (p , 113, 1. 13  14), the size of N is not the only issue here: by (iv) on p. lIS, a 'further idea' was alleged to be needed to extract a bound from Littlewood's proof, for example, by the switch from TI to W. This impression is refuted, beyond a shadow of doubt, by the proof theoretic analysis which makes a ~outine appl ication (to Littlewood's original proof for TI) of a gene~aZ (logical) method. Historical Remark. To be precise, the ideas involved in the general proof theoretic procedure were applied, not the procedure itself. For one thing, such a procedure operates on a formalized proof, and, of course, Littlewood's original proof was not. But also, at the time the relevant procedures had not been worked out for formal systems very close to the language of ordinary analysis. It turned out that what were obviously the only critical steps in Littlewood's proof could be transcribed into the formal ism of first order predicate logic, to which thencurrent proof theory appl ies; cf. Remark 5.Z p. 171 of Kreisel (1958). A significantly more systematic use of proof theory here would have to make the passage from informal proofs to their formal izations a principal object of study: this was premature before the advent of highspeed computers (at least, if the general impression is right that formalizations of such proofs as Littlewood's are too complex for humans); cf. §4(b). (i i) Unwinding a p~oof of Roth. in a conversation about Roth's theorem: \fn\fa3q (\fq;;'q )\fp(ia _ p/q[> q o
0
A. Baker brought up the following point  (Z+.!..)
n),
where a ranges over the irrational algebraic numbers (and the other variables over the natural numbers). Baker felt morally certain that the proof in Roth (1955) could be 'unwound' to yield some bound for the numbe~ of exceptionally close approximations to a, that is, the number of the set E (of rationals r): 1 n,a (Z+)
{r:[a rl,,;;; q
n}
where r ~ p/q (in its lowest terms),
the bound depending on n and the height of a. But he also felt that this bound would be insufficient for the use he made in Baker (1964) of the bound by Davenport and Roth (1955), which requires a 'further idea.' As it happened the matter of unwinding had been considered in the 1iterature, in Kreisel (1970), pp. 135136, as an application of Herbrand's analysis of logical theorems of the form 3x\fy R (x,y), R(t l, Yl) v R(t Z' yZ) v ... R(t k, Yk) , where the terms ti do not contain any variable Yj with j ;;. i. This appl ies in an obvious way to Roth's theorem for fixed n and a; with q in place of x and the pair (q,p) in place of y. Inspection then shows that k ?ields a bound on the
THEORY OF PROOFS
115
the number of En u' This supports the first part of Baker's impressions. Modulo the uncertainty about the passage from Roth's informal proof to its formal ization, discussed at the end of §la(i), I have convinced myself that the kn a suppl ied by Herbrand's analysis are indeed too large for the application in Baker (1964). Incidentally, this example provides one of the few useful appl ications of Herbrand's own formulation, specifically, in contrast to the nocounter example  interpretation which introduces fuctionals; in the case of 3x Vy R(x,y) above, a new function symbol f is introduced (as a 'counter example' \fx,R[x,f(x)]) and one gets Terms T containing f, such that i,
It will not have escaped the reader's notice that the interests of both Littlewood and Baker were so to speak contemplative, not activist. They had bounds  and (as mentioned) there are better ones than those suppl ied by the original proofs considered. Contrary to activist propaganda, those interests are no less permanent than the search for better results, which, after all, will be superseded, too. (b) Hilbert's 17 th problem: a case of a knotty unwinding. Artin's solution of Hilbert's problem was stated for archimedean formally real, that is, orderable fields K in which every positive element is a sum of";;; k squares. If P is a polynomial with coefficients in K and p p
=
~(Pi/qi)2,
1";;;
~
0, then
i ,.;;; N, where Pi and qi are also polynomials with
coefficients in K  How large is
N?
Because of the algebraic character of Artin's proof, it was clear that some such bound for N would depend only on k, the number n of variables and the degree of p; and not on, say, arithmetic properties of K (which might, however, permit 'better' bounds; cf. the end of §2). Historical Remark. In seminars Artin had raised the problem of finding bounds, ever since his original proof in the twenties. He mentioned the problem to me some 20 years ago. As it happened, around the same time, the matter was also considered by A. Robinson, but in model theoretic terms. His proof appeared in Robinson (1955), not long after the proof theoretic solution had been found. Model theory suppl ied a recursive method of determining a bound (by trial and error appl ied to derivations in predicate logic), proof theory a bound involving nfold exponentiation. As in §l (a), or, for that matter, also in comparable uses of model theory in the fifties, Artin's proof was first transcribed into (first order) predicate logic, but with this difference: two not altogether trivial modifications had to be made, (i) Restatement of Artin's theorem. In contrast to the cases in §! (a), Ar t l n ' s formulation has to be modified since the condition that K be archimedean is not of first order. Inspection of Artin's argument showed that it was only used to ensure that p ~ 0 in all real closed extensions of K. So it could be replaced by this weaker condition. Incidentally, as observed in Henkin (1960), given this replacement a more elegant formulation is obtained for ordered K without the assumption that positive elements are sums of";;; k squares: U~der the new hypothesis, or, equivalently, if p ~ 0 in the (unique) real closure K of K, then c.sK I
and
c. > O. I
As.usual, the proof was uniform for given nand d. So there are polynomials Pi J, qi J in bo~h t~e variables and coefficients of the general polynomial p, and polynomials coJ ,c i J in the coefficients only (J ,.;;; j ,.;;; j) such that, for each j,
G.KREISEL
116 p
•
~
•
.
• 2
I(c.J/c J) (p.J/q.J) . and if P > 0 in I 0 I I J
K,
for some j, each (c.j/c j) > 0 I a (ii) Restatement of Artin's proof. Very much in contrast to §l (a), it can hardly be claimed that the passage from Artin's own proof to a proof in the language of predicate calculus is altogether obvious, even after a reformulation of his theorem is given,as in (i) above. Artin uses an infinite tower of field extensions; determining a bound on the relevant finite portion of it, is tantamount to finding N: and one needs that finite portion before one can even begin to formalize a version of his proof in predicate logic.  Actually two quite different (first order) proofs of (i) are sketched in Kreisel (1960). One keeps quite close to Artin's proof, the other combines the algebraic identities used in his proof with the completeness of the axioms for real closed fields and the socalled uniformity theorem of logic, a corollary of Herbrand's theorem. The last five years have seen a considerable extension of proof theory; in particular, cutel imination was extended to (impredicative) type theories in such a way that all cutfree proofs of first order theorems are themselves of first order. In other words, to an arbitrary (higher order) derivation of a first order theorem is associated a particular first order derivation: its cutfree (normal) form. Cutelimination is then the procedure which mechanizes the mathematician's unwinding of proofs (or is, at least, a candidate for such a procedure). But the matter is not yet settled. Sure, if one insists on replacing Artin's ordinary (algebraic) language by the 'logical' language of type theory mentioned in the last paragraph, the passage from his own exposition to a formal ization of his proof in type theory is pretty unambiguous; certainly as much as for the parallel passages in §l(a). However, this insistence may involve a strategic mistake, in accordance with the ordinary mathematician's instinctive resistance to the language of type theory and to other 'foundational' languages. This issue is a principal topic of §4(b) below.
Remark. At least at present the issue is also contemplative, and not activist (in the sense of the remark at the end of §2(a) above). In particular, numerically better bounds for N itself have been obtained in Pfister (1967) independently of d when K itself is real closed. (Both the proof theoretic and model theoretic methods yield bounds simultaneously for N and the degrees of everything in sight.) More interestingly, for suitable topologies on (suitable) K, one would expect topological versions of Artin's solution, where the p.,q.,c.,c depend continuously on the coefficients of p or where c and q~ Aev~r ~anish 0 I at al I when p ~o in K. Such results are certainly not to be obtained by unwinding Artin's proof. ELABORATIONS AND DIMINISHING RETURNS 3. The main proof theoretic results on first order' predicate calculus used in §2 are. cut el imination and functional interpretations. They have been extended to stronger systemS (including infinitary ones), usually by use of some kind of transfinite iteration or hierarchy, for philosophical reasons discussed in (a) below. When the process which is iterated is 'elementary', the dOminant factor is ti) the 'length' or ordinal of the iteration; tacitly, together with a 'control' at limit ordinals by means of socal led fundamental sequences. When already each step of the process involves some kind of 'collection', one has (i i) functionals of higher type; tacitly, together with some defining schema for the latter.  NB. The distinction between (L) and (t i ) is quite well illustrated by the familiar (i) constructible and (i i) natural hierarchies of sets generated by iterating the socalled predicative, resp. the full power set operation; 'quite well', not perfectly since in the case of proof theory there are additional relations. Specifically, the computability of terms (for functionals) in (ii), is proved by transfinite induction on (orderings with) the ordinals in (i); conversely, 'definitions' by transfinite recursion in (i) are, demonstrably, satisfied by suit
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able terms in (ii).  NB. Many of the elaborations contain material of considerable mathematical charm, as illustrated very well by the lectures of Girard and Ershov at this conference, which state results in a form independent of the original proof theoretic aims. Make no mistake about it: though perhaps exceptionally polished, Girard's material is quite typical of the kind of ordinal structures developed in proof theory (categories of not necessarily wellfounded orderings with suitable maps and functors between them), and Ershov's is typical of  what are nowadays cal led  cartesian 'closed categories used for functional interpretations when certain informal continuity properties are needed (made precise by Ershov's topology or by appropriate limit space structures); cf. footnote 1 of Kleene (1959) or the 'principal result' (2.4 and 5.1) of Kreisel (1959). Two obvious questions are prompted by what has just been said: (i)
Have we simply not elaborated enough? not put enough resources into the traditional aims? or,
(ii) Have the elaborations only heuristic value? by having led (some of us) to ideas which are effective only when divorced from the original, misconceived aims? Before giving  what seems to me convincing evidence for (i i) in the case of current proof theory, readers may wish to have some
Examples concerning (i) and (ii) from successful parts of logic.
Ad (i): Mcintyre's lecture at this conference illustrates very well (my) misjudgments on the use of elaborations in model theory, specifically, in connection with Morley (1965), usually considered to be the first piece of 'hard' model theory. Some ten years ago, I had high hopes of glamorous appl ications; cf. p. 152 of the original (French) version of Kreisel and Krivine (1971). These were not fulfil led in the next five years, and so the reference to Morley (1965) was dropped from (p. 186of) the latest (German) version. Mcintyre's results more than fulfill the original hopes, when the new 'hard' model theory is combined with results for suitable mathematical structures (groups, not, e.g., rings, as Mcintyre mentioned), and emphasis is shifted away from simpleminded 'logical' cardinal ity properties to stability and the like. I  Ad (ii): Gadel 's work on the constructible hierarchy, L, mentioned earl ier, provides an 'excellent object lesson here. L is an (elegant) extension of the ramified hierarchy introduced by Poincare and Russell for their programme of predicative foundations. But as stressed in the clearest possible terms on pp. 146147 of Gadel (1944), his work drops a requirement which is absolutely essential to that progr9mme, since the relevant properties of the iteration process are not proved predicatively (but in axiomatic set theory itself, which is enough for Gadel's relative consistency results). A somewhat similar twist w.r.t. fundamental sequences for 'proof theoretic' ordinals was used successfully in Jensen (1972). Gadel 's somewhat nostalgic description of his twist, loco cit. as having primarily mathematical, not its original, .philosophical interest seems (to me) to overlook the principa) philosophical issue: Is the predicative programme, admittedly a philosophical affair, also of philosophical interest? And one test is to compare the results obtained by modifying it (as Gadel did) with those obtained by respecting it, as in certain autonomous transfinite progressions; cf. the survey in Feferman (1968). Of course, outside logic there are much more impressive examples of (i i). for example. from the famil iar theory of ideal fluids, The need for such combinations is also fami1 iar from applications of '50ft' model theory (padic fields) and of recursion theory (finitely generated groups), and of course from §2 above.  Warning. The 'contemplative' results of §2 are to be compared to appl ications of model theory in the fifties which consisted in easy (general) proofs of easy (general) theorems, not to the more substantial applications in the sixties. In the fifties, model theorists familiar with the material of §2 found it to be as promising as thencurrent model theory (an impression which has not been justified so far); cf. A. Robinson's review of Kreisel (1958).
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mentioned in the Introduction. This theory  or, as one says, this' ideal ization'  is simply not adequate for its original aim, for hydrodynamics. But the ideas that came from the development of the theory, especially in the two~dimen sional case, have permanent value provided they are suitably separated from the original aim (functions of a complex variable or harmonic functions, used to describe the potential and the flow of  hydrodynamically pretty useless  ideal fluids). After these illustrations concerning the alternative between (I) and (iii auuve we return to our principal concern, proofs. Here, as promised, is evidence for (i i), both w.r.t. (a) the particular hierarchies mentioned at the beginning of §4, and (b) traditional proof theoretic aims in general. (a) Two strategic assumptions in the construction of hierarchies. The first concerns justifications (of principles P of proof), and is restrictive: P should be justified 'from below', via a hierarchy, by some kind of reduction to more elementary principles than P. The second is permissive: with any (reduction) step, an arbitrary finite iteration is taken to be 'given' too; in short, witerations are not counted, nor their witeration, and so forth; for a fairly, but by no means absurdly broad sense of 'and so forth', cf. Girard!s lecture.  NB. Trivially, idealizations are involved here. This Is not the issue at all. What is questionable is the impl icit assumption that they are even approximately adequate, for example, for studying rel iabil ity of proofs. In fact, there is a radical alternative: Don't we do better by reversing the strategy altogether? specifically, by not building up P from below at al I, but by reducing length, the (finite) number of iterations of anyone step. For example, suppose that, for some given P, the passage, in §2, from derivations d to explicit real izations td becomes compl icated. What use is then the possibility of such a passage? One would actually look for P+ which permit  real istically  simpler proofs than P, at the  real istically negl igib!e  cost of losing that possibi lity altogether. A less extreme example is famil iar enough from elementary arithmetic, where it is certainly futile to reduce numerical terms to numerals (0, sO, ssO, •.. ); instead one looks for new, more efficient notations; for an instructive appl ication in 'advanced' arithmetic, cf. Feferman (1971) and its review where the aspect relevant here is pointed out.  For reference in §4: Statman (1974) reverses the  usual  aim of eliminating cuts in order to reduce not length, but genus. More generally, one might try to mechanize a good deal of the related, fami! iar routine of introducing suitable lemmas for 'cleaning up proofs': after al I, we learn to do this sort of thing almost automatically. Viewed in the I ight of the considerations above, the successes of §2 may well constitute a kind of I imit to useful apptications of the guiding ideas of current proof theory; a kind of optimal value for the ratio: additional information/additional effort in a traditional proof theor.etic analysis. To be a I ittle more specific, we conclude PaAt I of this article with some general ities about proof theoretic aims. (b) Mathematical reasoning and mathematical objects: a discovery. Trivially, the moment we make it our business to be selfconscious about our knowledge (of anything!), the socalled subjective elements of this knowledge become most prominent: they are thought of as particularly close to the thinking subject. In the case of mathematical knowledge, definitions and proofs  as opposed to the objects defined or to the theorem proved  are among those elements. As a matter of historical fact, whenever some branch of mathematics began to be analyzed, the first distinctions that came to mind concerned methods: projective and metric methods in geometry, algebraic and differential ones in analysis, and the like. It was a discovery that the particular differences mentioned were more profitably interpreted
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by reference to the different, possibly novel notions or structures for which results proved by different methods are val ido 2 In other words, we have discovered an unexpected adequacy of 'objectivist' analysis. Historical Remark. Traditionally, and more dramatically, one speaks here of confl icting views of the nature of (mathematical) real ity, of a grand conflict between: objective and subjective. This becomes much less dramatic when specialized to famil iar examples: after all, there are projective and metric planes on the one hand, and there are projective and metric methods on the other. Far from being presented with a confl ict, with a choice between different views on what there is, we have a very close relation between methods and objects, so close that the objects concerned can be characterized in terms of the methods; cf. the distinction between those physical objects which are, and those which are not visible to the (ideal i zed ) naked eye. The distinction is objective enough (and not particularly hard to make precise). But  on present evidence  it is weak simply because visibi 1ity is not a significant factor in most physical phenomena at all (for which we have viable theories). In short, there is a very real issue here, but much more subtle than the hackneyed business of real ity. Do the reservations (a) and (b) above finish the subject? (of proof theory) Surely not, provided we look for phenomena of mathematical reasoning in which proofs are  1 ikely to be  principal factors; in short, if proofs are to be principal objects of study; if we do not insist on standing on our heads, and think of proofs principally as a means, for example to analyze the 'meaning' of theorems. 0
PART II.
A FRESH START
INTRODUCTION To continue the view of scientific progress presented in the Introduction to P~ T, readers should recall that, at least in existing sciences, the mathematical elaboration of early conceptions and the restriction to imagined experiments soon reached a point of diminoishing returns. Generally  this seems to be a fact of scientific 1 ife  progress in the successful sciences really picked up only when striking laws were discovered which had been in doubt or had not even been suspected. In other words, they were found by genuine, not merely imagined experiments. For the present purpose it is not necessary to distinguish between such experiments and what are called observations: the latter concern phenomena that turn up in the course of nature, while experiments involve observations of phenomena in specifically designed situations. Of course, as in all matters of knowledge, observers are not passive: in experiments the external circumstances are manipulated (so to speak' interfered with' before the observation), in observations a selection is made among the raw data. This selection involves, in effect if not by intention, .the notions to which early speculations, discussed in the Introduction to p~ T, had drawn attention; in particular, in the branches of physics considered there, one looks for phenomena exhibiting 'perfect' shapes, 2 For readers interested in constructive mathematics, the corresponding reinterpretation concerns the particular (new) species of operations for which the theorems hold  as opposed to the methods of proof used to establ ish that the identities hold which the operations are claimed to satisfy.  Occasionally, there are candidates for the reverse procedure, for example, when Brouwer came up with a characterization of functions F on choice sequences f in terms of his 'fully analyzed' proofs of Vf3n R(f,n), and F satisfies the identity VfR[f,F(f)]. As at the end of (a) above, one expects only a narrow class of cases where the possibility of 'extracting' F is useful: in general, F must be so simple that it is worth writing out a definition in full.
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resp. properties of 'ideal' fluids. Incidentally, even if this normative terminology, of perfection and ideals, is inappropriate if taken to mean what 'ought' to be the case, it occasionaly makes good sense as a hint on what ought to be studied. In short, early speculations come up with such hints.
Natural history wil I be used below for those parts of science which formulate laws close to our ordinary conception of the phenomena involved. This is somewhat broader, but more appropriate than the usual meaning of 'natural history' which excludes all but the most rudimentary mathematical developments of the (ordinary) conception.
As is well known, some early speculations introduced also extraordinary views. The best known such view goes back to the Greeks: the atomic structure of matter. (It is so far removed from our ordinary views that we do not see the world as atomic even after we have absorbed atomic theory). However  and this is perhaps the principal point of p~ II  the mere introduction of this extraordinary view was sterile. What was needed was massive progress in natural history,specifical ly the discovery of laws relating data belonging to sciences which are quite different for our ordinary conceptions, for example, laws in spectroscopy relating data from optics and chemistry. (Massive progress was needed, and not merely isolated relations of this kind; for example  the substance with the composition of  glass has long been known to be transparent). Naturally, a systematic science, covering phenomena from different branches of natural history has to use extraordinary conceptions(or combinations of ordinary conceptions, which are so compl icated that the difference in degree beCOmes enough of a difference in kind; enough to be consistent with the assumption in force, that we have to do with different branches of natural history). In general, laws relating different branches do not determine a particular extraordinary conception. But they pinpoint an area where such a conception may be tested, The simple observations above will guide the exposition and the interpretations of p~ II. PROOFS AS PRINCIPAL OBJECTS OF STUDY:
NATURAL HISTORY
4.
Explicit definitions have turned out to be fruitful here. They provide the sort of striking phenomena (in the area of mathematical proof) which are needed for natural history in the sense of the Introduction above, And, as required at the end of §3, these phenomena cannot be analyzed in 'objectivist' terms, that is, by anything 1 ike current reinterpretations of the theorems proved.  NB. This article confines itself to val id proofs though, presumably, fallacious arguments are also important fora study of reasoning (as illusions are for perception). Incidentally, fallacies are often a~equately analyzed in objectivist terms, for example, in those paradoxes which come about when some of the axioms and inferences are val id for one notion, say, of 'set' or 'predicate', others for another notion; cf. the discovery discussed in §3(b). Uses of explicit definitions are conveniently grouped into those where (a) notions are introduced by means of such definitions, and (b) already understood notions with many famil iar properties are analyzed by being shown to be equivalent to others that are expl icitly defined (in some given language). Roughly speaking, in (a) it will often be obvious, in (b) some 'foundational' work is needed to show that the expl icit definitions can be el iminated (at all): but the quantitative differences between (a) and (b) will turn out to be quite del icate.
Examples. (a) An important strategy in applying modern abstract mathematics, say group theory to number theory, is this: (i) An operation 0 is defined number theoretically, say on lZ.xlZ. or modulo some prime. (ii) 0 is shown to satisfy the group laws. (iii) Special izing general facts about groups to 0 leads to number theoretic results. As a matter of empirical fact this strategy has often
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been strikingly successful; probably the most spectacular use is in Weil (1967) which brought to I ight a computational oversight in Siegel (1952). At the same time, here and elsewhere in current practice, the strategy is logically tpivial, that is, el iminable in the following precise sense: Proofs of (ii) have turned out to be formal izable in elementary number theory (though, by incompleteness, this has to be checked), and the general facts used in (i ii) are naturally proved in conservative extensions of that theory (by completeness of predicate logic this is automatic if the facts involved are of first order). Thus the uses of this strategy in current practice are candidates for the~rogramme at the end of §3.  (b) Much of traditional mathematics, for example, Eucl id's geometry was dominated by the search for correct definitions of famil iar notions (circle, tangent) in Eucl idean terms, for which fami 1iar (geometric) properties of those notions can be formally derived from Euclid's axioms. Clearly, these properties correspond to the facts about groups in (i i i) above, though  in contrast to the case of groups  geometric properties were used long before there was an 'official' list of axioms for those familiar notions. Bibl iographical remarks. The examples (a) and (b) above playa basic role in the particular brand of natural history (of mathematics) described, somewhat grandly, on p. 42 of Bourbaki (1948) in terms of 'intuitive resonances' (to mathematical structures). The discovery of (expl icit) definitions which are easy to grasp and to handle provides of course critical evidence for such 'resonances' (and for the significance of the notions defined): evidently, this appl ies to groups in (a) above. Presumably in order to stress the (obvious) similarities between the two kinds of uses of explicit definitions in (a) and (b), Bourbaki play down the differences. Finally, with the relentless logic of epigones, some followers want to suppress altogether the uses quoted in (b) above, for example, in the school curriculum.  Less grand, but no less emphatic are the references to the importance of expl icit definitions in Wittgenstein (1976), for example. p. 33 or p. 111. The idea, in (b) of pecognizing the equivalence between an understood notion and an explicit definition,that is, of the correctness of the definition is as much anathema to Wittgenstein as to Bourbaki. It is perhaps worth noting that  apart of course from the obvious literary differences  there are several other significant similarities in the general views of Bourbaki (1948) and Wittgenstein (1976), above al I, concerning the superficial character of the properties of proofs stressed in traditional foundations. Wittgenstein illustrates his points by means of examples from the most elementary kind of numerical arithmetic, Bourbaki from 'advanced' mathematics. Wittgenstein's skepticism extends to set theoretic foundations, while Bourbaki (seem to) pay I ip service to the language of set theory on p.37. However, on p. 40, footnote (2), they stress the importance of their basic structures without eve~ mentioning the definability of those structures in set theory. These matters wi 11 be elaborated in a review of Wittgenstein (1976) for the Bull. A.M.S., including the errors of both Bourbaki and Wittgenstein due to their ignorance of logic (of the sixties). Granted then the  obviously  striking ro~e of expl icit definitions, one looks for specific factors involxed in the effective use of such definitions; tacitly, always together with the pertinent axioms; cf. (ii) in Example (a) above. The reader should recall here from the Introduction (to p~ II) that superficial or 'phenomenological', not 'hidden' factors are treated in (any) natural history; further, that their relevance is not expected to be establ ished purely mathematically from fami 1iar experience: the latter only helps one choose a sensible candidate for a (relevant) factor.  NB. As in other scientific practice, the facts will be stated in terms of any notions we have learnt to use, without insisting on a premature analysis (of 'how' we know, for example, that a particular spectral I ine is blue); an extreme case is the use of 'metal I ic look' in early mineralogy. (a) Genus of proof figures and the particular use of expl icit definitions illustrated in Example (a) above. Fi rst of all, it is to be remarked that none of the famil iar measures of length is a particularly decisive factor here. The number
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of steps (nodes) is reduced only (roughly) linearly by introducing new notions by explicit definitions. The number of symbols (in formulae) can be reduced exponentially, but striking uses of the strategy in Example (al occur also without such shortening. In any case many arguments are strikingly affected by mere rearrangement, if thereby, say intricate cross references are avoided. This observation is, qual itatively, consistent with using genus as a factor, introduced in Chapter 1 of Statman (1974) where details are to be found. (Roughly speaking, the genus of a proof figure is the least genus of any surface in which the figure can be embedded without crossing). Here are a few rather general features of genus which are not discussed in Statman (1974), but illustrate the kind of considerations needed for the natural history of proofs (and which tend to be disturbing to someone brought up on traditional mathematical logic). (i) Genus is sensitive to the style of formal ization. Specifically, all derivations in a calculus of sequents have genus 0, in contrast to natural deductions (provided the nodes are joined where an assumption is introduced and used).3 Far from being a defect, this sensitivity allows one to choose between the styles; cf. the shift to hel iocentric coordinates in astronomy at the time of Kepler (which was essential for natural history, despite all the business about the relativity of space). (ii) Genus is hard to calculate from the data: one needs a computer even for quite elementary derivations (and a good deal of effort to convert ordinary proofs into data suitable for highspeed calculation). An anecdote concerning such calculations: At Stanford a body of (natural) deductions had been stored in a computer, left over from the homework by students in a computerassisted course on axiomatic set theory; information on the course can be found in Suppes (1975). Tarjan's algorithm was efficient enough for dividing this readymade material lnto planar and nonplanar deductions. The job would be formidable if instead written homework for an ordinary course had to be prepared for the computer. Far from being a drawback, (i i) seems (to me) promising, granted that genus is a significant factor at all. For one thing, since computers are new, only a short while ago it would have been simply premature to study our subject empirically at all. Less trivially, (ii) reduces a, if not the principal complication in studying human behaviour, whether it be psychological or physiological (which is in practice rarely an important difference). Specifically, knowledge of a theory is 1iable to influence the subject of the observation; in contrast to other sciences where of course the influence of such knowledge on the observer, not on the object of the observation, can be critical, and where one needs safeguards against conscious or unconscious faking (by the observer). If a theory involves notions 1ike genus, which are hard to calculate, there is a better chance that the subjects will not even try  and so simply will not know the theoretical prediction (even if they know the theoretical Jaws). At the very least, the need for statistical tests is reduced in the case of such intractable theories. Finally  and quite superficially  the phenomena (of reasoning) considered here strike us as obscure: so if a theory uses superficial (phenomenological) notions 1ike genus such obscurity would go well with a difficulty in actually applying the theory (to the data); cf. also the last paragraph of the Introduction to Panx II.  Incidentally, the actual computations of genus for the corpus of proofs by Stanford students mentioned earlier, were not particularly conclusive; at least partly because the whole material was just a bit too elementary to present any really striking differences to start with. What has been said so far is quite sufficient to show that there are plenty of data for a natural history of proofs; in fact, whenever we find striking phenomena 3 This distinction, in Statman (1974), pinpoints for the first time a measure for the difference between the two styles, though it had long been clear that there was some significance to that difference (as suggested in a general way by Gentzen's terminology 'natural deduction').
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which do not invite analysis in objectivist terms (in the sense of §3). The role of explicit definitions discussed above is of course only one of many such phenomena. As observed in the last paragraph, the presentation of those phenomena for theoretical treatment, that is, the choice of data, is quite critical, especially if notions 1ike genus are involved which are so hard to compute that only automatic dataprocessing is real istic. 4 For a conventiona1 exposition of elementary logic, one of the best known 'reductions' is the expl icit definition of an arbitrary propositional operation (tacitly, together with proofs of its axioms) in terms of (the rules for) say ~ and v. What is lost by such reductions? Statman (1974) shows (actually for the Impl icational calculus, not for , and v) that the genus can be increased unboundedly. In a lecture at ClermontFerrand (in 1975) he considered propositional quantifiers, and showed how length is increased too; those quantifiers allow expl icit definitions since \fpP is 'short for' p[p/T] " p[pll].
Example.
Correction. Though Statman (1974) refers to Kreisel (1973), tacitly, p, 267, his (successful) treatment followed totally different 1ines from what I intended at the time, that is, at the congress at Bucharest in 1971. My idea was that the choice of axioms for explicit definitions would be made with essential help of some kind of traditional philosophical analysis of the meaning of the defined notions, and the combinatorial side would look after itself as it were. Indeed, as J stressed expl ici tly on p. 258 and in the PS, I was looking for sustained philosophical analysis, in contrast to its use in current foundations where it is completely overshadowed by the mathematics (needed to develop the onel iners expressing those analyses). Statman concentrates on combinatorial, not philosophical analysis.  Incidentally, five years ago I did not think (consciously) of Bourbaki nor of Wittgenstein in connection with expl icit definitions, though J had read Bourbaki (1948), and I knew Wittgenstein in the forties. Actually, even today I do not remember his talking about the topic in my presence. Evidently he must have done this in view of the quotations from Wittgenstein (1976) given earl ier, and equally evidently, I was not ripe to profit from those remarks. (b) Foundational languages: artifacts. Trivially, the effectiveness of an expJ icit definition of a given notion will depend on the choice of language (to be used in the definiens). What seems to be less weI Iknown is that the famil iar foundational languages tend to introduce artifacts, in particular, when notions from geometry including descriptive set theory are defined in the usual way in the language of higher order arithmetic, The examples below come from my own misjudgments. (i) There are two standard proofs of the theorem of CantorBendixson; one identifies the perfect kernel of a (closed) set F (of reals) with its set of condensation points, the other with the I imit of its derived sets. If one insists on transcribing those proofs into second order arithmetic, when F is coded by its set of complementary intervals (with rational end points), then the difference between the proofs is most obviously expressed in model theoretic terms: the former uses IT 1  comprehension (appl ied to the predicate of being a condensation 2 point), the latter uses only IT] 1  comprehension; cf. Kreisel (1959A).5 This difference is illusory l nasrnucb as a sotospeakmoregeometric formalization of the first proof (with additional primitives for real numbers, etc.) can be modelled
4 For example, in contrast to an exposition based on normal ization where the
data include normal ization rules, and so a 'reduction' must preserve normal ization steps, too. 5 The main result of Kreisel (1959A) does not concern the difference between the two proofs mentioned, but the relative complexity of F and of (any second order definition of) the perfect kernel of F.
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by use of TIll  comprehension too. This was discovered by Friedman in recent developments (to third order theories) of his careful formal izations in Friedman (976). Corollary. The (striking) difference between  careful formal izations of  the two proofs above does not invite analysis in objectivist terms after all, and is thus a candidate for the programme at the end of §3.
Remark concerning the logician's (so far practically unsubstantiated) impression of a 'need' for higher types in ordinary mathematical practice. Of course, by a footnote 48 in Godel (1931) mentioned already in §l (a), the possibility of a
logical need exists: some arithmetic theorems of the (usual impredicative) theory of type n + 1 just aren't provable in the theory of type n , But the actual: basis for that impression may not be logical at all, but structural; cf. the structural improvements introduced by Friedman's third order language with axioms that are weaker (than those in the original second order formalization). It is certainly of interest for the natural history of proofs to present alternative hypotheses for such impressions (as was done above by distinguishing between logical and structural needs). (i i) Many theorems which are formulated in the language of first order logic (and therefore, by completeness, aan be proved by the usual rules of predicate calculus), are in fact proved by use of geometric and/or analytic methods. An obvious problem for the natural history of proofs is this: Are these mathematical methods needed to make the proofs manageable (by reduction of length, genus or whatever)? In Kreisel (1976) a candidate from the theory of real closed fields was suggested: the theorem establ ished by schut te and van derWaerden (1953) in their solution of Newton's problem of the 13 points (on the surface of the unit sphere). My specific suggestion was to formal ize their solution in usual type tyeory where, in particular, trigonometric functions are expl icitly defined. But closer inspection shows that elementary facts about geometric and topological notions (area of spherical triangles, or Euler's theorem on polyhedra) have quite unmanageable proofs in the language I had in mind. In other words, a representation in that language introduces complexities that are not at all intrinsic to the solution of Newton's problem.
Remark.
It cannot be assumed that to every solution of Newton's problem in a suitable geometric language there corresponds a particular proof in first order predicate calculus (in contrast to solutions in type theory where such a correspondence is provided by normal ization); cf. the transcription of Artin's own work on sums of squares into first order logic discussed in §2(b), as an issue where the 'instinctive resistance' of ordinary mathematicians to foundational languages is relevant. PROOFS AS PRINCIPAL OBJECTS OF STUDY:
SYSTEMATIC SCIENCE
5. To put first things first: How much natural history of the kind suggested in §4 do we need? Can we not learn from the history of successful sciences how to find a systematic scheme without wasteful detours via natural history? Historical Remarks. Much work in natural history was indeed scientifically wasteful; it was superseded by later, fundamental science without being used. One example was already menioned in (the bibl iographical remark of) §4: before Xray crystallography, the natural history of minerals was preoccupied with shapes and colours or with the distribution of particular minerals in odd corners of the globe. The information obtained was sound enough, but simply does not lend itself to theoretical study. (The same appl ies to much of botany or zoology). A more topical example is provided by the subject of 'natural languages'. It has not inspired much confidence, perhaps because  as somebody said  these particular languages do not tell us much about the genuine possibilities (for human language), and even less about the actual world which is supposed to be described by language.  Cer
THEORY OF PROOFS
125
tainly, there have been bright ideas in the parts of natural history mentioned (perhaps as imaginative as much more successful ones); for example, d'ArcyThompson's on growth and form, or Chomsky's on transformational grammars. Not despite, but rather because of their familiarity and general ity, these ideas do not even smell 1ike germs for a systematic theory.  As a memorable contrast: The superficially very special idea in genetics, to concentrate on bacteria and viruses, was not only successful for a systematic science of genetics, but immediately convincing: here one could observe  in a humanly short time  50 many more generations that the difference in degree could be expected to produce a difference in kind (of the data involved). Despite all the excitement about the discovery of formalization, at present there do not seem to be any even mildly promising ideas for a systematic or fundamental science of proofs. Worse still, the two pillars of all current work, (a) and (b) below (which  at the present stage  are surely sound for the natural history of proofs), seem  to me  quite weak. (a) We look for conscious elements in (mathematical) reasoning, or, at most, the very mild extension to those elements which we are able to make conscious to ourselves. As usual, traditional philosophy discusses pedantic doubts (here: doubts about the reliabil ity of introspection, existence of mental objects and the 1ike), and thereby  consciously or unconsciously!  draws attention away from the critical issue,namely this: Do those (sub)conscious elements constitute adequate data for anything 1ike a systematic theory? cf. the situation in mineralogy before Xray crystallography (in the Historical Remarks above), and more generally the inadequacy, for systematic physical theory, of the domain of phenomena that can be made visible. (b) Impressed by its unquestionably distinctive features, we separate mathematical reasoning from other intellectual activities; in humans, and of course in other species. Again, the issue is not the (official) alleged difficulty of making some such distinction precise, and even less the  equally often allegedprejudice against assuming intellectual abilities in subhuman species (after all, before very recent advances in electronic technology, we could not even begin to record the more intimate features of their behaviour). The issue is, once again, whether this particular part of intellectual activity, which as it were strikes the mind's eye particularly vividly, is right for (building a fundamental) theory. To repeat: (a) and (b) do not cast doubt on the natural history of proofs, where we make do with with a comparison between stages in the study of nature of) proofs and of matter. The comparison tion; of course, it is not suggested that proofs
poss i b iii ty of progress with the what we have.  Let me conclude (what is usually called: the seems quite effective for orientaare material substances. 6
A parallel: What is proof? and What is matter? (i) An important step in the development of the atomic theory was the discovery of chemically pure substances, among the mostly impure substances which dominate natural history. This allowed the distinction between atoms and molecules. Bourbaki (1948) mentioned already in §4(a), sounds very much as if their structures  meres (basic structures) are to be compared to chemically pure substances, in particular, to atomic ones. But as stressed in §4(b), foundational analysis has not yet taken the discovery of those analogues to chemical purity into account: By insisting on one foundational language, such as set theory or type theory, one enforces a 1iteral, but purely
6 The comparison is also useful for expanding the discussion, at the Introduction, of the heuristic role of traditional philosophy. After idea of an atomic structure is one of the most striking contributions lative philosophy  perhaps matched by the idea of the relativity of time, made prominent by critical philosophy.
end of the all, the of specuspace and
G.KREISEL
126
formal unity on mathematics. (On pp. 3637 Bourbaki too begin with a fantare on the 'unity' achieved by means of their basic structures  but eight pages later they stress that there are several such structures, and also that quite a lot of mathematics, for example, what is left out of their treatise, is not 'composed' of those structures anyway). Bourbaki, surely quite properly, do not mention in this context the hackneyed business about an objective or subjective view of mathematical real ity. After all, even granted the objectivity of those basic structures, the fact that they contribute to the 'profound intelligibil ity' of mathematics (p. 37) concerns specifically our intellectual equipment. (ii) Building on the discovery in (i), chemical atomism could be developed, and the molecular diagrammes of elementary texts in chemistry. This required only a quite crude idea of atomic structure, the valency of particular atoms; in fancy language, needed for a pun (on the representation of proofs by graphs, as in §4(a) above}, molecules were represented by a purely graph theoretic structure. This can hardly be said to tell us what matter is (possible, as it were), since it leaves open an enormous number of combinations of atoms which never turn up at all. (i ii) A really significant advance was made by relating valency to the intepnal structure of atoms, in Rutherford's theory refined by use of quantum theory. This not only made 'sense' of valency (as is well known). It did much more when Paul ing derived from it a metric structure, specifically, the lengths of and angles between chemical bonds. This additional structure cut down the number of possible combinations sufficiently to make it an effective scientific tool; probably, the most spectacular uses were made in molecular biology.  It would be a I ittle too facile to compare the use of an internal structure of atoms to going beyond conscious elements in (a) above; or the quite essential information provided by such unfamil iar elements as radium to the need for broader experience in (b). However, it seems quite clear that in one direction the current graph theoretic representation of proofs has to be enriched (even though, presumably not by a conventional metric) since a proof is rarely convincing if the conclusion depends on something which is too 'distant' in our memory. In short, one expects that some hypothesis about memory structure is needed here (and, as mentioned already, the difficulties are much the same whether one thinks of memory in physiological or psychological terms). But also, after we have grasped a proof of a proposition, we use the proposition more easily: as somebody said, we know more when we have proved a proposition than when we merely know it is true. The functional interpretations mentioned in §§23 make expl icit some such additional information (which, by §2, is occasionally useful). What is missing is a convincing test for deciding whether this particular addition even approximates the factors which are actually operative. As so often, the trouble is not at all that we cannot think of any theory; if anything, we can think of too many which are roughly consistent with famil iar experience (of mathematical proofs). HEURISTIC VALUE OF TRADITIONAL AIMS:
A OISCLAIMER
6. The view of scientific progress adopted in this article and described in the introductions to p~ I and II disregards the principal specific claims of traditional philosophy  and the conscious or unconscious hope that something like this tradition would provide a shortcut to a fundamental systematic science, without detours via natural history. In particular, the view disregards the claim that traditional analysis is needed to coprect eppors in our ordinary conceptions, and the associated hope that corrections would sotospeak automatically lead to the extraordinary conceptions needed for a basic systematic science. In the case of proofs, our idea of a valid argument is claimed to be in need of correction (or at least of some kind of analysis). And the hope mentioned is impl icit in the socalled logical priority of validity which, being the 'essence' of proof, is assumed to be a basic element of any systematic theory of proofs. (A pun may be involved
THEORY OF PROOFS
127
here too, since unquestionably, discussions of val idity were temporally, that is. 1iterally prior to theories of proof.) Granted all this. the hope rests on the further assumption that an analysis of val idity (tacitly. in terms real istically available at the present time) would be rewarding. Of course. the parallel to validity in the case of physics is the business of real ity. which , in its originally intended generality  has not been of much consequence for the progress of physics.  The claims just mentioned are disregarded in this article, but not dismissed nor rejected. After all, we have evolved in the world in which we 1ive. So why shouldn't our builtin socalled a priori conceptions be a very good guide in science, both efficient and re l iable, and not primarily a I imitation, an obstacle between us and the Ding an sich? Certainly, a mild form of paranoia is needed to concentrate only on the limitation  either with pride in our impotence or in horror of it. ACKNOWLEDGMENT My colleague, P. Suppes, has helped me with careful criticisms of an earl ier draft of this article. REFERENCES: Baker, A. (1964).
Acta Math. 111, 97.
Bishop, E. (196]). Foundations of constructive analysis. (McGrawHi 11, New York); reviewed by A. Stolzenberg (1970). Bull. Amer. Math. Soc. 76, 301. Bourbaki, N. (1948). Pp. 3547 in: Les grands courants de la pensee mathematique ed. F. LeLionnais. (Cahiers du Sud, Paris). Davenport. H., and Roth, K. F. (1955). Mathematika 2. 160. Feferman. S. (1968). Pp. 121135 in: Logic, Methodology and Philosophy of Science III. (NorthHolland, Amsterdam). (1971). Actes Congr~s Int. Math. 1970, Vol. I. 229; rev. J. Symb. Logic 40, 625. Friedman, H. (1976). J. Symb. Logic 41, 558. Gentzen, G. (1969). The collected papers of Gerhard Gentzen. (NorthHolland, Amsterdam). Godel, K. (1931). Monatshefte Math. Physik 38, 173. (1944). Pp. 123153 in: The philosophy of Bertrand Russell (Northwestern Univ. Press, Evanston and Chicago). Henkin, L. (1960). Pp. 284291 in: Surnrna r i es of talks presented Summer Inst. 5ymb. Logic, Cornell ·Univ. 1957. (Inst. Defense Analyses, Princeton). Ingham, A. E. (1932). The distribution of prime numbers. (Cambridge University Press, Cambridge). Jensen, R. B. (1972). Ann. Math. Logic 4,229. Kleene, S. C. (1959). Pp. 81100 in: Constructivity in Mathematics. (NorthHolland, Amsterdam) • Kreisel, G. (1952). J. Symb. Logic 17, 43. (1958). ibid. 23, 155; reviewed by A. Robinson (1966). ibid. 31, 128. (1959). Pp. 101128 in: Constructivity in Mathematics. (NorthHolland, Amsterdam) •
128
G.KREISEL (l959A) . Bu11. Acad. Sc , Po1. ?, 621. (1960). Pp. 313320 in: Summaries of talks presented Summer Inst. Symb. Logic 1957 (Inst. Defense Analyses, Princeton). (1970). Springer Lecture Notes 125, 128. (1973). Pp. 225277 in: Logic, Methodology and Philosophy of Science IV. (NorthHolland, Amsterdam). (1976). Acta Phil. Fennica 28, 166. (in press). Proc. Fourth Scand. Logic Symp. (NorthHal land, Amsterdam).
Kreisel, G., and Krivine, J.L. (1971). Elements of Mathematical Logic. (NorthHolland, Amsterdam). Lehman, R. S. (1966). Acta arithmetica
11,397.
Littlewood, J. E. (1953). A Mathematician's Miscellany. (Methuen, London). Morley, M. (1965). Trans. A.M.S. 114,514. Pfister, A. (1967). Inventiones Math. 4, 229. Robinson, A. (19556). Math. Ann. 130, 257. Siegel, C. L.
(1952). Math. Ann. 124, 17 and 364.
Statman, R. (1974). Structural complexity of proofs. (Dissertation, Stanford University). Suppes, P. (1975). pp , 173179 in: Computers in Education, IFIP, Part 1 (NorthHolland, Amsterdam). Wei1, A. (1967). Acta Math. 113,1. Wittgenstein, L. (1976). Lectures on the Foundations of Mathematics, Cambridge, 1939 (Cornell Univ. Press, Ithaca, N.Y.).
R. Gandy, M. Hyland (Eds.l, LOGIC COLLOQUIUM 76
© NorthHolland Publishing Company (1977)
THE FOUNDATIONS OF MATHEMATICS IN POLAND AFTER WORLD WAR II by
, "
W. MAREK (WARSZAWA) ,
In the period after world war two, Foundation of Mathematics developed in Poland in three main directions:
Set Theory, Hodel Theory, and Recursion Theory.
Obviously the Foundations of Mathematics in Poland were not created in 1945; the roots of interest in these particular domains of sciences one must seek in the investigations of great mathematicians of the between the war period, 19181939 and in particular of Banach, Jaskowski, Kuratowski, Lesniewski, Lindenbaum, . ~ukasiewicz, Sierpinski, Tarski and Ulam.
Among those mentioned only Jaskowski,
Kuratowski and Sierpinski lived and worked in Poland after the war.
Banach,
Lindenbaum and Lesniewski vanished or died in the war, and~ukasiewicz, Tarski and Ulam carried over their activities outside of Poland.
It is worth mentioning
that Tarski has maintained contact with Polish scholars up to the present. From the historical point of view, the oldest among the fields listed above is set theory and its foundations.
Immediately after the war, important investi
gations in this field were carried out by Sierpinski [35], who had already proved during the war the result (previously announced by Lindenbaum and Tarski) that the generalized continuum hypothesis implies the axiom of choice; he subsequently investigated equivalents of the continuum hypothesis. Immediately before the war, Mostowski discovered a general method of independence proofs in set theory with atoms (urelements).
This method, which is nowadays
called the FraenkelMostowski method, is based on the following construction: Let
A
be a set of individuals (i.e. objects which are not sets).
a hierarchy of sets over these individuals as follows: A A U (R~ n P(R~»), for t; > 0 Finally set R = A ; Rt; O \!
We construct U
t;EORD
*The opinions expressed in this paper as well as judgments of the importance of particular papers reflect only the author's opinions and not those of persons and institutions with which he is connected.
These opinions are probably affected by
the author's own mathematical interests and his own ignorances.
"While preparing this paper the author was partially supported by NRC Grant No. A3040.
129
w.
130 Let
G be a group of permutations of
MAREK A ; then for
E G we define the exten
(~ permutes VA) as follows: ~(a) =
of
which are hereditarily symmetric where a symmetric element is one for which G x set
{
U
is an element of the filter
A of individuals, the group
G, and the filter
of a class which is a model of set theory.
Thus the choice of the U leads to a construction
Now appropriate choice of the above
parameters yields "pathological" models in which normally the axiom of choice is invalid although some of its consequences may hold.
Numerous papers by Mostowski
and other mathematicians using this method provided deeper insight into the role of the axiom of choice in mathematics.
Among them one is compelled to mention
the investigations of Mostowski on the finite axioms of choice [13].
An interest
ing feature of these investigations was an application of the methods of finite groups to the investigation of interconnections between the axioms of choice for families of nelement sets (for various n). Investigations in the foundations of set theory were conducted at that time mostly by Mostowski.
In evaluating the developments in the early fifties, one must
mention two of his results.
Seeking a semantic proof of the incompleteness of
Peano arithmetic [15], Mostowski proves the following theorem (and also finds another undecidable statement) which nowadays is generally called "Mostowski's contraction lemma"  one of the basic tools for modern investigations in the foundations of set theory. This theorem states the following. 2 where E C A , with the following properties: 1°
is extensional, i.e., (x) (y) [(z) (zEx
2°
is well founded, i.e., there is no infinite Edescending sequence
Then there is a transitive set
B
such that
~
zEy)
Given a structure
x = y]
~
~
r
E
I
B> (where E is the
membership relation). This lemma is a generalization of the following fact:
Every wellordering is
similar to some von Neumann ordinal. It is no exaggeration to say that every paper investigating transitive models of set theory uses t.hi.s lemma.
It's simply not stated anymore.
Another basic result due to Mostowski [16], a result with farreaching consequences, is a collection of facts on seemingly paradoxical properties of the GodelBernays theory of classes.
Mostowski proves that in that theory the full axiom of induc
tion (in the form of a schema: formulas
p), is not provable.
ones in which formula
~
~(O)
& (n) (P(n)
P(n+l)) =
(n)~(n)
, for all
The unprovable instances are impredicative, i.e.
contains quantifiers ranging over all classes.
FOUNDATIONS OF MATHEMATICS IN POLAND
131
A consequence of this surprising phenomenon was the formulation of the socalled KelleyMorse theory of classes (sometimes called the impredicative theory of classes) which was intensively investigated much later in Warsaw.
The last
modern system of set theory is based on the following intuitions, which originated with Cantor and subsequently von Neumann. possible kinds:
The settheoretical objects are of two
"small" ones, which are able to belong to another object  these
are sets  and "large" ones, which do not belong to any object  these are proper classes.
Such an approach eliminates semantical paradoxes (in this scheme
BuraliForti paradox just states that the ordinals do not constitute a set while Cantor's paradox means that the class of all sets is not a set).
These purposes
(elimination of paradoxes) are achieved by the GodelBernays theory of classes but a distinguishing feature of the KelleyMorse theory of classes ([23]) is a convincing formalization of Frege's principle of the existence of extensions of properties.
This is the following scheme of class existence: (EZ) (x) (x E Z = X is a set & ¢(X»
where the only limitation on
¢
is that
Z
,
does not appear in it.
This schema
simply states that each property of sets determines a class (let us notice that Mostowski in the aforenamed paper shows that the scheme of class existence is unprovable in GodelBernays theory of classes).
Note that in
¢
some quantifiers
may range over all classes; thus, in a sense, this scheme is "selfreferring".
On the other hand, the consistency of the KelleyMorse theory of classes is provable in
ZFC + "there exists an inaccessible cardinal" (this seems to be a
fairly weak set theory in times when measurable and strongly compact cardinals seem to appeal to a lot of people).
Investigations of the KelleyMorse theory
of classes have been conducted in Warsaw in recent years by Mostowski and his collaborators [12], and it seems that KelleyMorse theory of classes isrecentlyquite popular among topologists and category theorists. Among other results connected with foundations of mathematics one must mention those of Kuratowski [9], who found interesting consequences of GOdel's axiom of constructibility in descriptive set theory. In the late fifties and the beginning of the sixties, the foundations of set theory was in a sort of standstill. Cohen's breakthrough.
Thus the explosion of results following
His method of forcing led to numerous independence proofs
in set theory without individuals.
In Poland investigations of the subject were
conducted by Mostowski and a group of his younger collaborators.
A survey of
investigations on forcing is Mostowski's monograph "Constructible sets with applications".
The presentation is based on the following topological inter
pretation (due to RyllNardzewski andindependentlyto Takeuti). the method is based on the following construction:
Let
Roughly speaking
M be a countable,
132
W. MAREK
transitive model of
ZFC
set theory.
We try to make it "thicker" by adding new
elements but in such a way that it will not become "longer" i.e. no new ordinal is added. from all
M a
Let and
a
by closure of the set
M U {a}
It turns out that the set of
dual in Cantor set topology for each axiom such that
generic over
M[a] M.
q,
The model
M[a)
under some operations.
are suitable for our purposes namely we need
a model of ZFC.
a
a f M
be a set of natural numbers,
a
a
a
such that
such that
M[a]
of set theory.
satisfies all axioms of ZFC.
The set
a
M[a)
F
arises But not
q,
is again is resi
In this way we find is called Cohen
Using topologies other than Cantor set topology and considering
other topological spaces leads to models in which the axiom of constructibility, the continuum hypothesis or other interesting statements do not hold. Another major idea in the foundations of mathematics emerges in the beginning of the sixties, the socalled axiom of determinacy of Mycielski and Steinhaus. G (where Xc P(w». X In this way a 01 sequence a
Let
us imagine the following game
Players I and II choose a
number 0 or 1 in turn.
is formed (i.e. a subset of
w).
The subset thus constructed will eventually be in
agree that the player I wins if see that the game X = {a : 0 E a} sequence.
a EX.
X
or outside of it.
Otherwise player II wins.
We
It is easy to
G is determined for some X  at least, for instance if x then clearly I wins, just by putting 1 at the oth place of the
Similarly whenever
X
is denumerable, II has a winning strategy:
consecutively omitting elements of
X
(these results are far from optimal 
recently A. Martin proved that for every Borel set statement:
For every
X ~~w)
, G
X, G is determined). The x is determined (i.e. for each xc P(w) either
X I or II possesses a winning strategy in
G is called the axiom of determinacy. x) This statement implies (Mycielski [27], [28]) that all sets of reals are Lebesguemeasurable, thus contradicting the axiom of choice.
But on the other hand, it
implies some weaker forms of the axiom of choice (i.e. for families of denumerable sets of reals).
For these and other reasons (it was proved by Solovay that the
axiom of determinacy has very great metamathematical strength) the determinacy is being investigated very intensively. Now we shall discuss model theory, which is the newest and most intensively studied branch of foundational research. Basically the theory of models was initiated by Tarski.
In Poland modeltheoretic
research was started by Mostowski in his study on direct products.
An important
paper by Mostowski [17] was later generalized by Feferman and vaught.
Their in
vestigations were later considerably extended by a group of mathematicians in Wroclaw centred around RyllNardzewski. the following problem. U({Ai}iEI)
A typical example of their interests is
Given an operation
U(·)
on families of structures (i.e.
is a structure), to find in what way the theory of the structure
U({A.}. )depends on the properties of I and of the family J. J.EI . additional parameters may be involved).
{Th (Al.')J.'EI}
(some
FOUNDATIONS OF MATHEMATICS IN POLAND
133
Throughout the fifties Rasiowa, Sikorski and others were developing an algebraic approach to the foundations of mathematics.
The algebraic methods are based on
the application of the theory of Boolean algebras (or other distributive lattices  in the case of nonclassical logics) to an algebra of formulas of the language. A fundamental achievement of this method is an algebraic proof of completeness theorem, based on the socalled "RasiowaSikorski lemma" [32]. algebra
B be given and let
assume moreover that
a. a
{a,
a
!1
JEI
i
that there exists an ultrafilter i.e. if
,L~ Eco
~J
ij
u
'EI ,J i
Let a Boolean
B
be a "matrix tl of elements of
exists for all
iECil
in the algebra
B which preserves the inf's
{aij}jEI, ~ U then also
a, E U a,
(for all
Then the lemma states
iECil).
~
The usefulness of algebraic methods in investigations of classical logic and subsequent extensions of these methods so they become useful also for the study of nonclassical logics started a broad current of research of a group of mathematicians (mainly in Warsaw) centred around Rasiowa and Sikorski. A surprising contribution to algebraic logic was presented in the late sixties by Scott and Solovay in their work on Booleanvalued models of set theory.
It
turns out that the construction (described above) of a set of natural numbers generic over
M may be viewed as an application of RasiowaSikorski lemma.
The
investigations of the socalled Martin's axiom, one of the most tempting alternatives for the continuum hypothesis, are also connected with Rasiowasikorski result.
The investigations of Rasiowa and Sikorski were summed up in their mono
graph "The Mathematics of the Metamathematics" (and later in Rasiowa's "An Algebraic Approach to Nonclassical Logic") grouping and classifying the algebraic methods and the results proved using these methods. Another important branch of model theory is Ehrenfeucht's results on elementary equivalence and the characterization of this notion by the methods of game theory (so called Ehrenfeucht's games [2]). been applied in abstract model theory.
The methods originated there have recently Other important results of Ehrenfeucht's
was socalled "omitting types" theorem, which provides an essential means of constructing models. At approximately the same time (the late fifties), basic results on categoricity in power were obtained by ~s and RyllNardzewski.
We say that the theory
categorical if  up to isomorphism  it possesses just one model.
T
is
For instance,
Peano arithmetic with a second order (nonelementary) induction axiom is categorical.
However no elementary theory with infinite models is categorical, so for
elementary theories we consider "categoricity in power". categorical in power
k
if any two models of
T
of power
A theory k
T
is
are isomorphic.
For instance, the theory of dense linear orderings without endpoints is categori
W. MAREK
134 cal in power \~o
Los [10] gave the following sufficient condition for an
0
elementary theory to be complete (this is so called the If a denumerable theory
T
power, then it is complete. denumerable theory
T
~osVaught
criterion):
with no finite models is categorical in any infinite ~os
formulated the following conjecture:
If a
is categorical in some nondenumerable power, then it is
categorical in all nondenumerable powers.
This conjecture, proved in 1962 by
Morley, initiated a line of research which recently has been very actively pursued. RyllNardzewski found very important criterion for categoricity in power '\'0' [34]. Let
~l
'
be formulas with free variables among
~2
is equivalent over
T
with
~2
T ~ ~l ~ ~2'
iff
xl, ••• ,x
We say that n' RyllNardzewski theorem
states that a complete theory without finite models is categorical in power iff for all
n, the number of equivalence classes of formulas with variables
~l
o
is finite. It is impossible to overestimate an importance of the notion of ultraproduct, introduced by Zos, [11].
Let
{Ai}iEI
the sake of simplicity we assume U be an ultrafilter in lence relation f
~
g iff f
~U
and
I.
Ai
In the product
by stipulating g
be a family of similar structures (for
R> i
where
i~I Ai
R ~ Ai x Ai) and let i we introduce an equiva
f ~U g ~ {i : f(i) = g(i)} E U
are equal "almost everywhere").
In
(i.e.
Ai 1"11
TI
we
iEI
introduce a relation
R as follows:
R(f/~U ' g/~U) ~ {L : R
(f(i) , g(i»} E U • i The fundamental result on ultraproducts, proved by Los, is the following: A sentence
~
holds in
{L : Ai ~ ~} E U. products
0
TI
iEI
A.I~U ~
iff it is true "almost everywhere"
i.e.
The above theorem is the key to applications of ultra
It is difficult to describe the numbp.r of papers devoted to this
construction and its applications. books on ultraproducts.
It is enough to say that there are entire
Important results by Tarski, Scott, Keisler and Shelah
(chronologically) established the ultraproduct construction as a basic tool of model theory. Another 'important result and research technique introduced at the same time is the method of "indiscernibles" of Ehrenfeucht and Mostowski.
Their paper [3J on
models with automorphisms contains a method applied later in numerous studies in model theory (in particular in
Silve~'s
papers on large cardinal numbers).
shown by Ehrenfeucht and Mostowski, for any complete theory
As
T without finite
models there is a model which has a large number of automorphisms (its automorphism group contains given group).
As it often happens the most important result
for applications turned out to be the main lemma on which the theorem is based.
FOUNDATIONS OF MATHEMATICS IN POLAND
135
It states that for a theory as above and an arbitrary linear ordering < A there is a model
M
of
T
sequences (in the sense of
and an imbedding '"
and~)
~
: A
~
of element of
M
r
'"
>
such that increasing of the same length
~(A)
have exactly the same properties. Recently "abstract" model theory has been studied in many places.
In particular,
the study of generalized quantifiers was initiated by Mostowski [19] who considered the generalized quantifier "There exists infinitely many
x
such that ••• " •
Another important result by Mostowski which motivated abstract model theory was his paper [22], where he proves that Craig's and Beth's theorems fail in some extensions of first order logic.
The important results of Mostowski on quantifiers
(also [25]) were the inspiration for the investigations of Fuhrken and Vaught and, later, of Keisler and Krivine and McAloon. The field of Recursion Theory is represented by research on hierarchies of sets and functions, decidability and metamathematical investigations on second order arithmetic (analysis).
Of basic importance was the early work of Mostowski [14]
on the arithmetical (KleeneMostowski) hierarchy.
In the late forties and the
fifties Mostowski stUdied the incompleteness of Peano arithmetic.
Apart from
obtaining important extensions of GOdel results, Mostowski wrote a monograph "Sentences undecidable in formalized arithmetic" which is one of the most widely studied monographs of the subject. An important result of the forties was the proof by Szmielew [36], of the decidability of abelian groups. Later research in recursion theory was conducted by Grzegorczyk both on computable functionals and on the hierarchy of recursive functions [6], [7].
Grzegorczyk
also did some important work on decidability. Research in the metamathematics of analysis (on the borderline between recursion theory and set theory) were initiated by a paper of Grzegorczyk, Mostowski and RyllNardzewski [8].
Formalized analysis or the theory dealing with properties
on integers and reals, like every first order theory, has numerous models.
Among
them one discerns models with a faithful interpretation of the natural numbers, these are the so called Wmodels.
The study of these models by the authors named
above, in COmbination with later deep results by Kreisel, Baiwise, Kripke, Platek, Sacks and others, led to the development of a completely new field; generalized recursion theory.
A slightly different line of research initiated by Mostowski
in [20], was investigation of so called Bmodels of analysis, which were intensively stUdied by a group of mathematicians around Mostowski in Warsaw [26], [1],
[25].
When we sum up the activity in the Foundations of Mathematics in Poland after W.W. II, it is quite clear that the individual prevailing over the development
W. MAREK
136
of this domain of mathematics in Poland was Andrzej Mostowski.
His efforts led
to the establishment and education of a large group of mathematicians dealing with foundations and the mathematization of the subject.
The lines of research
initiated by Mostowski were and are actively developed in foundational studies. Throughout the postwar period, foundational research was connected with the intensive investigations in computer science conducted by Ehrenfeucht and Pawlak and later by Blikle and Mazurkiewicz.
Inspirations from each side led to
interesting developments in both fields. Another important aspect of Foundations in Poland was an exchange of scientific thought with mathematicians from abroad. periods in Poland one lists:
Among logicians who spent longer
Addison, Benda, Hinman, LopezEscobar, Prikry,
Sayeki, Scott, Suzuki and many others. International activity of Polish logicians is also evident in the organization of international conferences and meetings.
Apart from the "Infinitistic methods"
symposium (Warsaw 1959) and an ASL meeting (Warsaw 1968) there was the Logical Semester at the Banach Center in Warsaw (Spring 1973) and more recently a series of three conferences on set theory and hierarchy theory (Sudety Mountains, 1974, 75, 76) was organized by Wroclaw group. The editorial activity of Polish authors  apart from the above mentioned monographs of Mostowski, Rasiowa and Sikorski  consisted in publishing handbooks and monographs including:
"Set Theory" by Kuratowski and Mostowski, "Mathematical
Logic" (in Polish) by Grzegorczyk. Foundational actiVity is conducted in Poland mainly in two centers:
Warsaw
and Wrocraw (although almost every University in Poland hires logicians and Logic and Foundations is a part of the courses required by teaching institutions). Recent research in Warsaw concerns chiefly algebraic logic, foundations of geometry, and foundations of set theory, whereas in
Wroc~aw
foundations of set
theory and model theory are the main subjects of interests. REFERENCES This bibliography does not exhaust important papers written in the postwar period in Poland.
It is just pertinent to the main lines of the article.
[1] Apt, K.R., Marek, W. (1974). Second order arithmetic and related topics Annals of Math. Logic ~, pp. 177229. [2] Ehrenfeucht, A. (1961). An application of games to the completeness problem for formalized theories. Fund. Math. XLIX, pp. 129141. [3] Ehrenfeucht, A., Mostowski, A. (1956). Models of axiomatic theories admitting automorphisms, Fund. Math. XLIII, pp. 5068. [4]
Grzegorczyk, A. (1953).
[5] Grzegorczyk, A. (1955). 202.
Some classes of recursive functions, Diss. Math IV. Computable functionals, Fund. Math. XLII, pp. 168
FOUNDATIONS OF MATHEMATICS IN POLAND
137
[6] Grzegorczyk, A. (1956). Some proofs of undecidability of arithmetic, Fund. Math. XLIII, pp. 166177. [7] Grzegorczyk, A. (1974). Warszawa  Dordrecht.
Outline of Mathematical Logic, PWNReidel,
[8] Grzegorczyk, A., Mostowski, A., Cz. RyllNardzewski (1958). The classical and wcomplete arithmetic, Journal of Symb. Logic 22, pp. 188206. [9] Kuratowski, K. (1948). Ensembles projectifs et ensembles singuliers, Fund. Math. XXXV, pp. 131140. [10] ~s, J. (1954). On the categoricity in power of elementary deductive systems and some related problems, ColI. Math. III, pp. 5862. [11] bas, J. (1955). Quelques remarques, theoremes et problemes sur la classes definissables d'algebres, In: Math. interpretation of formal systems, NorthHolland, Amsterdam, pp. 98113. [12] Marek, W., 11ostowski, A. On extendability of the models of ZF set theory to the models of KM theory of classes, In: Springer Lecture Notes 499, pp. 460522. [13] Mostowski, A. (1945). pp. 137168.
Axiom of choice for finite sets, Fund. Math. XXXIII,
[14] Mostowski, A. (1947). XXXIV, pp. 81112.
On definable sets of positive integers, Fund. Math.
[15] Mostowski, A. (1949). XXXVI, pp. 143164.
An undecidable arithmetical statement, Fund. Math.
[16] Mostowski, A. (1951). Some impredicative definitions in the axiomatic set theory, Fund. Math. XXXVII, pp. 111124. [17] Mostowski, A. (1952). Logic 17, pp. 131.
On direct product of theories, Journal of Symb.
[18] Mostowski, A. (1952). Sentences undecidable in formalized arithmetic, North Holland, Amsterdam, 117 pp. [19] Mostowski, A. (1957). pp. 1236.
On generalization of quantifiers, Fund. Math. XLIV,
[20J Mostowski, A. (1960). Formal system of analysis based on an infinitistic rule of proof, In: Infinitistic Methods, PWN  Pergamon Press, Warszawa  London. [21J Mostowski, A. (1965). Fennica12, 180 pp.
Thirty years of foundational studies, Acta Phil.
[22] Mostowski, A. (1968). Craig interpolation theorem in some extended systems of logic, In: Logic, Methodology and Phil. of Sciences III, NorthHolland, Amsterdam, pp. 87103. [23] Mostowski, A. (1967). Constructible sets with applications. NorthHolland, WarszawaAmsterdam, 269 pp. [24J Mostowski, A. pp. 220282.
An exposition of forcing.
In:
PWN
Springer Lecture Notes 450,
[25] Mostowski, A. (1975). Observations concerning elementary extensions of Wmodels I, Proceedings of MiS Symposia in Pure Mathematics XXV, pp. 349355. [26] Mostowski, A., Suzuki, Y. (1969). Fund. Matho LXV, pp. 8393. [27] Mycielski, J. (1964). pp. 205224.
On wmodels which are not Smodels,
On the axiom of determinateness I, Fund. Math. LIII,
W. MAREK
138 [28] Mycielski, J. pp. 203212.
(1966).
On the axiom of determinateness II, Fund. Math LIX,
[29] Mycielski, J., Cz. RyllNardzewski, (1968). II, Fund. Math. LXI, pp. 271281.
Equationally compact algebras
[30] Pacholski, L., Cz. RyllNardzewski, (1970). products I, Fund. Math. LXVII, pp. 155161.
On countably compact reduced
[31] Rasiowa, H. (1974). An algebraic approach to nonclassical logics. NorthHolland, WarszawaAnsterdam.
PWN
[32] Rasiowa, H., Sikorski, R. (1950). A proof of the completeness theorem of Gadel, Fund. Math. XXXVII, pp. 193200. [33] Rasiowa, H., Sikorski, R. (1963). PWN, Warszawa.
The mathematics of metamathematics,
[34] RyllNardzewski, Cz. (1959). On the categoricity in power < ~' Bull. Acad. Pol. Sci. Sere Sci. Math. Astronom. Phys. VII, pp. 545 2 548. [35] Sierpinski, W. (1947). L'hypothese generalisee du continu et l'axiome du choix, Fund. Math. XXXIV, pp. 15. [36] Szmielew, W. (1955). XLI, pp. 203271. [37] W)!glorz, B. (1966) • pp. 289298. [38] WtglOrZ, B. pp. 993.
(1967).
Elementary properties of abelian groups, Fund. Math. Equationally compact algebras I, Fund. Matho LIX, Equationally compact algebras III, Fund. Math. LX,
POSTSCRIPT>I< Professor A. Tarski and Dr. S. Givant pointed to us certain inaccuracies and important omissions in our text. As this could lead to misunderstanding we notice most important of them: (1) Reporting the domains of foundational studies we omitted several branches of investigations such as:
al
Foundations of Geometry investigated in Warsaw and other places by Borsuk, Szmielew, Szczerba and others.
bl Universal Algebra as developed in RyllNardzewski and others.
wroc~aw
by Marczewski, Mycielski,
(2) We generally omitted reference to parallel work of other investigators like in case of Ehrenfeucht games reference to parallel Fralsse work on elementary equivalence or in case of RyllNardzewski result on ~ocategoricity independent work of Engeler and Svenonius. (3) We erronously stated that the Axiom of Determinataness implies the principle of dependent choices. As pointed by Professor J. Mycielski the axiom of determinateness implies only the existence of choice function for countable families of reals.
* Added on
March 20, 1977
R. Gandy, M. Hyland (Eds.), LOGIC COLLOQUIUM 76
© NorthHolland Publishing Company (1977)
A TRIBUTE TO A.MOSTOWSKI Helena Rasiowa Institute of Mathematics University o~ Warsaw Warsaw, Poland It is my privilege to speak about Andrzej Mostowski, a teacher, a colleague, and a friend of mine, and to pay homage to the work he left behind him which constitutes a vital contribution to the development of mathematical logic and the foundations of mathematics. Andrzej Mostowski was one of the greatest spirits and a cofounder of the Polish school of logic and the foundations of mathematics. He was one of those researchers inspired to undertake logical investigations in mathematics itself, yet at the same time his papers are shot through with philosophical ideas. He always maintened a lively interest in algebra and in this field also he wrote a number of books and expository articles. Professor Mostowski was the author or coauthor of 118 scientific publications, including monographs and textbooks, which form a lasting contribution to Polish and world science. The papers of Mostowski opened up new avenues of research and the results contained in them are among those most of~en cited in the literature. I will touch on only some of Mostowski s results which were the source of new lines of investigation. Mostowski's permutation models (1939) which he obtained by making precise an idea of Fraenkel have found wide application in proofs of independence results for set theory and of course in particular in the proof of independence of the axicm of choice. The method of permutation models is still important even after the discovery of forcing. The socalled Mostowski contraction lemma l1949) is one of the basic metamathematical results of set theory. The hierarchy of arithmetical notions, devised by Mostowski during the second wordl war independently of S.C.Kleene and published only in 1947, after the appearance of Kleene's paper on the same subject, is considered as one of Mostowski's greatest achievments. It became the source of many discoveries and the starting point of a very wide literature. Mostowski's paper on nondeducibility in intuitionistic functional calculus (194e), in which he proposed an algebraic interpretation of quantifiers, inspired later researc~ of other authors on an algebraic approach to various predicate calculi • The theory of indiscernible sets created jointly with A.Ebrenfeucht (1956) yielded one of the most frequently employed methods for constructing models and has found numerous applications both in the theory of models and in other parts of the foundations of mathematics. 139
140
HELENA RAS IOWA
His paper on generalized quantifiers (1957) had a strong influence on the beginnings of socalled soft model theory and also found application in research on second order arithmetic. The procedure invented by Mostowski in his paper "On direct products of theories" (1952) for reducing the truth of a sentence in a direct product to questions about the truth of related sentences in components of the product, was later extended in stages by other authors and applied many times in model theory. The paper "The classical and the ~ complete arithmetic" (1958) written jointly by A.Mostowski, A.Grzegorczyk and C.RyllNardzewski initiated the etudy of wmodels of arithmetic. MostolVski's paper" Formal system of analysis based on an infinita~~ ~~~n~fo~~~~f~ri~~~~i~~fundamental for the study of ~ models The monographs and textbooks of which Mostowski was the author or coauthor , characterized by their preciseness will always arouse the reader's interest. His first book "Mathematical logic" ,wri tten in Polish immediately after the second wordl war and publishea in the series Monografie Matematyczne in 1948, was an excellent textbook. He refused permisson for its translation and publication in other languages. In 1946 appeared "Outline of Galois theory" ,written in Polish, as an appendix to the textbook "Higher algebra" of W.Sierpinski. In 1952 appeared the monograph "The theory of sets" ,written in Polish, the joint 1V0rk of K.Kuratowski and A.Mostowski. The second edition completely revised was published in 1966 and an English translation appeared in the NorthHolland series Studies in Logic and the Foundations of Mathematics in 1967. The third edition, completely revised and widely extended, will appear. Professor Mostowski finished the preparation of this edition in May 1975A Russian translation was published in Moscow in 1970. In the NorthHolland series Studies in Logic and the Foundations of Mathematics also appeared the following monographs of MostolVski: 1952 "Sentences undecidable in formalized arithmetic" a very successful exposition of the work Qf G~del on the undecidability of arithmetic; 1953 "Undecidable theories" written with A.Tarski and R.M.Robinson, which presented a method for prOVing the undecidability of many mathematical theories a method used for many years afterwards; finally in 1969 appeared "Constructible sets with applications" the most complete acoount of the theory of constructible sets and of relative constructibility, inclUding independence results in set theory, which has appeared upto the present time. From amongst his more expository works one should also mention "Thirty years of Foundational Studies", Acta Philosophica Fennica (1965), "Mod~les Transitifs de la theorie des ensembles de ZermeloFraenkel", University of Montreal (1967), and "Models of Set Theory", C.loM.E. Lectures, Varenna (1968). With M.Stark , Mostowski wrote in Polish a threevolume textbook "Higher Algebra" (1953,1954,1954) as well as the textbooks "Linear
A TRIBUTE TO A.
~lOSTOWSKI
141
Algebra (1958) and "Introduction to Higher Algebra" (1955), of which the last named was published in English, Oxford (1963). One of the great contributions of Mostowski to Polish science was the substantial influence he exercised on the work of postwar Polish logicians the majority of whom were his students. Mostowski's influence was also felt in other countries. Young specialists from allover the word1 came to work with him. Among nonPolish logicians who spent some time in their careers with Mostowski are: J.W.Addison, M.Benda, M.Dickman, P.Hinman, M.Machover, K.Prikry, H.Sayeki, Y.Suzuki and G.Wilmers. Andrzej Mostowski was invited to lecture at many universities and gave invited addresses at many international congresses and symposia. All wished to talk with him and hear his advice. Professor Mostowski kept in touch with many of the most distinguished specialists in the wordl. He heard of the most interesting results before they were published directly from the authors. He knew all the problems in logic and the foundations which were currently the centrebf interest. He himself was an inexhaustible source of information in this field due to his exceptional memory and the ability to concentrate on problems discussed with him. Mostowski was not one of those scientists who confine themselves to their own specia1ity.Besides mathematics and philosophy he had a lively interest in physics, history, literature, painting,music and the theatre. Thomas Mann and Charles Dickens were two of his favourite authors. His favourite literary work was "The Pickwick Papers" from which he often quoted. As a teacher he was demanding and most demanding of himself. He did not rate highly his own achievments and successes. He hated any kind of ceremonial or even collegial expressions of the recognition he so richly deserved.When his colleagues at the University of Warsaw organized a reception in honour of his sixtieth birthday he did not turn up. The next day he felt very abashed and apologized to the organizers. Andrzej MostowsMi was born October lst,19l3 in Lwow. His father StanisIaw Mostowski an assistant in physiological chemistry at the University in Lwow died a year later. By the family and friends Andrzej Mostowski was called Staszek as his father. Besides the early childhood spent in Lw6w and Zakopane, a mountain resort in the Tatras,Mostowski's almost who~e life was connected with Warsaw. There he went to school from 1923 to 1931 and there studied mathematics at the University from 1931 to 1936. At that time at the University of Warsaw worked mathematicians and logicians some already with a wordlwide reputation, some young and talented, cofounders of the Polish school of mathematics and logic: Wac1aw Sierplnski, Stefan Mazurkiewicz, Kazimierz Kuratowski, Karol Borsuk, Jan Bukasiewicz, Alfred Taraki, Adolf Lindenbaum, stanis~w Lesniewski. The scientific atmosphere of this environment and the trends of research, being focussed mainly on the theory of sets, topology~ mathematical logic and foundations of mathematics, shaped the scientific interests of Andrzej Mostowski.However, as a student he strove to obtain the widest possible mathematical education going beyond the programme of studie~
142
HELENA RASIOWA
The strongest influence on the creative work of Mostowski was exercised by his teacher Alfred Tarski. The years 1936, 1937 Mostowaki spent abroad studying mathematics at the Universi try of Vienna and at the Federal Polytechnic in Z~ich. In Vienna he was a student of Kurt G~del whom he regarded with warmth and admiration until the end of his life. In ZUrich he attended the lecture~ of Herman Weyl on ay,mmetry. Weyl'a personal charm, the beauty of his lectures and his treat work "Raum, Zeit, Materie ", from which Mostowski learned the theory of relativi try, left a permanent impression. After the return to Poland Andrzej Mostowski began a new phase of intensive research. At that time he was intrigued by problems surrounding the idea of finiteness. In 1938 at the University of Warsaw he defended his doctoral dissertation which was devoted to various forms of the definition of finiteness of a set. After his doctorate, being unable to find a position in the University because they were none vacant, he started work in the Gove~ ment Meteorological Institute whose director was the distinguished physicist Jan Blaton. During the occupation of Poland Andrzej Mostowski remained in Warsaw working as assistant to the accountant in tarpaper factory. During the years 19421944 he gave lectures in the underground university, the University of Warsaw having been closed by the. invader. After the Warsaw uprising all were forced to leave Warsaw whose destruction had been ordered and which already lay in ruins. Mostowski escaped from one of the transports by which the inhabitants of Warsaw were moved to Germany and located in concentration camps. Expositions of numerous results obtained by Mostowski during the war were burned when his appartment was destroyed. A large part of those results has never been reconstructed. Among them were results on decidability and on descriptive set theory parallel to those due to J.W.Addison. Immediately after the liberation of Poland Andrzej Mostowski had for several months a position at the Sl~sk Polytechnic which had temporary quarters in Krakow. He attained the rank of docent at the Jagiellonian Universi~ in 1945 on the basi3 of his habilitation thesis about the independence of the axiom of choice, which initiated research on the independence of various forms of the axiom of choice for finite sets. A year after his habilitation he returned to the University of Warsaw to become head of the section concerned with the philosophy of mathematics. At that time he was the only logician in the Department of Mathematics and Natural Sciences remaining from the strong prewar group of research workers. On him fell the responsibility for the future of the Polish school of logic and the foundations which before the war had attained", Lead.i.ng position in the wordl. Andrzej Mostowsk1, the young mathematician, eagerly threw lumaelf into teaching and research. He gave lectures and seminars in his o,vn speciality and at that tims also lectured on group theory and Galois theory which he always p'.J.rticularly Li ke d, His talent and personal charm attracted young mathematicians and students to him.
A TRIBUTE TO A. MOSlOWSKI
143
gladly discussed scientific problems with them, he always had time and patience. He suggested questions of interest, discussed them in an atmosphere of humour and good will, and kindled the enthusiasm of youth. He won the hearts and shaped the thoughts of his students. I am happy to recall that in 1946 I became his first assistant, and that in 1950 Andrzej Grzegorczyk and I graduated together as his first doctoral students. The number of his students quickly increased.In the early years were Henry Hiz, the talented logician Antoni Janiczak who died in 1953, Andrzej Ehrenfeucht,and later many others.
~e
Polish mathematics between the wars was characterized by concentration on certain chosen fields of mathematics: the theory of sets, topology, functional analysis, real analysis, mathematical logic and foundations of mathematics. In the period of reconstruction the Polish mathematical community felt an urgent need to develop every subdiscipline and in particular algebra. Understanding this need in 1953 professor Mostowski became head of the algebra section in the University of Warsaw and continued ~his function until 1969. The awakening of interest in algebra in Poland and the creation of a cadre of algebraists of international calibre was in considerable measure due to Andrzej Mostowski. The algebra textbooks which he wrote with Marceli Stark were basic material for our students for many years and are still in use together with others which appeared later. From 1969 to the end of his life Andrzej Mostowski was head of the section of the foundations of mathematics in the University of Warsaw and in his last years was also vicedirector of the Institute of Mathematics. From the opening of the Government Mathematical Institute in 1949, later transformed into the Mathematical Institute of the Polish Academy of Sciences, Andrzej Mostowski was head of the section concerned with the foundations of mathematios. In 1956 Andrzej Mostowski beoame a oorresponding member of the Polish_Aoademy of Sciences, and in 1963 attained full membership, the highest scientific honour in Poland. Twice he received Government Prizes for his mathematical achievments. He received also Irzykowski prize being Polish American prize for d1stinguis~ ed Polish scientists. In 1973 he was elected to the Finnish Acadeny of Sciences an honour which he prized highly. Professor Mostowski fulfilled many administrative functions in Polish institutions and scientific societies. He also fulfilled a number of functions in the administration of the Association for Symbolic Logic. From 1964 to 1968 he was VicePresident, Division of Logic Methodology and Philosophy of Science, of the International Union of History and Philosophy of Science. From 1971 to 1975 he wa President. Andrzej Mostowski also found time to undertake many editorial duties.He was editor of the Bulletin of the Polish Academy of Sciences for mathematics, astronomy and physics. He was a member of the editorial oommittees of Fundamenta Mathematicae,Dissertationes Mathematicae, Studia Logica, the Journal of Symbolic Logic and others. From the beginning of fifties he was in a olose connection with NorthHolland Publishing Company cooperating first with M.D.Frank and later with E.Fredriksson. From 1966 he was one of editors of NorthHolland series Studies in Logic and the Foundations of Mathematics. He was also one of cofounders and editors of Annals of Mathematioal Logio.
144
HELENA RASIOWA
Of Mostowski's numerous visits to the scientific centres of other countries I will mention only the longest. The academic year 1948/ 1949 he spent at the Institute for Advanced Study in PrincetOn where he renewed contact with K.G~del. Ten years later he was a Visiting professor at the University of California, Berkeley,where he worked once again with A.Tarski. In the academic year 1969/1970 he was a member of All Souls College, Oxford. Andrzej Mostowski spent the summer months of 1975 at the Universi4Y of California, Berkeley. Hlis lectures awoke lively interest in his listeners as always, and he mentioned with pleasure the satisfaction they had given him. On the way from Berkeley to London,Ontarw, where he intended to participate in the Vth International Congress of Logic Methodology and Philosophy of Science, he stopped in Vancouver at the invitation of professor Alistair Lachlan of Simon Fraser University. On the 20th August at 3:30 he gave his last lecture "Set theory with classe~'. He spoke beautifully and as usual laced his presentation with flashes of humour. None of those present amongst whom were A.Lachlan, R.Harrop, S.Thomason, D.Pinc~ M.Dubiel and I, sensed that this would be his last lecture. Almost immediately after the lecture he suddenly fell ill and died two days later. At the Congress in London, Ontario, there was a special memorial session in his honour as part of the General Assembly of the Division of Logic Methodology and Philosophy of Science of IUHPS. Andrzej Mostowski was an unusually modest man of great personal charm and unfailing kindness. His modesty and simplicity made him universally liked. All who knew him personally found his death a great loss. REFERENCES Mostowski,A.(1939).Uber die Unabh~ngigkeit des Wohlordnungsatzes von Ordnun~sprinzip. Fundamenta Mathematicae XXXII,pp.20l252. Mostowski,A.(1947).On definable sets af positive integers.Fundamenta Mathematicae XXXIV,pp.81l12. Mostowski,A.(1948).Proofs of nondeducibility in intuitionistic functional calculus.Journal of Symbolic Logic 12,pp.193203. Mostowski,A.(1949).An undecible arithmetical stalement.Fundamenta Mathematicae XXXVI,pp.143164. Mostowski,A.(1952).On direct product of theories.Journal of Symbolic Logic 17,pp.131. Mostowski,A.·and Ehrenfeucht,A.(1956).Models of axiomatic theories admitting automorphisms.Fundamenta Mathematicae XLIII,pp.5068. Mostowski,A.(1957).On a generalization of quantifiers.Fundamenta Mathematicae XLIV,pp.1236. Mostowski,A.,Grzegorczyk,A.~dRyllNardzewski,Cz.(1958).The classical and GVcomplete arithmetic. Journal of Symbolic Logic ~, pp.1882Q6. Mostowski,A.(1960).Formal system of ~alysis based on an infinitary rule of proof.Proceedings of Warsaw Symposium on Infinitistic Methods, WarsawLondon,pp.141166.
R. Gandy, M. Hyland (Eds.l, LOGIC COLLOQUIUM 76
© NorthHolland Publishing Company (1977)
BOLZANO'S CONTRIBUTION TO LOGIC AND PHILOSOPHY OF MATHEMATICS Jan Berg Department of Philosophy Munich Institute of Technology Munich, West Germany
The Wissenschaftslehre (1837) by Bernard Bolzano (17811848) is one of the masterpieces in the history of logic. In this encyclopedic work Bolzano intended to construct a new and philosophically satisfactory foundation of mathematics. The search for such a foundation brought forth valuable byproducts in logical semantics and axiomatics. For example, Bolzano introduced the notion of abstract, nonlinguistic proposition and described its relations to other relevant notions such as sentence, truth, existence and analyticity. Furthermore, he studied relations among propositions and defined highly interesting notions of validity, consistency, derivability and probability, based on the idea of "replacing" certain components in propositions. In set theory, he stated the equivalence of reflexivity and infiniteness of sets and considered isomorphism as a sufficient condition for the identity of powers of infinite sets. He conceived of a natural number
as
a
property characterizing sets of objects, even though he did not base his development of arithmetic on this notion, and analyzed sentences about specific numbers in a way reminiscent of Frege and Russell.
In a posthumous manuscript from the
1830's (recently pUblished) he developed a theory of real numbers, which differs from those of Dedekind, Weierstrass, Meray and Cantor. BOlzano's real numbers may be identified with certain sequences of rational numbers. Logic in BOlzano's sense is a theory of science, a kind of metatheory, the objects of which are the several sciences and their linguistic representations. This theory is set forth in Bolzano's monumental fourvolume work Wissenschaftslehre (hereafter referred to as 'WL') .
JAN BERG
148
Bolzano's very broad conception of logic with its strong emphasis on methodological aspects no doubt accounts for the type of logical results which he arrived at. The details of his theory of science proper are given in the fourth volume of the WL and belong to the least interesting aspects of his logic. On the other hand, Bolzano's search for a solid foundation for his theory of science left very worthwhile byproducts in logical semantics and axiomatics. His theory of propositions in the startingpoint of these results. Bolzano became more and more aware of the profound distinction between the actual thoughts of human beings and their linguistic expressions on the one hand, and the abstract propositions and their components which exist independently of these thoughts and expressions on the other hand. Furthermore, he imagined a certain fixed deductive order among all true propositions. This idea was intimately associated with his vision of a realm of abstract components of propositions constituting their logically simple parts.
* For the following presentation of Bolzano's theory of propositions I have to define some terms. A concrete sentence occurrence is a sequence of particles existing in space and time, arranged according to the syntactic rules of a grammar, and contrasting with its surroundings. A simple sentence shape, on the other hand, is a class of similar concrete occurrences of simple sentences. A compound sentence shape is built up recursively from simple sentence shapes by means of syntactic operations. Not every compound sentence shape has a corresponding concrete sentence occurrence. Two compound sentence shapes may be considered identical if they are built up from identical simple sentence shapes in the same way. Two simple sentence shapes are identical if they contain the same sentence occurrences. Now consider the compound sentence containing the following concrete sentence occurrence:
'a simple sentence shape is a class of similar
sentence occurrences or it is not the case that a simple sentence shape is a
cl~ss
of similar sentence occurrences'. In another sense
one could say that this sentence shape, which is an abstract logical
149
BOlZANO'S CONTRIBUTION TO lOGIC
object outside of space and time, contains two sentence occurrences, i.e .• two abstract "occurrences" of the simple sentence shape containing the following concrete inscription:
'a simple sentence shape
is a class of similar sentence occurrences'. In the following, I will use the expression 'sentence occurrence' exclusively in the first, concrete sense.
* Bolzano's notion of abstract nonlinguistic proposition (Satz an s i cn ) is a keystone in his philosophy and can be traced in his' writings back to the beginnings of the second
decade of the 19th centu
ry. I shall try to characterize Bolzano's conception of propositions by means of certain explicit assumptions. These assumptions also give information about the relation between propositions and other logically interesting objects. In his logic Bolzano utilizes a concept which is an exact counterpart of the modern logical notion of existential quantification. Therefore, he could have stated that (1)
There exist entities, called 'propositions', which fulfill the following necessary conditions (2) through (15).
r cr ,
WL §§30
ff. ) Thus, propositions possess the kind of logical existence developed in modern quantification theory. However, (2)
A proposition does not exist concretely in space and time (WL § 19).
According to Bolzano, both linguistic and mental entities such as thoughts and jUdgments are concrete (WL
§§
34, 291). Hence, proposi'
tions could not be identified as concrete linguistic or mental occurrences. Furthermore,
(3)
Propositions exist independently of all kinds of mental entities (WL
§
19).
Therefore the identification between propositions and mental dispositions sometimes made in medieval nominalism cannot be applied to
150
JAN BERG
propositions in BOlzano's sense. A proposition in BOlzano's sense is a structure of ideasassuch. Hence, an ideaassuch (Vorstellung an sich) is a part of a proposition which is not itself a proposition (WL
48). But to be able
§
to generate propositions we have to characterize ideasassuch independently of propositions. This is in fact implicit in Bolzano. He worked extensively with the relation of being an object of an ideaassuch, which corresponds in modern logic to the relation of being an element of the extension of a concept.
In terms of this relation,
taken as a primitive by Balzano, certain postulates may be extracted from his writings which concern the existence and general properties of ideasassuch. Independently of human minds and of linguistic expressions there exists a collection of absolutely simple ideasassuch. As examples Bolzano mentions the logical constants expressed by the words 'and',
'some',
'to have',
'to be',
'ought'
(WL
§
'not',
78); but he admits
being unable to offer a more comprehensive list. He seems to mean that each complex idea A can be analyzed into a sequence ~ (A) of simple ideas which would probably include certain logical constants. I shall call this sequence .::r(A) the
'primitive form'
of A.
The man
ner in which a complex idea is built up from simple ones may be expressed by a chain of definitions. So it appears that some complex ideas behave somewhat like the open formulas of a logical calculus. Bolzano assumes that two ideas are strictly identical if and only if they have the same primitive form (WL
§§
92, 119, 557).
We can now state two conditioqs for propositions, which are peculiar to Bolzano. The first one reads essentially as follows:
(4)
Not all propositions are simple. A compound proposition is built up recursively from simple parts by means of primitive operations (WL
558).
§
Furthermore, to each Bolzanian proposition P there corresponds uniquely a proposition i(p) containing only simple ideas. Such an f(P)
I shall call the
'primitive form' of P.
::rep)
may be found by
reducing all complex ideas of P to simple ones by means of chains of definitions. We can now express Bolzano's principle of individuation for propositions:
BOLZANO'S CONTRIBUTION TO LOGIC (5)
151
Two propositions are strictly identical if and only if they have the same primitive form (WL
§§
32, 558).
We may assume that we have a language of a more or less specified structure and that we know what a sentence shape is in this language. Bolzano apparently identified all linguistic entities with concrete occurrences (WL
§§
49, 334) and did not say anything explicit about
the abstract notion of sentence shape. It is possible, however, to reconstruct a concept closely related to our notion of linguistic shape and based entirely on concepts within Bolzano's philosophy. For if an object of the idea A is a linguistic expression in Bolzano's sense, then A may be a counterpart in Bolzano's theory of our notion of abstract linguistic shape. In particular, if the objects of A are sentence occurrences, then in some cases A may be an abstract sentence shape. Whereas a shape is usually taken as a class in modern logical syntax, Balzano's notion could be described as a corresponding intensional property. From the postulates (4) and (5) it appears that propositions in Balzano's sense behave somewhat like the closed formulas of a logical calculus. The following assumption, on the other hand, would seem to agree with Bolzano's intentions:
(6)
No linguistic entity should necessarily be a component in propositions.
Hence, it would not be possible to identify propositions as sentence shapes or as any other objects essentially involving sentence shapes or occurrences, such as, e.g., ordered pairs of sentence shapes and the interpretations of a language. Had Bolzano been faced with the notion of abstract linguistic shape, he would probably have approved (cf. WL
§
410) of the following gen
eral assumption concerning sentences (whether shapes or occurrences):
(7)
There are more propositions than sentences.
If we suppose that the number of atoms in the universe is finite, then we have to accept the fact that there are only finitely many
JAN BERG
152
concrete sentence occurrences. From the definitions of simple and compound sentence shapes given above it follows that there are denumerably many sentences (whether shapes or occurrences). The distinction between the denumerably and superdenumerably infinite was not yet clear to Bolzano, although he had one notion of continuum that applied to the set of real but not to the set of rational numbers and had a clear understanding of the fact that isomorphism is a sufficient condition for the identity of powers of infinite sets. However, for a modern development of a system of nonlinguistic Bolzanian propositions it is natural to presuppose that there are nondenumerably many such entities. Consider, for example, a denumberable domain of individuals and the nondenumerable infinity of subsets of this domain, and take any of these subsets M, and any individual x. We then have the proposition (true or false) that x is
an element of M.
We call a notion of truth for linguistic or nonlinguistic semantic entities Y intuitively satisfactory if its definition implies the classical Aristotelian condition that X is true if and only if where
Y,
'X' stands for a name or description of the substituend of 'Y'.
Now Bolzano would require that (8)
An intuitively satisfactory notion of truth is definable for propositions (WL
§
19).
With respect to the relations between propositions and sentences the following four postulates hold: If a declarative principal sentence (shape or occurrence) expresses something,
P,
then P is a proposition (WL
§§
19, 28).
(10)
If the sentences 8, and 8 2 express the same proposition, then 8 1 and 8 may have different structures (WL § 127). 2
(11 )
If 8
and 8 express the same proposition, then 8 is 1 1 2 logically equivalent to 8 2"
The latter assumption underlies Bolzano's theory of the reduction of sentences to certain canonical forms. The converse does not hold, however, in Bolzano's system:
BOLZANO'S CONTRIBUTION TO LOGIC ( 12)
If the propositions P and Q are expressed by 8 respectively, and if 8
153 1
and 8
is logically equivalent to 8
1
P need not be identical with Q (WL § 32, Note).
2
2,
then
As a consequence, propositions cannot be identified as classes of logically equivalent sentences. In the WL Bolzano used a partly formalized language which embraces an ordinary language extended by constants, variables and certain technical expressions. He also investigated the relations of his semiformalized philosopical language to colloquial language (WL
§§
121146, 169184). He believed that most sentences of collo
quial language were "reducible" to sentences or sets of sentences of certain canonical forms expressed in the philosophical language. When Bolzano speaks of a sentence being reducible to another sentence, he seems to require that the sentences be synonymous in the sense of expressing the same proposition (WL §§ 127, 171). In Bolzano's theory of reduction, most sentences of ordinary language are reducible to sentences obtained by inserting expressions of particular ideasassuch for
'A'
and expressions of particular
abstract ideas of second order (Which have a property as the sole member of their extension) for
'b'
in one of the following two
expressions (WL §§ 127, 136): (i)
'[The idea of],
A has b
(i i)
'[The idea or]
A has La c kr o f'rb
!
,
! ,
In a formalization of Bolzano's philosophical language, the vocabulary would include as predicates both substituends of 'b' and corresponding instances of 'lackofb'
in (i) and (ii), respective
ly. Hence, the contradictory of a sentence of the form of (i) is another sentence of the form of (i), namely (iii)
'P(i) has falsity',
where P(i) is the proposition expressed by the particular instance of (i)
(WL
§
189).
JAN BERG
154
An example of Bolzano's application of his theory of reduction to ordinary language would be the set of sentences obtained by inserting a particular expression of an idea for
'A'
in 'There is an A'
and 'Nothing is an A'. These sentences are reducible to the canonical sentences 'The idea of A has nonemptiness' and 'The idea of A has emptiness', respectively (WL
§§
137, 170).
Had Bolzano's theory of reduction been completely developed it might have resulted in the construction of an ideal language for philosophical analysis. In this ideal language sentences of the canonical forms of (i) or (ii) would not play the same role, however, as the atomic sentences of modern
~uantification
theory on the basis of
which more complex forms are built up. It seems, on the contrary, that Bolzano intended even.the most complicated sentences to have canonical form or to be reducible to a set of such sentences. Yet there is a fundamental vagueness in Bolzano's theory of reduction, for nowhere in his work does he give any systematic rules for the construction of the extremely complicated names that would occur as substituends for the variables
'A' and 'b'.
Now the interesting point for the discussion of propositions is Bolzano's implicit assertion: (13 )
Sentences of the canonical form of (i) or (ii) mirror their correlated propositions in the sharpest way.
Hence, we have to presuppose that ( 14)
For all A and b within the appropriate ranges, the transition from A and b to the propositions expressed by (i) and (ii) must be explained.
In other words, it has to be explained how two concepts can be combined to make a statement. For propositions corresponding to sentences of the canonical forms of (i) and (ii) the notion of truth is defined substantially as follows
(WL
§
24):
BOLZANO'S CONTRIBUTION TO LOGIC (i+)
155
P(i) is true if and only if every object of A has the property which is the object of b,
(ii+) P(ii) is true if and only if every object of A does not have the property which is the object of b. Here A and b are indicated by the sUbstituends of 'A' and
'b' re
spectively. Balzano apparently presupposes that the set of sentences of
collo~uial
language and of his own philosophical language could
be mapped into the set of propositions, so that an indirect definition of truth for these sentences would be forthcoming. Balzano elaborates his theory of reduction by imposing a necessary condition for the truth of sentences of the form of (i) and (ii). If A is nonempty (i.e., if there is at least one object of A), then the corresponding propositions, P(i) and p(ii), are also said to be nonempty. And if A is empty (i.e., if there is no object of A), then P(i) and p(ii) are said to be empty (WL P(ii) is true, then A is nonempty (WL ( 1 5)
§
§
146). Now,
if P(i) or
225.4). Hence,
If A is empty, then both P(i) and p(ii) are false.
Since the entity indicated by the substituend of 'A'
in (i) and (ii)
can be a class concept, Balzano's condition implies an existential interpretation of classical syllogistic. And the very same condition shows that Bolzano was heading in substance for a philosophical language without existence assumptions, where even empty domains are permitted. An attempt to grasp Bolzano's theory of propositions must comply with the assumptions (1) through (15). These assumptions function as an axiom system for which a model is sought. A domain of relatively wellknown objects, which behave in the same manner as the entities to be explained, has to be described. The fulfillment of the conditions (1) through (15) also means that a philosophical, ideal language must be constructed, which differs considerably from the usual languages of logic. The reduction of the sentences of ordinary language to sentences of canonical form could be accomplished, for example, if the distinction between predicates and function symbols were abolished and if all expressions were considered as the result
156
JAN BERG
of applying a function symbol to a corresponding number of arguments. Furthermore, the assumption (15) would be satisfied if BOlzano's ideal language were syntactically constructed and semantically interpreted in such a way that a nontrivial existence predicate could be introduced.
* Bolzano observed that a proposition may change its truthvalue upon variation of certain components. For example, the true proposition: (i)
Bolzano is mortal
can be turned into the falsity: (i i)
Bolzano is omniscient
by changing the predicate idea. But if the sUbject idea Bolzano in (i) is changed into other ideas within the (usually explicitly presupposed) range of variation, proposition (i) cannot be turned into a falsity. Thus we see that Bolzano implicitly utilized a replacement operation for propositions, sending Pinto P(A/B), where the latter proposition is similar to P except that it contains the idea B at all places where P contains the idea A.
This operation can be extended to a
simultaneous replacement in a proposition of the distinct ideas A" ••• ,A
n
by B"
••• ,B
n,
respectively, thereby sending Pinto
P(A , •.• ,An/B, , ... ,B The replaced ideas A,,'" n). 1
,An have to be
distinct, of course, whereas the replacing ideas B"
... ,B
n
need not
be distinct. Each idea occurring in a proposition has its particular range of variation. In doubtful cases this range may be indicated in the proposition itself by means of a description adjoined to the sUbject idea. For example, the proposition: The human being Bolzano is mortal is an explicit counterpart to proposition (i) above.
BOLZANO'S CONTRIBUTION TO LOGIC
157
The variation of ideasassuch is an original and very fruitful conception. Bolzano exploits his innovation with great thoroughness in a logical theory which, on decisive points, anticipates notions expounded (independently by others) in the philosophical discussion of our time, especially in Logical semantics and axiomatics. First of all, the notion of validity of propositions is introduced essentially as follows (WL § 147):
('6)
P is valid with respe~t to the distinct ideas A" ... ,An in P if and only if P(A" sequences B, , ..• ,B
n
... ,An/B, , .•• ,B
n)
is true for all
, where every idea B. belongs to the range J
of variation of A .. J
Contravalidity is defined analogously: (17 )
P is contravalid with respect to A" P(A, , ... ,An/B1 , ... ,B
n)
... ,An if and only if
is false for all B , ... ,B 1 n.
Here we have to assume that P contains no defined ideasassuch. Otherwise an unabbreviated proposition could be valid, while a corresponding proposition obtained by definitional abbreviation of certain ideas could be formally nonvalid according to (16). Take P as the proposition: 7 + 5 < 7 + 6, and Q as: 7 + 6 c 13, obtained
=
5. Now p(7/N) is true for all natural number ideas N, whereas Q(7/8) is false; hence P but not from P by the definition:
13
def.
7 +
Q is valid with respect to the idea 7. Yet the conception that an unabbreviated proposition could be found for any given conceptual proposition (the content of which embraces concepts only and no intuitions, in the German sense of 'Anschauung') permeates Bolzano's whole philosophy. with regard to certain relations among propositions, Bolzano distinguishes the special logical case, where the ideas varied embrace all nonlogical ideas of the propositions (WL We then define:
§§
148.3, 223).
158
JAN BERG
('8)
P is logically valid if and only if P is valid with respect to the sequence of all nonlogical ideas of P.
Bolzano also considers logical properties of sets of propositions. He then utilizes what is basically a replacement operation on sets of propositions, sending the set {P"P 2" {PI ,P 2 , · · .}(A" P
2(A"
... ,A
,An/B,,'" ,B
n/B"
n),
.• } into the set
,B n), i.e., (P,(A,,'"
,An/B,,'"
,B n),
... }. That a set K of propositions is true
means, of course, that each member of K is true.
We may then extend
Bolzano's definition of validity as follows:
('9)
{P"P , . · . } is valid with respect to A,,'"
2
{P, ,P 2 , ... HAl" B1, ... ,B
n
.. ,An/B, , ..• ,B
, where every B.
J
n)
,An if and only if
is true for all sequences
belongs to the range of variation
of A .• J
The important notion of consistency is a generalization of this relation (WL
'54):
§
{P"P , ... } is consistent with respect to A , ... ,A if and 1 2 n
(20)
only if (P"P , ... }(A 2
sequence B , · .. ,B 1
1,
... ,A
n/B"
... ,B
n)
is true for some
n•
A corresponding relation between sets of propositions can be introduced:
(2')
{Pi} and {Qi} (i A"
... ,A
n
=
',2, ... ) are consistent with respect to
if and only if (P"Ql,P 2 , Q2 " " } is consistent
with respect to A"
... ,A
n•
As a special case of consistency in the sense of (2'), Bolzano now introduces the relation of derivability between sets of propositions (WL
§
,
55 ) :
BOLZANO'S CONTRIBUTION TO LOGIC (22)
159
{Qi} is derivable from {Pi} with respect to A,,'"
,An if
and only if {Pi} and {Qi} are consistent with respect to A".oo,A n and if {Qi}(A"oo.,An/B"
... ,B n) is true for any
sequence B".oo,B n that makes {Pi}(A"
... ,An/B"
... ,B n)
true.
Bolzano distinguishes even the special logical case of derivability, which is defined essentially as follows (WL (23)
§
223):
{Qi} is logically derivable from {Pi} if and only if {Qi} is derivable from {Pi} with respect to the sequence of all nonlogical ideasassuch of the propositions Pi and Q i (i
= , ,2, ... ).
Bolzano's relation of logical derivability is one of his most impressive discoveries and a highly interesting counterpart of the modern notion of logical consequence introduced by Tarski. There are certain differences, however, between the two relations. First of all, modern notions of consequence are defined for sentences within formalized languages, whereas Bolzano's derivability holds between abstract nonlinguistic propositions expressed in a natural language extended by variables and certain constants. The difference is of vital importance for the study of the relationships between consequence and other logical notions. For example, the equivalence of logical consequence and syntactic provability within elementary logic could not be imagined without a highly developed conception of a formal calculus. Only after logic had become completely formalized by Frege was it possible to formulate a precise notion of syntactic provability. Secondly, BOlzano's condition that the sets {Pi} and {Qi} should be consistent with respect to the sequence of all nonlogical ideasassuch of every Pi and Q has no counterpart in the definitions i of modern logical consequence. As a result of this consistency clause, Balzano's theory is less general and more complicated than the modern theories of consequence. For example, consequence relationships with contradictions of the type: P and not P, as
ante~
cedent must be represented in Bolzano's theory by stating that the
160
JAN BERG
disjunctive proposition: P or Q,
is logically derivable from P,
where Q is the original consequent. Moreover, consequence relationships with contradictions of the type: A is not identical with A,
cannot be represented at all in Bolzano's system.
Thirdly, Bolzano's implicit semantics differs in a fundamental respect from Tarski's semantics. Following Tarski, we say that a formula S is a consequence of the set r of formulas if and only if every interpretation over any nonempty domain that makes all elements of
r true also makes S true. Here we generalize universally
over both interpretations and domains. An interpretation over a domain D is a function sending symbols for individuals into D and predicates onto sets of ntuples of elements of D. 1\ semantics based on such functions may be called an
'interpretation semantics'.
It is
possible, however, to base a semantics exclusively on functions sending formulas of a language onto truthvalues. We then get what may be called a
'truthvalue semantics' or an
'evaluation semantics'.
Now, S is an evaluationsemantic consequence of a (possibly infi' nite) set r of formulas of
if and only if there is a finite subset 6
r such that S is true under any evaluation for which all elements
of 6 are true. The decisive difference between an interpretation semantics and an evaluation semantics lies in the different treatment of the quantifiers,
For example, a formula ~ is interpreta
tionsemantically true for the interpretation J Over D if
~
is true
for all interpretations J/over D which assign at most to ~ a different value from D than does J.
But under an evaluation seman
tics, ~A is true under the evaluation E if A~ is true under E for if. all possible substituents if. of the variable ~. In spite of this essential ontological difference, however, the notions of consequence with respect to both kinds of semantics are equivalent at the level of elementary logic, even if there are infinitely many antecedents.
Bolzano never referred to domains of individuals in
his semantics and never thought of combining quantification over domains with his generalization over ideasassuch.
Instead, his
quantification over sequences of ideasassuch would, if applied to a formalized language, correspond closely to the evaluationsemantic conception of quantification. Thus, Bolzano may be considered the first philosopher who characterized an evaluationsemantic notion of consequence with a finite number of antecedents.
*
BOlZANO'S CONTRIBUTION TO lOGIC
161
In connection with his definition of validity Bolzano also introduces the function degree of validity. We may write 'g(P,A"
.... A
n)'
to denote the value of this function for the proposition P upon variation of the ideas A,,'" call Q a P(A, ....
.A
n.
To simplify definitions I shall
'variant' of P with respect to A"
,Ani
B,,""
,B
... ,A
n
if Q equals
for some sequence of ideas B, . . . . . B Now n•
n)
Bolzano's definition of degree of validity can be expressed as follows (24)
(WL
§
'47):
g(P,A, , ..• ,A
equals the proportion of the number of true
n)
variants to the number of all variants of P with respect to
the
The domain
of~logical
meaSure function g is restricted to proposi
tions with a finite number of variants. In order to extend the domain of g. Bolzano adds the further restriction that variants differing only in'ideasassuch with the same extension should be identified. Within the domain of the function g it obviously holds that P is valid with respect to A"
... ,An if and only if g(P,A"
.. " ,A n
)=' .
Furthermore. P is contravalid with respect to A, •... ,An only in that case where g(P,A"
••. ,An)=O. Now it is possible to generalize
the relation of derivability within the domain of the function g. To say, then. that A,,'"
P
is derivable from {Qi} with respect to
,An is tantamount to saying that the conditional proposition:
If {Qi} is true then P is true, is valid with respect to A, , ... ,An in other words that
th~
, with respect to A,,""
degree of validity of this conditional is ,An'
It is natural, then, to consider weaker
cases of derivability where the conditional has a lower degree of validity.
In this way Bolzano is led to his notion of probability.
Let 'P(P.{Qi},A"
... ,A
n)'
denote the probability of the hypothesis
P relative to the set of propositions {Qi} and the sequence of ideass
a si s u c h A"
... ,An' Then Bolzano' s partial definition of the notion
of probability can be expressed as follows
(WL
§
'6'):
162 (25)
JAN BERG
If {Qi} (i
= , ,2, ...
,m)
is consistent with respect to ... ,A n) equals the proportion o}~e
A" ... ,A n, then P(P,{Qi},A"
number of true variants of {P,Q,,'" variants of {Qi} with respect to A
,Qm} to the number of true
1.···,An•
In analogy to BOlzano's treatment of the notion of derivability. we can even distinguish the special logical case of probability, p*, of P relative to {Qi}:
(26)
p * (P,{Qi}) equals P(P,{Qi}.A 1,··.,A n), where A" . . . . An are all the nonlogical ideas of P and {Qi}'
Bolzano's notion of probability is explicitly conceived of as a relation among propositions and their constituents. The notion of logical probability immediately reconstructible within Bolzano's system is the logical relation of an hypothesis to its evidential support. These features and Bolzano's measurement of probability by the technique of variation are strongly reminiscent of Carnap's recent theory of regular confirmation functions.
Thus, Bolzano was
probably the first philosopher who characterized inductive probability.
* Modern logic distinguishes the three questions: ,
a consequencE of the formula $' (ii) Is ,
• in the theory TI
(i) Is the formula
formally deducible from
(iii) Is the sequence F of formulas a deduction
in T from. to '1 The theory of derivability developed within Bolzano's logic of variation yields a method of handling the first question. As waS pointed out earlier, however, Bolzano had no precise way of dealing with questions of the second kind. Instead he turned to a theory which was intended to answer certain very special questions of the third type when they are transferred to the nonlinguistic level of propositions. Bolzano's relation of ground to consequent (Abfolge) involves the notion of being directly deduced in a preexisting system of true propositions, where the meaning of 'deduced' is not confined to
BOlZANO'S CONTRIBUTION TO lOGIC
163
purely logical relations but embraces even direct inferences based on causal and deontic implications. The Abfolge is an asymmetric and intransitive relation between sets of true propositions (WL
§§
203, 204, 213). We employ the notation '{Qi}=¢. pI to indi
has the Abfolge relation to P. Bolzano further ~nd {U } ,.> P i assumes, inter alia, that {Qi}~ P~imply the identity of the two antecedent sets (WL
206), and that P is neVer a member of {Qi}
§
if
{Qi}~
P holds (WL
If
{Qi}~
P holds, the members of {Qi} may be conceived of as
§
204).
immediate s u b s i d i a r y truths of P, and for each of these members, in turn, we may look for its immediate subsidiary
trut~.
ceed in this manner until we reach what Bolzano calls i.e., true propositions P having no (WL
§
We can
pro~
'basic trutru;,
set {Qi} such that
{Qi}~
P
214). In this way we may discover a preexisting unique
Abfolge hierarchy or "prooftree" for P (WL
§§
216, 220). The
general notion of a subsidiary truth of P is now forthcoming as a representation of all truthsQ distinct from P and included in a prooftree for P (WL
217).
§
In his logic of variation Bolzano was aware of an inference rule corresponding to Gentzen's "cut rule": from the proposition that P
2
is derivable from P
we can deduce that P
4
1
and that P
4
is derivable from P
is derivable from P
1
and P
3
(WL
2
and P
3,
155.24).
§
Gentzen's Hauptsatz asserts that under certain conditions any proof in elementary logic can be reduced to a normal form by eliminating all cuts. Bolzano is aware of a similar kind of "normal" proof in his logic of variation. This idea is carried through in his theory of the Abfolge structure. Here it is shown that the number of simple ideasassuch involved in the propositions of a branch of a prooftree'will in general increase downwards (WL
§
221).
It further follows from the assumptions of the theory that no unnecessary ideasassuch are introduced into branches of the deductive hierarchy of propositions.
*
JAN BERG
164
After completing the WL, Bolzano returned in the 1830's to pure mathematics and started projecting a new encyclopedic work, the GroBenlehre, in analogy to his earlier treatments of theology and logic. It was intended that this great work should establish the foundations of mathematics, give a detailed exposition of most known branches of that science, and also pursue original investigations and inaugurate new domains of research. Bolzano's mathematical manuscripts contain a vast body of drafts of the various sections of the GroBenlehre, which deal with the elements of logic and semantics, with artihmetic, algebra, mathematical analysis, set theory and geometry. The second volume of the GroBenlehre is called 'The pure theory of numbers'. It ranges from an analysis of natural numbers to various attempts at constructing a theory of real numbers. It has recently been published in volume 8 of series IIA of the Bernard BolzanoGesamtausgabe (Stuttgart: FrommannHolzboog, 1976). At the end of the 19th century, Frege's definition of natural numbers as classes of concepts with isomorphic extensions appeared in print. And a few years later Russell, independently of Frege, defined natural numbers as classes of isomorphic classes. But half a century before Frege, Balzano in the first part of his "Pure theory of numbers» defined natural numbers as concepts of isomorphic classes. However, Bolzano did not base his construction of arithmetic on this abstract notion of natural number. In the WL there is another approach to the notion of natural number, which is the outcome of the analysis of sentences about particular numbers, such as the following: (a)
There are exactly
~
objects falling under the concept C.
Bolzano's analysis of such sentences is strikingly reminiscent of Frege's and Russel's corresponding analyses (WL
§§
139, 243). Now,
if (a) is definable by ¢(C), then the abstract number
~
may be
constructed as the concept of all C such that ¢(C). In the second half of the 19th century, theories of real numbers were put forth by Dedekind, Weierstrass, Meray and Cantor. In the
BOlZANO'S CONTRIBUTION TO lOGIC
165
last part of his "Pure theory of numbers" BOlzano made a different kind of attempt in which he based the system of real numbers on a purely arithmetical foundation. According to Bolzano, a rational number is definable by means of finitely many applications of rational operations to natural numbers. He then considers those cases in which there are infinitely many applications of rational operations. Among the examples mentioned are the following: 1+1+ .•.
b
in info and
. Such constructions are called 'infinite number
in info
concepts'
1 + 2 + 3 +
and their expressions are called
'infinite number
expressions'. In his proofs, Bolzano computes infinite number concepts by applying formal arithmetical operations to infinite b
number expressions.
'1' q
For example, he subtracts 1+1+ •.. in info
and transforms the result into
'( 1+1i". . . q( 1+1+.
in inf. )gb' in inf.)
from
. This
procedure is possible if the infinite number expressions are conceived of as designations of sequences of rational numbers and if the arithmetical operations are applied to the members of these sequences. Under this interpretation the infinite number expressions ,
b
'1+2+3+ ... in info' and 1+1+ ... in info correspond to the sequences 1
<2n(n+1»
b
and <;>, respectively.
The sequence <1 £>, the members of which are >0 when n > bq, q
n
corresponds to the expression of the result of the abovementioned subtraction.
Such expressions Balzano calls
'essentially positive
number expressions'. Hence, an essentially positive number expression can be represented by a sequence <sn> which there is an no
o~
rational numbers, for
such that sn>O for all n>n
o'
Such a sequence
we may call an 'essentially positive sequence'. Now Bolzano considers those infinite number concepts S such that, for every natural number q, there is an integer p such that the following equations hold:
166
JAN BERG
E.q
S
+ Rand
S
I!.:t.l. q

R" •
Here Rand R"are essentially positive number expressions or R denotes O. Expressions of such infinite number concepts are called 'measurable number expressions'. Under the interpretation adopted here, Bolzano's equations state identity of sequences. The numerator p depends on the denominator q and the sequence S. To make this explicit, we may write
'p(q,S)'.
Furthermore, the sequences Rand R"depend on the sequence S, and from later proofs of Bolzano's it also appears that they depend on the denominator q. To make this explicit, we can write 'R(q,S)' and 'R"(<:<,S)'. 'Let S be the sequence <s
n
and let
>
n
and
be the corresponding essentially positive sequences. Then
BOlzano's equations may be stated as follows: For all q and all n it holds that s
n
=~ q
+ R (q, S) n
and
s
n
pig,S)+1 _ R"(
n q,
q
S).
Bolzano's measurable number expressions, then, correspond to sequences S (= <sn»
of rational numbers such that, for any natural
number q, there in an integer p(q,S) such that for all n:
s
n
~ q
+
where either Rn(q,S)
R (q,S) n
=
p(g,S)+1 _ R~( q
n
q,
0 for all n or
positive sequence and where
S)
,
is an essentially
is an essentially positive
sequence. We may call such a sequence S a
'measurable sequence'.
In the first version of his manuscript on real numbers, Bolzano calls a measurable number S, for which it holds for all q that p(q,S)
= 0,
an
'infinitely small positive number'. Hence, an
infinitely small positive number may be identified as a null sequence of positive terms. An infinitely small negative number is the opposite of an infinitely small positive number, i.e., a null sequence of negative terms.
BOLZANO'S CONTRIBUTION TO LOGIC
167
In the first version of his manuscript on real numbers, Bolzano neglected the null sequences with terms of alternating sign. This of course has disastrous consequences for his original theory, for by means of such sequences it is possible to give counterexamples to many of his alleged theorems. For example, under Bolzano's original definition of 'infinitely small numer', it is not true that the difference between any inifinitely small numbers equals zero or an infinitely small number. a c
1
ti'
n
1 =2n(2n1)'
2n
B:
b 2n1

1
2'
1,
=
=
1 b 2n' 2n
' B
Let A
1 2n1 ' c 2 n 1
=
, C
 1
3'
...
,
and
c.

1
1
2 ' 2'
n
> and
1 2n(2nl)'
We then have the sequences A: 1,
1  4'
=
1  12'

1 1  4' ;;1 ,  3'
... ,
1
12'
In Bolzano's notation A and B would look like this:
1 and 1 1+1+ ... in info 2+13+13+ ... in info
, respectively. Now C is
identical with A  B without being measurable. For suppose that C is measurable. Then it must hold that
where Rn(q,C) ~ 0 and R~(q,C) > 0, in both cases for sufficiently large n. From this there follows:
c n_
l2..l9.....tl, q
p(g,C )+1 q

c
n
But C is a null sequence with terms of alternating sign. Hence, we have the contradiction: p(q,C) < 0, p(q,C) > 1. Bolzano observed this gap in his theory, however, in a later amendment to his manuscript, in connection with the definition of equality between measurable numbers.
In the first version of his
168
JAN BERG
manuscript, Bolzano defines the measurable numbers A and B as equal if and only if it holds for all q that p(q,A) = p(q,B). It follows from the properties of the identity relation and the universal quantifier of elementary logic that this relation of equality is an equivalence relation. However, this notion of equality does not satisfy the law of trichotomy. to hold if and ~nly if A 
(A > B is said
B is positive and not infinitely small.)
A counterexample emerges from Bolzano's own text: let A = 1 
1+1+ ...
~..i n f". and B
~n
and since for all q, to B.
p(q,A)
= 1. Then neither A > B,
=q

= q,
1 and p(q,B)
not B
>
A,
A is not equal
In the amendment to his manuscript, Bolzano proposes instead
the fOllowing definition: (28 )
A is equal to B of and only if
iA 
Bi
is infinitely small.
This notion of equality is an equivalence relation and satisfies the law of trichotomy and, moreover, saves many otherwise false statements in Bolzano's theory. This definition also shows that Bolzano was aware of a wider notion of an infinitely small number, Which may be identified as
the general concept of a null sequence.
The generalized notion of an infinitely
s~all
number requires a
change in the definition of a measurable number. This was in fact supplied by Bolzano. According to a late amendment to his manuscript, a measurable number expression corresponds to a sequence S (= <sn»
of rational numbers such that, for any natural number q
there is an integer p(q,S) such that for all n: (29 )
s
n
~ + q
R
(
n q,
S)
= P ( 9 , S ) +m 
q
R
(
n q,
S)
,
where m is a natural number parameter and
is an
=
1 for all
number expressions Which correspond to sequences converging toward an irrational number.
For number expressions which correspond
to sequences converging toward a rational number it suffices to take m = 2. In his theory of natural numbers Bolzano distinguishes between concrete numbers and abstract numbers.
In his theory of real
BOlZANO'S CONTRIBUTION TO lOGIC
169
numbers he considers only concrete numbers explicitly. However, a corresponding abstract notion is hinted at in various places. In Bolzano's theory the abstract real numbers could be identified as concepts or classes of equal measurable infinite number concepts.
* During the 1840's it became clear to Bolzano that the completion of the projected work on mathematics would be too much for him. He then concentrated on publishing in installments in an attempt to arouse the pUblic interest in the posthumous work he hoped his disciples would bring before the pUblic. The work on the Paradoxien des Unendlichen was undertaken in accordance with that policy. This monograph contains ideas on the foundation of set theory. In the WL (§ 87.3) Bolzano essentially defines a set as finite if it is an instance of a natural number. Bolzano's notion of natural number is implicitly based on the notion of isomorphism.
In the
Paradoxien des Unendlichen (§ 20) Bolzano describes what we call 'isomorphism' between the sets M and M as follows: Every x in M, 1 2 can be paired with a y in M in such a way that any element of M, 2 or M is a member of at least one pair, and no element of M, and M 2 2 is a member of more than one pair. Bolzano now observes that an infinite set can be isomorphic to a proper subset of itself, and he asserts that all infinite sets have this characteristic. We call a set
'reflexive' when it is isomorphic
to a proper subset of itself. Hence, Bolzano states in effect that all infinite sets are reflexive. And he is also aware of the fact that all finite sets are nonrelexive (§ 22). Yet he does not posit reflexivity as a defining property of infinite sets, as was done later, e.g., by Dedekind. In the Paradoxien des Unendlichen, and implicitly in the WL too, Bolzano does not consider isomorphism to be a sufficient condition for the identity of powers of infinite sets. Instead, he subscribes to the doctrine that the whole is greater than its parts beyond the finite case. Accordingly, finite isomorphic sets are equinumerous, whereas the isomorphism of infinite sets does not imply their equinumerousness. As a result, a number of statements follow in the
JAN BERG
170
Paradoxien des Unendlichen which do not correspond to the modern view on this subject. In the WL (§
102) Bolzano considers the series (8,) 1,2,3, ...
4
1,16,81, ... ,n , ••• ; ... 3) Obviously, he says, each of these series has infinitely many terms, ... ,n, ..• ; (8)
1,4,9, ... ;
(8
. He contends that and the terms of 8 are among the terms of 8 i l i the number of terms of 8 "exceeds infinitely" the number of terms i . However, in a letter written a few months before his death, of 8 i_ l precisely this conclusion is sharply criticized by Bolzano. He has
become aware of the fact that 8"
82' 8 have the same number 3, of terms. Unfortunately, death came before he could develop the implications of this promising insight.
Hence, Bolzano stated the equivalence between reflexivity and infiniteness of sets, and in the end confined the doctrine that the whole is greater than its parts to the finite case, accepting isomorphism as a sufficient condition for the identity of powers of infinite sets. These are two achievements of major importance in Bolzano's investigation of the infinite.
* The WL was intended merely as a prelude to Bolzano's work on mathematics. His main ambition was to recreate the whole body of contemporary mathematics in accordance with the vision of an abstract hierarchy of true propositions. For Bolzano this task implied the creation of entirely new foundations for certain branches of mathematics, as may be seen from his highly interesting efforts directed toward basing geometry on topological concepts. In carrying out this program, most of the means of expression of modern quantification theory were in essence available to Bolzano. He came very close to modern notions of satisfaction, logical truth, consistency and logical cons€quence.
On the other hand, the formal
deductive machinery of quantification theory is practically nonexistent in Bolzano's works. This
synta~tic
machinery appears
only in Frege, who created the first strictly logistical system at the end of the 19th century. Bolzano's lack of interest in developing particular logical calculi most probably stems from his at times almost excessive preoccupation with the methodological
BOLZANO'S CONTRIBUTION TO LOGIC
171
aspects of logic and mathematics and of science in general. The notion of calculus in the modern logistical sense was first clearly considered by Leibniz. His basic dream was of an effectively decidable, interpreted calculus embracing all "eternal" truth. Bolzano was justifiably critical of this overambitious program and presented instead his own theory of the Abfolge structure of nonlinguistic propositions, thereby taking his stand away from that line of development in logic which leads to modern syntactic concept formation. A reasOn for Bolzano's general lack of interest in questions of logical syntax was no doubt his profoundly intensiQnal, nonlinguistic approach to logic. Bolzano's central thesis, that there are abstract objects which differ from both mental occurrences and all kinds of linguistic expressions, has been advocated by later philosophers of the Germanspeaking countries, inter alia by Lotze, Brentano in his earlier period, Meinong and Frege. Lotze and Frege never refer to Bolzano's work, though, and the others protested their independence of Bolzano. Husserl admits that he received vital influences from Bolzano, but his notions of "ideal" Objects derive from Lotze's and not from Bolzano's logic. Among the great Western philosophers Bolzano is perhaps the least influential. In epistemology, logic and mathematics his most fervent disciples were not able to propagate his ideas with sufficient vigor. His keen criticism of German idealistic philosophy and his important discoveries in logic,' semantics and mathematical philosophy silently died away. A contributing cause of Bolzano's lack of influence on the development of the philosophica1 disciplines was, of course, the fact that most of his works were, for political reasons, published anonymously in editions not easily accessible. Furthermore, an immense number of unpublished manuscripts in a partly almost indecipherable handwriting is to be found in archives in Prague and Vienna. Several unfruitful attempts have been made in the last 150 years to bring out more or less complete editions of Bolzano's works. It is to be hoped that the latest venture launched in Stuttgart, West Germany, will prove more successful.
R. Gandy, M. Hyland (Eds.). LOGIC COLLOQUIUM 76
© NorthHolland Publishing Company (1977)
THE SIMPLE THEORY OF TYPES
R. O. Gandy Mathematical Institute, Oxford.
In the first part of this paper I review the early history of the simple theory of types; in the second part I discuss some questions raised by the work (published and unpublished) of A.M. Turing on type theory. EARLY HISTORY I have nothing new to say about sources.
A rather superficial skimming through
Church's bibliography and the'known sources suggested that the papers which are usually cited, by Russell, Chwistek and Ramsey, are indeed the sources from which the simple theory of types was developed. graphy for this paper.
These papers are listed in the biblio
But many of the historians, or logicians making historical
remarks, seem to me to have misunderstood and misreported these early papers. So a review, a REview, may be of some use. The simple theory of types provides a straightforward, reasonably secure, foundation for the greater part of classical mathematics.
That is why a number
of authors (Carnap, Godel, Tarski, Church, Turing) gave a precise formulation of it, and used it as a basis for metamathematical investigations. straightforward because
i~
The theory is
embodies two principles which (at least before the
advent of modern abstract concepts) were part of the mathematicians normal code of practice.
Namely that a variable always has a precisely delimited range, and
that a distinction must always be made between a function and its arguments.
In
this sense one might claim that all good mathematicians had anticipated simple type theory.
[Indeed Turing made this claim for primitive man].
The claim goes
too far, but it does draw attention to the fact that the justification for the theory is not to be found by considering it as a formal device to avoid Russell's paradox.
This point is very clearly put by Church in his interesting paper
'Schroder's anticipation of the simple theory of types': Speaking in more general terms we may say that Schroder regarded a class as somehow built of its members, as a wall is built of brick or (to adopt Frege's simile) as a wood is composed of trees. From such a point of view, according to which members of a class are prior to the class, something like a theory of types does appear natural and inevitable.
173
R.O. GANDY
174
[In discussion at the conference Michael Dummett argued that Church claims too much for Schroder.
Schroder's stipulation that an individual in a domain D must
not, if it is itself a domain, have elements in common with D is, as Church recognises a technical necessity.
There is hardly a hint of the type structure,
let alone any suggestion that it might form a foundation for mathematics]. Mathematicians at that time did not have too clear a general notion of 'set', but the sets with which they were familiar were certainly constructed by 'assembling' previously given elements. I find it extraordinary  even after making allowance for the climate of metaphysical abstract in which, philosophically speaking, Bertrand Russell grew up  that nowhere in his writings (7) certainly nowhere in the Principia, does Russell adopt, or even refer to the kind of mathematical common sense described in the previous paragraphs.
And his blindness was inherited by many other logi
cians. I consider next the theory of types which Russell outlines, and rejected, in appendix B to the ramified theory. X is a
Principl~s
of Mathematics.
This is a simple rather than a
He defines, in effect:
~
if it is of the form {x:¢(x) is significant};
X is a range if it is of the form {x:¢(x)}; Russell is here very naive about what propositional functions ¢(x) can be used. He allows V = {x:x is a range} and U = {x:x=x} as ranges so that V sumably, though he does not remark the fact, U E V and V
E
U).
E
V (and pre
From his paradox
he concludes that the Russell class is not in the range of significance of x but gives no other reason for excluding it.
E
x
The reason which he gives for doubt
ing the correctness of this theory of types is sufficiently curious to be worth reproducing.
Let Prop = {x : x is a proposition}, and let ~ P stand for the (in
finite) conjunction of all the propositions in the set P. (a)
P:=' Prop ~
(b)
fA P
=
Mp
IX\ Q ~ P
E
Russell supposes:
Prop; Q.
Now let Q = {p
E
Prop::3 P
and let q = ~Q.
:=.
Prop. p
Then, using (b) q
& E
P
/I
P I. p},
Q * q I. Q. He then says:
It is possible of course to hold that propositions themselves are of various types and that logical products must have propositions of only one type as factors. But this suggestion seems harsh and highly artificial. A sentiment which may help to explain why it took him so long (5 years) to arrive at the ramified theory of types. I now consider Chwistek's contribution.
His motives in proposing a simple
theory of types were disingenuous; he did not believe that it could provide a true
THE SIMPLE THEORY OF TYPES foundation for mathematics. particularise.
175
To understand his point of view it is necessary to
In (1912) Chwistek claimed to have derived a contradiction from
the axiom of reducibility; the arguments are repeated in (1921) and (1922). is customary to dismiss this claim as based on a misunderstanding.
It
I think
Chwistek's argument deserves more attention than it is usually accorded; I will describe it briefly. In Principia the universe of individuals, the primitive relations over them, and the resulting atomic propositions are taken as given, and as not being part of logic.
Russell, supported by Wittgenstein's Tractatus seems always to have held
that in principle the universe of all existent things could be completely described by atomic propositions.
I call this universe the ultimate universe of discourse.
The logical constructions of Principia are built up from the primitive terms in the description of this universe, and the truth of nonlogical statements such as the axiom of infinity is determined by the atomic proposition which are true of it.
(Russell often refers to the truth of the axiom of infinity as a matter of
empirical fact.)
The range of a variable of any (ramified) type consists of just
those propositional functions which can be defined by the means appropriate to that type from the primitive relations of the ultimate universe of discourse. Now, to illustrate Chwistek's argument, suppose that there were only finitely many primitive relations.
(This might be false, but it is not absurd  indeed
both Carnap and Russell believed at one time that similarity was the only primitive relation.)
Then one could enumerate all predicative properties of the natu
ral numbers (construed, say, as classes of classes of individuals) by a single relation of appropriate type.
So, by diagonlisation, one could define a property of
the natural numbers which differed from all predicative properties, thus contradicting the axiom of reducibility.
So the axiom of reducibility (which appears to
be of a logical nature) has entirely nonlogical consequences  that there are not only finitely many primitive relations.
Indeed if the axiom of reducibility is
true there cannot be any logically definable enumeration of the primitive relations of the ultimate universe of discourse.
This last conclusion was totally unaccept
able to Chwistek, and that is.why he claims that the axiom of reducibility leads to contradiction.
I quote from (1921).
If we accept the Axiom of Reducibility we have to give up once and for all the logic of constructive propositions and functions, that is of such functions as may be constructed by means of letters or symbols given beforehand by enumeration, and we have to resort to functions which cannot be thus constructed. This seems to be inconsistent with the basic conception of logic which apparently should be independent of metaphysical assumptions such as the existence of nonconstructive functions. In his later work Chwistek went even further; the expressions (for constructive functions and so on) themselves became mathematical objects, rather than being treated metamathematically.
From all this it will be seen that the simple
176
R.O. GANDY
theory of types was eminently not Chwistek's' cup of tea.
But he did point out
that it was an alternative way of escaping from the unacceptable consequences of the axiom of reducibility.
In both (1921) and (1922) he remarks that if one is
ready to ignore the semantic notions involved in Richard's paradox, then one can jettison the hierarchy of orders and classify propositional functions simply and solely by the type of their arguments.
In (1921) he writes
It should be noted that the elimination of these elements from the theory of types of Whitehead and Russell would render this theory exceedingly simple and manifest. But,
apa~t
from asking (in (1922)) whether the resulting theory would be
consistent, he pursued the matter no further. Now I turn to Ramsey (1926).
The standard thing to say about Ramsey's
paper is that it distinguished between the settheoretic and the semantic paradoxes, and that it pointed out that the simple theory of types suffices to avoid the known instances of the former.
This is, of course, true.
But, like so many
of the received ideas trotted out by philosophers and logicians when they write about the foundations of mathematics, it misses the main point, the point which as the quotation below shows  Ramsey himself put first.
Ramsey's head was sometimes
in the clouds, but his feet remained firmly on the ground of mathematical experience.
In this paper the clouds include an extreme platonism ('our inability to
write propositions of infinite length which is logically a mere accident'), but through them he has a clear view of that creation of the mathematical imagination which distinguishes Cantorian set theory from the logic of classes: the concept of an arbitrary subset existing only in extension.
I quote.
The possibility of indefinable classes and relations in extension is an essential part of the extensional attitude of modern mathematics ... that it is neglected in Principia Mathematica is the first of the three great defects in that work. The mistake is made not by having a primitive proposition asserting that all classes are definable, but by giving a definition of class which applies only to definable classes, so that all mathematical propositions about some or all classes are misinterpreted. This misinterpretation is not merely objectionable on its own account in a general way, but is especially pernicious in connection with the multiplicative axiom which is a tautology when property interpreted, but when misinterpreted after the fashion of Principia becomes a significant empirical proposition. The contrast between Ramsey's approach and Chwistek's could hardly be greater.
But the whole weight of mathematical experience from the eighteenth
century onwards is surely on Ramsey's side.
Euler's view that one should allow
arbitrary continuous initial configurations for vibrating strings  as opposed to D'Alembert's insistence that they must be analytic: Fourier's work on discontinuous functions: the pathological counterexamples given by Weierstrass: Hilbert's work on Dirichlet's problem: Schwarz's theory of distributions: in each of these developments the important thing is not a mere increase in generality but the opening, by the enlargement of our concept of a function, of gates to new paths
THE SIMPLE THEORY OF TYPES of mathematical thought.
177
The most significant objection to the imposition of
criteria of constructibility on the foundations of mathematics as exemplified, for example, in the writings of Bishop and MartinLof is that it prevents us, or at very least strongly discourages us, from using our mathematical imagination to create or discover new, beautiful and applicable mathematics. (Cantor wanted pure mathematics to be called free mathematics.)
These flights of mathematical
imagination are not so high in the air that they have nothing to contribute to the concrete mathematics of computation. had successfully imposed his views.
Suppose, for example, that D'Alembert
Then there would have been no stimulus to
the development of the theory of Fourier series and integrals, and no need to look beyond powerseries for the solution of differential equations.
I am not
arguing that constructive ideas should not be taken seriously  for example, Godel discovered the constructible sets by taking the notion of predicativity seriously.
What I am saying is that mathematicians do well to disregard stern
moral injunctions dogmatically issued.
Better be a thief than an honest toiler
in a labour camp. To return to the simple theory of types.
Ramsey's contribution should be
seen then as a liberating, not a restricting, step in the development of a foundation for classical mathematics. Neither Chwistek nor Ramsey gave a detailed formulation of the theory, though evidently they could have done so if they had wished.
The first detailed
formulation is by Carnap (1929) who cites Ramsey but not Chwistek.
Meticulous
formulations are given in Tarski (1931) (who cites Chwistek and Carnap but not Ramsey) and in Godel (1931) (who cites only Principia Mathematica). A rather eccentric system, based on the simple theory of types, was published by Quine (1934); this certainly influenced Church's 1940 formulation.
[Indeed
Church gives Quine's paper a star (for importance) in his bibliography  surely one of the master's very few lapses from impeccable judgement.] The formulation in Church (1940) differs from the previous formulations in a number of important ways: (1) there is a type of propositions; (2) for any two types there is the type of functions from one to the other; (3) terms for functions are formed by lambdaabstraction.
The last feature allows one to dis
pense with the comprehension axioms and in conjunction with the rules of lambda conversion simplifies the procedure for substituting complex expressions for predicate letters. [A procedure which was ignored in Principia and incorrectly described in the first two additions of HilbertAckermann and in many other text books  including some published as late as the 1950's.]
A reformulation of
Church's system in Henkin (1963) has the engaging feature that the only primitive constants are the relations of identity for each type.
R.O. GANDY
178
TURING'S WORK ON TYPE THEORY First
give a brief account of the available material, published and un
published. [1]
Letters to M.H.A. Newman. There are 5 of these, handwritten, covering 31 sides; 4 were written in 1940, one in 1942.
[2]
(with M.H.A. Newman) 'A formal theory in Church's theory of types', J.Symb.Logic Vol.7 (1942) pp.2833, (received by the editors, 9 May 1941).
[3]
'Some theorems about Church's system', 20 quarto pages of typescript (1940?) .
[4]
Pages 3743, 6073 of an untitled quarto typescript (perhaps a continuation of an earlier draft of [3]).
[5]
'The use of dots as brackets in Church's system', J. Symb.Logic Vol.7 (1942) pp.146156, (received by the editors 17 June 1942).
[6]
'A practical form of draft of [10].
[7]
'A practical form of type theory II', 81 typescript quarto pages with some extra pages of handwritten insertions.
[8]
'Outline of proof of metatheorem p.38 of "Practical forms" " 20 pages manuscript, (the metatheorems referred to are labelled (1) and (2) on the last page of [10)).
[9]
'The reform of Mathematical Notation and Phraseology', 14 pages quarto typescript; there appear to be 4 pages missing in the middle.
[10]
'Practical forms of type theory', J.Symb.Logic Vol.13 (1948) pp.8094, (received by the editors Jan.6 1947).
t)~e
theory 1', 30 typescript quarto pages; a first
[Turing bequeathed the manuscripts to me.
They are deposited in the library of
King's College, Cambridge.] Turing had learnt about Church's type theory when he was at Princeton (19361938).
Almost all the work covered by the above material must have been
done between 1940 and 1946.
(As has recent ly been revealed, in 1940 he was en
gaged in breaking the code of the wartime Enigma encoding machine; later on (19441945) he had a less taxing job  building a speech encoder of his own invention.) Neither mathematically nor philosophically is there anything very surprising or exciting in the manuscripts.
[A fairly complete account will appear in
my edition of his collected work which is now, as ever, in active preparation.] In view of the current interest in normalization theorems it is perhaps worth recording that the first page of [3] contains a proof of normalisation for the typed lambda calculus.
[3) and [4] contain various simple reI ati ve consistency
proofs for additions to and modifications of Church's system.
(Also a purported
relative consistency proof of the axiom of infinity in the absence of extensionality: the howler in it is almost of the schoolboy variety.)
[7] gives a proof
THE SIMPLE THEORY OF TYPES
179
of the completeness of the cumulative type system of [10] when the axiom of infinity is replaced by axioms of finitude, and a proof of the equivalence asserted in section 3 of [10].
The topics discussed in [9] are (1) the distinction be
tween free and bound variables, (2) the use of hypotheses and the deduction theorem and the role played by variables which are restricted by hypothesis, (3) the application of the theory of types in ordinary mathematical practice. As might be expected from his other mathematical work, Turing's attitude to the theory of types is commonsensical and downtoearth; closer to the view attributed by Church to Schroder than to the philosophical positions of Russell Chwistek and Ramsey.
Some quotations will give the flavour.
From [6]:
The author wishes to repudiate any implication that may be suggested by this paper to the effect that he believes Russell's philosophy of mathematics to be the truest. He does believe however that it is the one which is most easily understood, and also that it describes most closely the accepted forms of mathematical thinking. From [9]: Actually it is not difficult to put the theory of types into a form in which it can be used by the mathematicianinthestreet without having to study symbolic logic, much less use it. The statement of the type principle given below was suggested by lectures of Wittgenstein, but its shortcomings should not be laid at his door. The type principle is effectively taken care of in ordinary language by the fact that there are nouns as well as adjectives. We can make the statement 'All horses are fourlegged', which can be verified by examination of every horse, at any rate if there are only a finite number of them. If however we try to use words like 'thing' or 'thing whatever' trouble begins. Suppose we understand 'thing' to include everything whatever, books, cats, men, women, thoughts, functions of men with cats as values, numbers, matrices, classes of classes, procedures, propositions, .•. Under these circumstances what can we make of the statement 'All things are not prime multiples of 6'. What do we mean by it? Under no circumstances is the number of things to be examined finite. It may be that some meaning can be given to statements of this kind, but for the present we do not know of any. In effect then t~e theory of types requires us to refrain from the use of such nouns as 'thing' 'object' etc., which are intended to convey the idea 'anything whatever' . [Notes of the lectures by Wittgenstein referred to have recently been published: 'Wittgenstein's lectures on the foundations of IDathematics', ed. Cora Diamond, Harvester Press 1976.] He goes on to propose the term 'noun class' for a permitted range of a variable and mentions operations (e.g., the powerset operation) which can be used to form one noun class from another.
In [10] a more formal definition of noun
class is given: any definable subset of some level in the cumulative hierarchy of types.
I cannot resist one further quotation (from [6]).
There is a close connection between the role that these formulae play and the use of nouns in ordinary language, so much so that one tends to feel that Russell's type theory was largely anticipated by prehistoric man. But this is going too far; everyday conventions for avoiding the use of un
R.O. GANDY
180
restricted variables are not sufficient if one wishes, for example, to handle transfinite ordinals and cardinals in a troublefree way. And this brings me to a last question: why did Turing prefer type theory to set theory as the ground on which the relations between mathematical practice and symbolic logic should be dIscussed?
1 think I can give a series of partial answers, which do add up
to an adequate answer. 1.
Turing, I am sure, was not familiar with Zermelo's cumulative type structure.
So set theory will have seemed to him (as to many others) merely a formal system, quite lacking the sort of quasiconcrete interpretation which he needed if he were to handle a subject successfully. 2.
In the same spirit he did not much care for 'general abstract nonsense'.
He liked the kind of mathematics which is concerned with specific, more or less concrete problems.
But this kind of mathematics, certainly then, perhaps still
today, can be formulated within the finite theory of types.
[In his paper on
ordinal logics he showed that the Riemann hypothesis can be formulated as a twoquantifier sentence of number theory.] 3.
One might be surprised that he seems never to have shown much interest in
the transfinite alefs. [This same lack of interest is manifest in Frege and Russell.]
One obvious possible reason is that he had genuine doubts about their
consistency.
But, at least equally important, is the fact that in the 1930's very
few interesting things were known about them, and there had not been, I think, any significant applications of them to other branches of mathematics. BIBLIOGRAPHY A. Sources F.W.K.E. Schroder (1890). Vorlesungen liber die Algebra der Logik, Vol.I, Leipzig 1890. B. Russell (1903). The principles of mathematics, Cambridge 1903. L. Chwistek (1912). Zasada sprzecznosci w swietle nowszych badan Bertranda Russella, Rozprawy Akademii Umiejetnosci (Krakow), WydziaJ historycznofilozoficzny, 2s. vol 30 pp.270334. (1921). Antynomje logikiformalnej, Przegl.fil.vol 24 pp.164l7l: translated in Polish Logic 19201939 (Ed. Storrs McCall), Oxford 1967, pp.338345. (1922). Uber die Antinomien der Prinzipen der Mathematik, Math. Zeitschrift vol 14 pp.236243. F.P. Ramsey (1926). The foundations of mathematics, Proc.Lond.Math.Soc.2 ser vol 25 pp.338384. R. Carnap (1929). Abriss der Logistik, Vienna 1929.
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A. Tarski (1931). Sur les ensembles definissable de nombres reels I, Fund.Math. vol 17 pp.210239. K. Godel (1931). Uber formal untentscheidbare Satze der Principia Mathematica und verwandter Systeme I, Monatsh.Math.Phys. vol 38 pp.349360. W.V. Quine (1934). A system of logistic, Cambridge, Mass. 1934. A. Church (1940). A formulation of the simple theory of types, J. Symb.Logic vol 5 pp.5668. L. Henkin (1963). A theory of propositional types, Fund.Math. vol 52 pp.323344. B. Other references A. Church (1936). A bibliography of symbolic logic, J. Symb.Logic. vol 1 pp.121218. (1937). Review of L. Chwistek's 'Uberwindung der Begriffrealismus' , J. Symb. Logic vol 2 pp.168170. (1939), Schroder's anticipation of the simple theory of types, Preprinted for 5th Int. Congress for the Unity of Science, Cambridge, Mass. as from Jour.of Unified Science (Erkenntnis) vol 9, 4 pages.
R. Gandy, M. Hyland (Eds,), LOGIC COLLOQUIUM 76 © NorthHolland Publishing' Company (1977)
SETTHEORETIC SEMANTICS Jean van Heijenoort Brandeis University
Two fundamental acquisitions of modern logic are the notion of formal system and that of interpretation. These two notions somehow complement each other: when we come to construct a formal language. we abandon the meanings of the symbols in order to deal with their syntactic relations; subsequently. we bring back the interpretation. The birth date of the first notion caa be given with complete precision; it is when Frege wrote. in his Begriffsschrift. that the rules he was introducing were rules 'for the use of our signs'. The history of the notion of· interpretation is more hazy. The notion of settheoretic consequence appears in Bolzano's Wissenschaftslehre (1837). but not for a formal language. Bolzano has no formal notationhe uses ordinary languagebut he takes a step toward objectivity by considering what he calls Satze and Satzformen instead of Urteile and Urteilsformen. these being too psychological. A proposition C follows (ist ableitbar) from the propositions ~l' ~ •••.• ~. with respe~t to certa~common constituents. if and only if every substitution.eaken in a certain class. for these constituents makes £ true whenever it makes ~l' ~. • ..• ~ true. Bolzano's discovery remained isolated. When. after 1870. the development of modern logic began to take some impetus. there appeared two streams that. until 1920, hardly seem to mix their waters. One stream is that of Frege and Russell (to whom we should perhaps adjoin Peano). and the other is that of Peirce. Schroder and Lowenheim. We could say. as an initial approximation. that the first stream is syntactic. while the second is semantic. This is not quite correct. since Frege and Russell have their semantics. these semantics being different from each other and from settheoretic semantics. Peirce, Schroder and Lowenheim consider domains of individuals. domains on which properties and relations are defined. and most of the time ignore formal proofs; their results~ which today we would characterize as modeltheoretic results. 'are obtained by semantic considerations. Schroder has axioms for Boolean algebras. but this does not go beyond the sentential calculus. Most striking in this respect is Lowenheim'ssystem in his famous paper (1915). There Lowenheim deals with firstorder logic with identity. But his system is not provided with a notion of proof; there are no axioms and rules. The approach is purely semantic. The notion that Lowenheim handles is that of interpretation, together with the companion notions of validity and satisfiability. We have formulas that are valid, valid in every finite domain. valid in a denumerably infinite domain. Lowanheim does not give an explicit definition of the semantic notions that he uses;' he takes them as wellunderstood; but the novelty and complexity of the results that he is able to obtain leave no doubt that he has a complete grasp of them. Besides his famous theorem. Lowenheim obtains. by these semantic method$. results about reduction and decision problems, problems in which Frege and Russell 183
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had shown no interest. To carry out this work Lowenheim introduces a number of technical innovations that will become standard tools of logic, especially in semantic arguments: reduction of a formula to prenex form, a prefiguration of the Sko1em functions (to obtain the satisfiabi1ity functional form of a formula), a prefiguration of Herbrand expansions (of course, Herbrand found his inspiration in Lowenheim's work), use of an argument that is, in fact, Konig's lemma. Lowenheim had learned his logic in the writings of Schroder, who in turn had codified and developed some of the work of Peirce (which branches off in several directions). As early as 1870, Peirce begins to represent binary relations by matrices, finite or infinite, whose elements are ordered pairs of individuals. In 1882 he takes a further step. A relative is represented as a Boolean sum of ordered pairs (taken in a certain domain), each pair being affected by a coefficient, which is 1 or a according as the relation obtains between the elements of the pair or not. The mathematical signs;S and 17 thus come to play the role of the existential and universal quantifiers. But these quantifiers are handled by Peirce in a way that is very different from the way Frege handles his quantifiers. There are no axioms about them. Their interpretation and handling are tied up with a specific domain, changeable at will. Moreover, they are generally used only in initial position. Another feature of the 'syntactic' stream in logic is that Frege and Russell are engaged in a total logical reconstruction of our world. They are interested in starting from a few primitive notions and building a universal system. Therefore their quantifiers range over all objects, that is, the range of these quantifiers is a fixed unique universe, the Universe. This is expressed by Frege in Begriffs~: 'All other conditions to be imposed on what may be put in place of a German letter are to be incorporated in the judgment.' (Frege l§Zi, § 11; ~ Heijenoort 1967, 24; the sentence is italicized in the original.) This means that, when we deal with a special class of objects, we use the method known as relativization of quantifiers. 'For every natural number ..• ' becomes translated as )(x(Nx :> ••• ), where lfx ranges over all objects. This is what Frege means by incorporating the conditionin the judgment. In Russell we have a stratified universe, but each stratum is a fixed domain. This universality of the FregeRussell logic is discussed in ~ Heijenoort ~ . With a fixed universe the semantic notions of validity and satisfiability lose their applicability, while with a variable universe they come to the fore. As early as 1885, Peirce gives the correct definition of validity for the sentential calculus: 'To find whether a formula is necessarily true substitute i and ~ for the letters and see whether it can be supposed false by any such assignment of values.' (Moreover, for testing validity Peirce suggests a method that foreshadows refutation trees.) If Frege and Russell corne to consider the formula, say, ~~(~ :> ~, it is only as a temporary equivalent of "f\f~(~ => Fx), and for this latter formula the notions of validity and satisfiability come to coincide and collapse into the notion of truth. Closely connected with these considerations is, of course, the question of firstorder logic as a separate system. Embarked in their grandiose logical reconstruction, Frege and Russell do not hesitate to go beyond firstorder logic. And they have to, since the notion of the ancestral is necessary for their definition of natural number. When Frege passes from firstorder logic to a higherorder logic (!§Zi, § 11; ~ Heijenoort l22L, 24), there is hardly a ripple. Not bound by largescale requirements, Peirce, Schroder and Lawenheim can more distinctly feel the ground under their feet, and then, of course, the difference in complexity between firstorder and higherorder logic is immediately apparent. The birth and development of model theory would hardly have been conceiva~l. without the separate consideration of firstorder logic.
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Hilbert's position is somewhat between that of FregeRussell and that of PeirceSchroderLowenheim. Like the former, he works with axioms and rules. With his mathematician's instinct, however, he is inclined to consider quantifiers ranging not over 'everything', but rather over welldefined collections of objects. He also feels that the jump frOm firstorder to secondorder logic marks an important change in complexity; and, though secondorder logic may be indispensable at a certain stage, it is important to see what can be done in firstorder logic. So, when the fusion of the two currents took place in the Twenties, Hilbert was one of its agents. The notion of a formal deductive system supplemented by an interpretation appears in Hilbert's logical writings, and in Hilbert ~ Ackermann 1928 the problem of the completeness of firstorder logic is posed in exactly the ~s in which it will be solved in Gadel 1930. Another agent was Skolem. He knew, of course, LOwenheim's work, which he had amended and generalized. But he had also read Principia. He had at hand the various arguments that constitute the proof of completeness, but the very import of the proof seems to have remained hazy for him. Rather than using axioms and rules, he kept looking for a proof procedure, perhaps similar to the one that Herbrand was going to propose. Skolem's work raises many delicate questions of interpretation, which I do not intend to discuss here. I will simply add that his work did not have, at the time, an impact comparable to that of Hilbert's; Skolem's papers, in the Twenties, were hardly read by the logicians that were then shaping logic. In 1930, when the proof of the completeness of firstorder logic with respect to settheoretic semantics was presented, there were two other seaantics, that of Brouwer and that of Herbrand. Brouwer was certainly concerned with meaning. In fact, a great part of his polemics against Hilbert turned around the notion of Inhalt. But neither he nor Heyting thought of systematizing the notion. I have ~ention of discussing intuitionistic semantics here. Herbrand, with his (feigned 7) acceptance of the Hilbert program, rejected settheoretic considerations that were not decidable. Instead of the settheoretic notion of satisfaction, he uses a notion based on an infinitely proceeding sequence of finite domains, existential quantifiers having a range different from that of the universal quantifiers. No doubt, Herbrand had the notion of settheoretic interpretation present in the back of his mind all the time. Only thus can his errors be explained. The semantic argument for Herbrand's theorem is incomparably simpler than the syntaqtic argument, so that, being convinced by the first, Herbrand did not bother to check carefully the second (in the same way that, when we have a simple geometric argument, we are prone to make a mistake in the complicated calculations of the parallel analytic argument, simply out of boredom). Compared to the glamorous axiom systems., which, with so few axioms and rules, claimed to encompass a large tract, if not all, of mathematical practice, semantic considerations seemed to be a homely breed. Gentzen, in 1935, still prefers to establish the constructive equivalence of his calculus of sequents with some existing system, rather than to prove directly the completeness of the new system (he may have thought that merely giving such a proof was a bit like cheating). It is Tarski's codification of settheoretic semantics, in the midThirties, that finally gave an honorable status to semantic considerations and launched them toward model theory. Model theory has bloomed into an extensive science. But this new discipline does not seem to care much about what was the original object of semantics, namely meaning. It has become an abstract mathematical science, whose philosophical implications are scant. More precisely, model theory does not seem to have anything to contribute to a theory of meaning, if such a theory has to deal with
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ordinary language. The only philosophical implications of model theory are at a highly technical level, where results in set theory may affect our views of whse is mathematics. Recently, attempts have multiplied to construct a semantics of ordinary language on a precise basis. In the last hundred years the views of the philosophical communi~ about natural language seem to have gone through a threestage dialectical process. Frege and Russell wanted to replace natural language with a formal language, because of the imprecision, vagueness, haphazard complexity of the former. The natural language is not considered to be a worthy tool of philosophical investigation, not even a worthy object of study. We may get suggestions from It, but it is no more than a ladder that we kick back, once we have been able to reach the formal language. As a reaction against this point of view, Wittgenstein, in his second period, wanted to leave natural language as it is, not replace it by anything else. The niceties and convolutions of natural language express subtle differences in thought, which we must examine with care and patience. Today, we have entered a third phase. Philosophers have been reminded by linguists that natural language, in its syntactic aspects, can be an object of exact study, a study that, with its generalizations, is reminiscent of that of artificial languages. Then, more recently, this exact study attempts to encompass also the semantic aspects of natural language. I do not intend to survey here the whole field, but to touch upon a number of points. In Tarski's wellknown paper on settheoretic semantics, the only philosophical remarks deal with the Aristotelian realism of his definition of truth. But the semantics rests on specific ontological assumptions that have hardly been made explicit: we have bare individuals; these individuals have no inner structure, they are mere pegs. That is why we often speak of 'elements', even sometimes of 'points'. We assume that we have a collection of such indiViduals, distinguishable by its cardinality. Properties and relations corne subsequently and do not contribute to the identification of the individuals. On the contrary, the indiViduals have to be individuated so that properties and relations can be introduced. How are the individuals individuated 1 This is left out of the account. What we have here is a highly abstract scheme. Can this dessicated ontology encompass everything we want to talk about 1 Is it applicable to material objects, to events, to acts 1 Just to question the universal applicability of settheoretic semantics I would like to raise here the question of mass terms. These terms, like ~ , gold, ~, are opposed in their syntax to the socalled count terms, like~, tables,~. With them we use the question How much 1, instead of How many 1 They obey peculiar syntactic rules. It is to be noted that in different languages, English and French for instance, the classes of mass terms are quite similar (that is, correspond under translation), while the syntactic rules that allow us to characterize these classes are quite different (that is, do not correspond under translation). This fact seems to give weight to the assumption that the class of mass terms is a stable and significant linguistic object. Mass terms can, in a sentence, be used before the copula (Iron is a metal) or after (This is gold). They can be u8ed with demonstratives (This water is dirty), possessives (My coffee is cold) or the defiaite article (The wine that you gave me was sour), to yield what I would call quasiindividual terms. They can also be used with socalled container words (three buckets of water, two pounds of sugar) in a variety of ways: sometimes the units are individualized, sometimes not. There is here a rich field for linguistic investigations. Concrete mass terms, like those we have mentioned so far, are important because of the linguistic problems they raise. But their importance is increased by the fact that abstract terms, like courage, wisdom, and so on, behave, linguistically, like them.
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Brave attempts have been made to deal with mass terms by means of settheoretic semantics. Some have taken them as denoting (scattered) individuals (Quine), some as referring to properties (Strawson). These attempts can be pushed up to a certain point, then end up by doing violence to the linguistic facts. The most natural way to deal with mass terms seems to be to provide them with their own semantics. As we have a universal domain of individuals for count terms, we now consider a universal lump for mass terms, a lump out of which we take slices. The two semantics may have to be used concurrently. First, because a sentence may contain both count terms and mass terms, as The man drank some water. Second, because the quasiindividual terms that I just mentioned, obtained from mass terms by a variety of linguistic means, require individuals for their interpretation. There is also, with mass terms, the problem of mixtures, both on the syntactic and on the semantic plane. Since quantifiers cannot be interpreted except in a discrete domain, the proper treatment of mass terms requires a variablefree system. But several such systems are at hand today, and there does not seem to be any difficulty in adapting one of them to the semantics of the universal lump. One can only wonder why all that has not been done before. It is ironic that Tarski's famous example, 'Snow is white', involves a mass term, and Tarski's semantics would have met with difficulties if sentences containing other uses of 'snow' had been considered. My purpose in raising here the problem of the semantics of mass terms has been to try to perceiW& the limits, and l~itations. of set~heoretic semantics. An ambitious attempt to apply to natural language the methods and procedures acquired in the study of formal systems was made by Richard Montague. His system went through several phases, and it is not my intention to discuss it here in its entirety. I would like to discuss a point on which it differs markedly from standard settheoretic semantics, namely, on the notion of individual, and I take Montague's doctrine as presented by Barbara Partee after his death ( ~ 1975). For Montague an individual is the collection of all properties that are true of the individual, in our sense. The semantic interpretation of John is the collection of all properties that hold of John. This conception ~ertainly different from our settheoretic semantics and offers an alternative semantics, which, moreover, has a long philosophical tradition (Leibniz). With this semantics, Montague is able to assign a semantic denotation not only to names (individual constants and definite descriptions), but also to expressions like every ~ or ~~. Or rather, these expressions become, in a way, names, on the same plane as~. The denotation of every ~ is the collection of all properties that hold of every man, that is, it is the intersection of the collections assigned to John, ~, and so on. The denotation of ~ ~ is the collection of properties that hold of at least one man, that is, the union of the collections assigned to John, Paul, and so on. Barbara Partee gives a list of contexts in which the substitution of expressions like every ~ or ~ ~ for ~ leads from a meaningful se,uence of English words to another. There are, however, limits to this parallelism, and these limits are so fundamental, linguistically, that the unification becomes doubtful. I now list three such limits. (1) Negation. The negation of John ~ is ~ did ~ ~ . The negation oE Every !!l.!!.!l ~ is not Ever:r: ~ did ~ ~ , but Not every ~ ~ . And so on. Examples could be multiplied. Negation is such an important linguistic and logical notion that these differences cannot be considered to be innocuous and chance peculiarities of the language. They are rooted in its functioning. (2) Scope. In John ~ Mary there is no problem of scope. But, with a and every, the problem of scope arises, as in ~ ~ loves every~. Here again
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examples could be multiplied. And here again we are confronting something basic in the functioning of the language. The problems of scope of quantifiers in English are of a degree of complexity that is no less than in quantification theory, as one can convince oneself by a few examples. Hence we cannot hope to get by through the use of a semantics that does not respect this complexity. (3) Inference. From::!2!!!l gave exactly 2!!~ ~ !:!l!!Y. ~ ~ to anybody else, we can infer ~ gave exactly ~ book. But we cannot infer this last ;er;tence from John gave exactly ~ ~ ~ every ~ ~ ~ to anybo~~. The substitutions mentioned above disturb inferences. And here again the same remarks can be made as for (1) and (2). This list is certainly not exhaustive. Other limits, on the linguistic plane, can be found to the unification proposed by Montague. On the philosophical plane, a number of objections present themselves. Without entering here into a full discussion of the problem, let me mention that Montague's conception, at least as presented so far, would make all statements analytic. If the denotation of ::!2!!!l is a collection of properties, then John came simply states that one specific property is in the collection in whi~twa; put. A great deal of the present activity in the field of the semantics of natural languages is devoted to the study of the relations obtaining among terms, like between fox and vixen, in the hope that, if we could codify these relations, we would be;ble to read off the logical entailments that obtain among sentences of a natural language like English. The entailment of (1)
by (2)
should emerge from the proper semantic analysis of mortal, to live and forever. And we should be able, progressively, to deal with ~nd;o~omplexexamples. What is attempted, or rather contemplated, is the construction of a dictionnaire raisonne. But this an old enterprise. Descartes was already asking for an enumeration of all the idees simples and their hierarchization. Such an enterprise worl~up to a certain point. There are in English perhaps a few thousand words that lend themselves pretty nicely to such a work of regimentation. But soon doubts and difficulties multiply. Try to unravel a skein of yarn; you will pull a good length of it, but then you will have a worse mess than before. One begins to grope for guidance and would like to have rules and principles. But these are notoriously lacking. The very idea of idee simple is not simple. Where is the line to be drawn between a dictionary and an encyclopaedia ? I do not want to claim that such work cannot be done. But, before it can be done (perhaps with the help of eomputers), certain fundamental questions have to be answered, questions that have not yet been formulated properly. A pervading assumption, implicit or explicit, underlying the construction of a rational dictionary is that we have to proceed from the simple to the complex. And how could it be otherwise? But the realities of a natural language force us to temper such a view. We do not simply proceed by addition, composition, conjunction. Consider the sentences ~ squeezed ~ ~ and He squeezed ~ hand. Don't we have here a kind of feedback effect, through which the meaning of the complement alters the meaning of the verb? Consider red in ~ ~ , red ~ and red hair. The meaning of the noun affects the meaning of the adjective. We do not simply have conjunction. When words are put together, they interact on one another, and there are effects that the inductive mode of definition, used for artificial languages, is not quite able to catch. Should we then consider words only in expressions? Should a dictionary be made up of examples, rather than of
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attempted definitions ? Consider now an example that has been much discussed by linguists. French translation of
The correct
(3 )
is
.ll traversa la riviere
(4 )
~ la nage ,
The word byword translation of (3), which is (5 )
a
has no clear meaning in French. We translate to swim across by traverser la nage. We have here something reminiscent of idiom;: ~e translate I im hungry by J'ai faim, we do not use a wordbyword transformation either, and linguists are prone to keep idioms out of their theoretical reconstructions (see, for instance, ~ ~ , page 50). However, he who has some practice in translating knows that the idiomatic aspect that reveals itself when we pass from (3) to (4) is no exception. One would be tempted to say that it is rather the rule. Once a translator has paired off a word of the target language with each word of the original text (perhaps with the help of a dictionary), he rereads the whole thing, catches the meaning, and writes down the translation. There is no upper bound upon the length of the rearrangement. We cannot isolate a definite class of idioms. The idiomatic effect pervades the whole language. But, if we do not have a rational dictionary, if we are not even sure of how such a dictionary can be written, does it mean that we cannot deal with entailments in a natural language, like that of (2) by (l)? Of course, we can. Logic, even today, allows us to deal with the semantics of a natural language locally. That is, we translate the sentences into the language of quantification theory, and then use standard settheoretic semantics. This is what we do when, in a beginners' course in logic, we give examples. It is a rough job. Many things get bulldozed. Given a sentence that we consider to be logically atomic, we pick one, two or three parts in it, which, we decide, work as individual terms, and we push everything else into the predicate. Adverbs suffer, and many other things. But, nevertheless, within the limits of what we intend to do, it works. If somebody claims that there is a relation of entailment among certain English sentences, we can check, by translating these sentences into quantification theory, whether the entailment obtains or not (check, that is, within the limitations imposed by the decision problem; but it is rare that an example suggested by natural language is not in a decidable fragment). If one and the same sentence occurs in two different sets of sentences, we may have to translate it differently if we want to reveal the entailment corresponding to each set, but in each case the operation is possible. There is nothing strange and mysterious about checking that (2) entails (1). We simply adjoin the side assumption (6)
For every
~, ~
is mortal if and only if
~
does not live forever.
If our dictionnaire raisonne had been completed, we would read (6) off the dictionary. As long as the dictionary is not at hand, we adjoin one or several side assumptions. If we understand the language, this presents no difficulty.
we could compare the natural language to a nonEuclidean space, for which there is, at each point, a tangent Euclidean space. In a neighborhood, the tangent space can take the place of the given space. If we go too far, the distortion becomes too great.
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The problem is to pass from the local treatment of the semantics of a natural language to a global one. One feels that it is possible and that it should be done. Why? Because the handling of words is an exact and precise activity. Whoever writes with some care knows that replacing one word by another, apparently close, may change the meaning of a sentence appreciably; even a comma, added or taken away, as its effect. The syntactic c~dification of such facts cannot be the full answer. Unless Godgiven, syntactic rules must be grounded somewhere. And where, if not in meaning? But between the exact working of a natural language, with the convolutions and intricacies of its meanings, and the logical skeleton representable in quantification theory there is a large gap, in which we are still groping.
References Ackermann, Wilhelm See Hilbert, David, and Wilhelm Ackermann. Bo1zano, Bernhard 1837 Wissenschafts1ehre. Fodor, J. A. 1972 Troubles about actions, Semantics of ~ language, edited by Donald Davidson and Gilbert Harman, 2nd edition, 4869. Frege, Gott1ob 1879 Begriffsschrift, ~ ~ arithmetischen nachgebildete Formelsprache des reinen Denkens; English translation in ~ Heijenoort 1967, 182. GiSde1, Kurt 1930 Die Vo11standigkeit der Axiome des logischen Funktionenka1ku1s,  Monatshefte fUr Mathematik ~ Physik 1L, 349360; English translation in ~ Heijenoort 1967, 582591. Hilbert, David, and Wilhelm Ackermann ~ Grundzuge der theoretischen Logik. Lowenheim, Leopold tiber MOg1ichkeiten im Re1ativkalkUl, Mathematische Annalen 76, 447470; 1915  English translation in ~ Heilenoort ~, 228251.Partee. Barbara 1975 Montague grammar and transformational grammar, Ligguistic inguiry £,   203300. van Heijenoort, Jean 1967 ~ Frege !:2. Gildel, A source ~ 1n mathematical .!2s.!.s.. 18791931. 1967a Logic as calculus and logic as language, ~ studies in ~ philosophy  of ~~, 440446.
R. Gandy, M. Hyland (Eds.l, LOGIC COLLOQUIUM 76
© NorthHolland Publishing Company (1977)
ON THE WORK AND INFLUENCE OF STANISLAW LEsNIEWSKI
Stanislaw J. Surma Jagiellonian University Krak6w, Poland
BIOGRAPHY OF STANISLAW LEsNIEWSKI /18861939/ The detailed biography of Stanislaw Lesniewski remains rather obscure. The Polish archives from which it could have been reconstructed were destroyed during the World War II and at least some information is known merely from hearsay sources. To some extend we must depend on trusting a few still alived witnesses. One of the sources is KOTARBInSKI /1965/. Numerous comments about Lesniewski and his work are contained in LUKASIEWICZ /1929/, KOTARBInSKI /1929/ and /1957/, TARSKI /1956/, CHURCH /1956/, SOBOCInSKI /1950/ and /1956/ and MOSTOWSKI /1948/. Compare also McCALL /1967/. A large number of details concerning historical and especially of philosophical aspects of Lesniewski's work can be found in LUSCHEI /1962/. Stanislaw Lesniewski was born on 28 March 1886 in Serpukhovo near the town of IvanovoVosniesiensk somewhere beyond Moscow, Russia, as the son of a locally employed Polish railway engineer. He attended grammar school/classical division/ somewhere in Siberia, probably at Irkutsk where his father was building a railway. As a schoolboy he was shown to be adherent to any principles and intolerant of exceptions. His student years he spent partly at least in Muenchen, Germany, where he was listening to Hans Cornelius and was trying to get to the bottom of his peculiar melange of positivism and Kantianism with its guiding conception.apparently called "das immanente Ding an sich". Then Lesniewski came to Lvov in order to complete his doctoral dissertation under Professor K.Twardowski, the leading philosopher of the University of Lvov at the time. He accomplished his Ph.D. in philosophy in 1912. While in Lvov Lesniewski was also studying mathematics under Professor J.Puzyna and partly also under W.Sierpinski appointed as a professor in 1910. For a moment being influenced by Professor L.Petrazycki's works Lesniewski conceived his future as lying in psychology but when he suddenly appeared in Lvov as a newcomer among K.Twardowski/s doctoral students the book he was under a strong impact was Marty's "Untersuchungen zur Grundlegung der allgemeinen Grammatik und Sprachphilosophie". Another work turning Lesniewski out of psychology and psychologism were E.Husserl's "Logische Untersuchungen"
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At the time Lesniewski was convinced that one had first to understand the meaning of words and only then one could philosophize at all responsibly. During the last years preceding the World War I Lesniewski threw himself into the whirpool of philosophy. He was endeavouring to overturn the principle of the excluded middle, employing the tools of analysis and criticism against the language in which philosophers argue about the foundations of logic, theory of cognition etc. Later, however, he abandoned it. There was a breakthrough in his attitude. What diverted the direction of his thinking was most probably the reading in 1911 a logical appendix to LUKASIEWIC~ /1910/. This appendix contained Lukasiewicz's explication of a general theory of objects, a branch of logistic as it was then often called, expounded by means of a symbolic method somewhat similar to that of COUTURAT /1905/. That was the end of Lesniewski the commentator and probable translator of Marty and the beginning of Lesniewski the enthusiastic adherent of modern, mathematizing formal logic. What Lesniewski learned from Lukasiewicz was the famous B.Russell's antinomy which fascinated and preoccupied him for the next eleven years. Although the Lukasiewicz's work was a turning point in Lesniewski's career from philosophy to mathematical logic and the foundations of mathematics, for a few subsequent years he remained still particularly reluctant to the use of the logicomathematical apparatus. EARLY WRITINGS Between 1911 and 1914 Lesniewski published in Polish seven papers. All of them are listed below /For bibliographical details see REFERENCES/. /1911/
A contribution to the analysis of existential propositions.
This is Lesniewski's doctoral thesis. /1912/
An attempt to prove the ontological principle of contradiction.
A direct respons to the influence of LUKASIEWICZ /1910/. /1913a/ Logical studies. Russian version of the two above essays originally published in Polish. /1913b/ Is truth only eternab or both eternal and temporal? A respons to the influence of Kotarbinski's "Problem of the existence of future" published in Polish in 1913. /1913c/ A critique of the logical principle of the excluded middle. Although written under an influence of Twardowski and Husserl the essay contains nevertheless a refutation of "platonism" and a defence of the nominalistic principle that there are only individuals. /1914a/ Is the class of classes not subordinate to themselves subordinate to itself? An exposition of Lesniewski's early views on Russell's antinomy written at the time when Lesniewski's own theory of classes had not yet been worked out. /1914b/ The theory of sets on the "philosophical foundations" of B. Bo rris t.e Ln ,
WORK AND INFLUENCE OF LEsNIEWSKI
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This is a review on B.Bornstein's paper "Philosophical foundations of the theory of sets" published in Polish in 1914. All these early writings were later repudiated by Lesniewski himself as immature or even philosophically unsound. Nevertheless, they are of some importance and, no doubt, they had influenced Lesniewski the philosopher. In particular, they encompass /1/ some interesting arguments against the existence of universals, /2/ a rather strong criticism of conventionalism, /3/ the first outline of the distinction between the language and the metalanguage, /4/ Lesniewski's comprehensive conception of semantic categories later opposed to Russell's theory of types, and /5/ the first analysis of the notion of set in the so called collective sense later developed within the system of mereology and applied to the solution of Russell's antinomy. During his lifetime Lesniewski read a number of papers. Between 1911 and 1915 he presented seven papers to the Warsaw Psychological Society, the papers dealing with the problems of existence, of contradictory objects, of creating the truth, of paradoxes of logic and mathematics, and of various aspects of the foundations of set theory. It happened just before the outbreak of the World War I that Lesniewski was to deliver a lecture on Russell's antinomy at a session of the Warsaw Psychological Society. They say that while preparing the lecture Lesniewski suddenly found an error in his criticism of the antinomy. As things were he decided to concentrate his attention to the utmost munching bars of chocolate for support. And thus chocolate gave birth to his new theory of classes later renamed mereology. It was with the idea of mereology in his mind that Lesniewski left for Russia to spend the years of the wartime there. He lived in Moscow from 1915 until 1918 and taught mathematics in a Polish grammar school /Mrs.Jakubowski's pension/. When in Moscow Lesniewski read a number of papers at meetings of various Polish groups and institutions. The list of the titles of the papers includes: Problem of noncontradictory set theory; Antinomies of formal theories and the Language; Basic problems of contemporary philosophy; and Philosophical foundations of Marxism. This period of Lesniewski's life and in particular his scholarly activities were observed by Professor W.Sierpinski who also spent the war years in Moscow. It was there and then that Lesniewski formulated and published in Polish the first outline of the formal system of m~reology. This publication is found to be somewhere between Lesniewski's early writings he later repudiated and the mature works to have come. THE FIRST OUTLINE OF MEREOLOGY /1916/
Foundations OL the general theory of sets. I.
This work deals with "parts", "ingredients", "sets", "classes", "elements", "subclasses" and certain interesting kinds of classes. Lesniewski developed and axiomatized here what he called the theory of parts or mereology. The work is written in colloquial Polish because Lesniewski had not yet overcame his distrust of technical symbolic notation and had not yet elaborated his own theory of symbolic reconstruction of colloquial languages. Part II of this work has never been published. THE MATURE ACTIVITY After the wartime Lesniewski returned to the liberated Poland. Well acquainted with mathematical logic yet having never had a formal
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higher mathematical educationhe joined a group of young mathematicians concentrating themselves on the foundations of mathematics. This group was led by a man of great mathematical gift, Z.Janiszewski soon to die, and S.Mazurkiewicz who were the very cofounders of the Polish school of mathematics. Before long Lesniewski was given a position at the University of Warsaw. That was the university where he was a professor of philosophy from 1919 until his death on 13 May 1939. In Warsaw a period of splendour in the field of mathematical logic and the foundations of mathematics had begun. Just then a new journal "Fundamenta Mathematicae" was created. In July 1920 the first volume of it appeared and this may be considered as a date of birth of the Polish school of mathematics. Lesniewski was a member of the editorial board of "Fundamenta" from the very beginning until 1928 when he resigned for personal reasons. Lesniewski's importance grew. Together with J.Lukasiewicz he was a cofounder of what in the course of time became known as the Warsaw school of logic, for long the biggest centre of logical research. Round Lesniewski and Lukasiewicz a large group of students were gathering: A.Tarski, A.Lindenbaum, B.Sobosinski, M.Wajsberg, M.Presburger, and later J.Slupecki, S.Jaskowski, Cz.Lejewski, W.Sadowski, J.Hossiason, H.Hiz and others. Tarski swiftly won a prominent position and joined Lesniewski and Lukasiewicz as one of the leaders. In 1927 Lesniewski published the first paper of a long series on the foundations of mathematics in which he redeveloped mereology first described in LEsNIEWSKI /1916/, elaborated it and incorporated other results from his earlier publications he foundworth preserving. Bibliographical description of the publication is given below. /19271931/ Foundations of mathematics. This series consists of twelve papers and it is divided as follows. /1927a/ Introduction. A brief characterization of Lesniewski's system as compared with other systems of logic and the foundations of mathematics, especially, from the point of view of antinomies and definitions. /1927b/ Section I. On certain questions concerning the meaning of "logistic" theorems. Lesniewski's account of initial difficulties with mathematical logic, and detailed criticism of ambiguities in "Principia Mathematica". /1927c/ Section II. On Rus s e Ll.f s "antinomy" concerning "the class of classes which are not elements of themselves". A brief description of Lesniewski's solution of Russell's antinomy. /1927d/ Section III. On different ways of understanding the words "class" and "set". A characterization of Lesniewski's collective conception of sets and a criticism of other interpretations /Cantor, Dedekind, Frege, Schroeder, Zermelo, Fraenkel, Hausdorff, Sierpinski, Russell, Russell and Whitehead/ . /1928/
Section IV. On the "Foundations of the general theory of sets. I".
A recapitulation of the work LEsNIEWSKI /1916/ with an improvement concerning the "existence of something" /In a number of previously stated mereological theorems the antecedents saying that objects are objects were added. It contains also a series of historical footnotes
WORK AND INFLUENCE OF LEsNIEWSKI
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and a criticism of Whitehead's theory of events. Concerning the criticism see, e.g., SINISI /1966/. /1929/
Section V. Further theorems and definitions of the "general theory of sets" from the period up to 1920 inclusive.
A long series of mereological theorems and definitions as well as of historical footnotes. /1930a/ Section VI. Axiomatization of the "general theory of sets" from the year 1918. A description of mereology in terms of "part" as the only primitive. /1930b/ Section VII. Axiomatization of the "general theory of sets" from the year 1920. A description of mereology in terms of "ingredient" as the only primitive. /1930c/ Section VIII. On certain conditions established by Kuratowski and Tarski, necessary and sufficient for p to be the class of a. Three sufficient and necessary conditions for P to be the collective class of a, one of them found by K.Kuratowski, the two remaining found by A.Tarski. /1930d/ Section IX. Further theorems of the "general theory of sets" from the years 19211923. /1931a/ Section X. Axiomatization of the "general theory of sets" from the year 1921. A description of mereology in terms of "exterior" as the only primitive. /1931b/ Section XI. On "singular" propositions of the type "A £ b". A discussion of the origin and character of Lesniewski's ontology. Section XI above ends with a misleading indication "To be continued" which, however, has never been done. In the meantime Lesniewski published also two papers, independent of his own systems, both on group theory. /1929a/ Ueber Funktionen deren Felder Gruppen mit Rucksicht auf diese Funktionen sind. A reduction of group theory to an adequate single axiom containing a single primitive term f/A,B,C/ meaning A.B = C. /1929b/ Ueber Funktionen, deren Felder Abelsche Gruppen in bezug auf diese Funktionen sind. A single axiom for Abelian groups is given. The next Lesniewski's paper contains a complete design of his most general logical system, protothetic, and it is the most important paper on protothetic. /1929c/ Grundzuege eines neuen Systems der Grundlagen der Mathematik. §1§1l. Its Introduction contains a brief characterization of the New System of the foundations of mathematics, consisting of three deductive theories: /1/ Protothetic, a generalization of the usual propositional logic; /2/ Ontology, a kind of a modernized traditional logic of names which resembles Schroeder's logic of classes enriched by the theory of individuals; and /3/ Mereology, outlined in LEsNIEWSKI /1916/. §1 includes a justification of equivalence as the sole primitive
196
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term /apart from quantifiers binding propositional variables and functions/. §2 characterizeS semantic categories as an alternative to the theory of types. §3 comprises a detailed completeness proof for pure equivalence based upon axioms /Al/ and /A2/, modus ponens for equivalence and substitution rule, and also the wellknown Lesniewski's decidability criterion for equivalence. §4§10 encompass a characterization of five systems of protothetic, denoted SI to S5, where the system S4 bases protothetic upon implication as the sole primitive. §11 describes successive simplifications of the single axiom for protothetic. In this connection compare SOBOCInSKI /1934/. The same paragraph contains also the wellknown Lesniewski's terminological explanations and directives for protothetic, and also a derivation of contradictions in CHWISTEK /1923/ and von NEUMANN /1927/, the contradictions being caused by a careless formulation of rules for introduction of new expressions which do not guarantee the uniqueness condition for decomposing expressions. /1930/
Ueber die Grundlagen der Ontologie.
It contains the terminological explanations for ontology, followed by brief remarks concerning alternative axiomatizations. In 1931 Lesniewski published also a paper on definitions in the theory of deduction which is like papers LEsNIEWSKI /1929a/ and /1929b/ independent of his own systems. /1931/
Ueber Definitionen in der sogenannten Theorie der Deduktion.
It includes a directive permitting addition to the propositional logic of definitions together with the conditions to be satisfied by such definitions. It contains also the appended terminological explanations and examples illustrating mutual independence of the conditions imposed upon definitions. /1938a/ Einleitende Bemerkungen zur Fortsetzung rneiner Mitteilung u. d.T. "Grundzuege eines neuen Systems der Grundlagen der Mathematik". The paper comprises of the resume of Sections IXI of LEsNIEWSKI /19271931/, followed by supplementary remarks on axiomatization of alternative "computative" systems of protothetic. It contains also Lukasiewicz's simplifications of the "deductive part" in Lesniewski's completeness proof for pure equivalence; a reduction to progressively shorter single axioms of protothetic /cf. SOBOCInSKI /1934//; a key to principles of Lesniewski's ideographic notation, the so called wheel and spoke notation /Concerning the subsequent use and adaptation of the Lesniewski's notation compare GOODELL /1952/ and MARTIN /1953//; further criticism of defective formalization of rules for introduction of new expressions, substantiated by additional proofs of contradiction in von NEUMANN /1931/. /1938b/ Grundzuege eines neuen Systems der Grundlagen der Mathematik, §12. This paper was to have followed the first eleven Sections of LEsNIEWSKI /19271931/ but after Lesniewski withdrew his manuscript from "Fundamenta Mathemaicae" in 1930 it did not appear until an opportunity was granted by Lukasiewicz and Sobocinski, the founders of a new journal "Collectanea Logica".
WORK AND INFLUENCE OF LEsNIEWSKI
197
In this paper more than 400 protothetical theorems are derived from the axioms /A1/, /A2/ and /A3/ given in LEsNIEWSKI /1929c/. LEsNIEWSKI AS AN ACADEMIC For Lesniewski his publications were not the only way of introducing his New System of the foundations of mathematics. He also attached a great importance to his university lectures though he lectured almost entirely about his own work. The following lecture courses were delivered by Lesniewski at the Univerity of Warsaw between 1919 and 1939. /1/ Foundations of the theory of classes /delivered during the acade
mic year 191920; influenced by discussions with L.Chwistek Lesniewski introduced here for the firat time a logical symbolism as simpler and more precise than colloquial languages/,
/2/ Foundations of arithmetics /delivered in 192021; in this course Lesniewski presented a number of systematic and detailed observations and results of ontology/, /3/ Axiomatic foundations of science /192021/, /4/ Foundations of threedimensional Euclidean geometry in the light of the new theory of classes /192124/, /5/ Foundations of logistic /192127/, /6/ Foundations of ontology /192527/, /7/ Axiomatics for group theory /192629/, /8/ Theoretical arithmetics /192627/, /9/ Introduction to mathematics /192728/, /10/ The problem of the primitive terms on the ground of arithmetics /192829/, /11/ Elementary outline of ontology /192932; students' notes of this lecture course form the basis for SLUPECKI /1955//, /12/ Directives of logistics and ontology /192931/, /13/ On the foundations of the co called theory of deduction /193031/, /14/ Chosen topics from the foundations of logistics /193032/, /15/ From the foundations of protothetic /193234; students' notes of this lecture course form the basis for SLUPECKI /1953//, /16/ Chosen topics concerning the axiom systems of geometry /193233/, /17/ Arithmetics of real numbers /193334/, /18/ Fundamental problems of mathematical logic /193435/, /19/ "Antinomies" of deductive sciences /193435/, /20/ Introduction to deductive sciences /193637/, /21/ Propositional calculus /193637/, /22/ On the so called manyvalued logics /193637/, /23/ Antinomies of semantics /193638/, /24/ Traditional "formal logic" and traditional "theory of sets" on the ground of ontology /193738/, /25/ Basic publications in the so called manyvalued logic /193738/,
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198
/26/ An outline of protothetic /193839/, /27/ An outline of ontology /193739/, /28/ Axiomatics for mereology /193739/. Apart from the lecture courses minars.
Le~niewski
also conducted several se
/1/ On the foundations of mathematics /191920; 193032; 193638/, /2/ On Cantor's set theory /191921; Le~niewski put here the problem of the axiomatization of the copula "is" as in the expression "A is b", and made the first observations on ontology/, /3/ On Zermelo's "Untersuchungen ueber die Grundlagen der Mengenlehre" /192021/, /4/ Philosophy of mathematics /193839/. Between 1918 and 1927 Le~niewski read several papers. Two of them, "On certain theorem from the theory of relations" in 1918, and "On the grades of grammatical functions" in 1921, were presented to the Logical section of the Warsaw Philosophical Institute. The second paper put forth the theory of semantic categories as a counterpart for the hierarchy of Russell's logical types. The first outline of ontology, another Le~niewski's system was presented in two further papers both under the same title "On the foundations of ontology". The firts one was read to the Warsaw Psychological Society in 1921 and the second one to the Warsaw Scientific Society in 1930. Le~niewski also gave two lectures, "On the foundations of ontology" and "On the foundations of logistic" to the Logical section of the second Polish Philosophical Congress held in Lvov in September 1927. LE~NIEWSKI'S
NEW SYSTEM
The threefold edifice of Le~niewski's New System of the foundations of mathematics consists of what he called protothetic, ontology and mereology. Mereology was developed between 1914 and 1916 and it was presented in written in LE~NIEWSKI /1916/. Then it was recapitulated and redeveloped in the Sections IX of LE~NIEWSKI /19271931/. Broadly speaking mereology may be understood as a theory of the partwhole relation. Starting point for the development of ontology was 1919. As a formal system logically pressuposed by mereology it was constructed in 1920. It first became known through copies of Lesniewski's university lectures and was presented in written only in LE~NIEWSKI /1930/ though the last Section of LE~NIEWSKI /19271931/ and the Introduction of LEsNIEWSKI /1929c/ contained quite a lot of details concerning the origin and character of onbology. A survey of Le~niewski's unpublished results based on students' notes collected after the last war by T.Kotarbinski is contained in S~UPECKI /1955/. A very good informal outline of Le~niewski's elementary ontology is included in KOTARBInSKI /1929/. Broadly speaking ontology is a theory of the copula "is" and it comprises the traditional parts of logic: the theory of predicates, of classes, and of relations, including the theory of identity. Protothetic as the most general Lesniewski's system logically pressuposed by ontology started 1921 although the idea of it came back to Le~niewski's early writings from 19121914. It was axiomatized in LE~NIEWSKI /1929c/, resumed afterwards in LE~NIEWSKI /1938a/, and
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199
developed in LEsNIEWSKI /1938b/. A survey of unpublished Lesniewski's reSults on protothetic based on students' notes can be found in SLUPECKI /1953/. Broadly speaking protothetic is the logic of propositions with quantifiers binding propositional and functional variables. Notice that for Lesniewski there was no need to introduce the usual first order predicate logic apart from protothetic and antology as "it can be proved that ontology contains not only the first order logic but also the worder predicate logic can be built up within ontology without additional axioms or rules of inference. PROTOTHETIC When the axiomatic foundations of mereology and then of ontology had been established Lesniewski turned to the problem of the logic of propositions pressuposed by the two theories. He wanted it to be as strong as possible. To construct it he introduced into the language of the usual propositional logic functional variables of arbitrary propositional types and allowed them to be bound by quantifiers and he called the resulting system protothetic or the theory of first principles. Thus protothetic is a kind of a theory of higher order propositional logic with quantifiers and at this point it resembles the higher order predicate logic. Here zerooder propositional logic is just the ordinary one and first oder propositional logic is the logic with quantifiers binding only propositional variables. historical remark concerning quantifiers in the proposit~onal logic. Russell was presumably the first who in his "Theory of implication" published as early as 1906 studies the possibility of introducing quantifiers binding the propositional variables into the propositional logic. But he had not developed this idea and made no use of it in "Principia Mathematica". A propositional calculus with quantifiers was subsequently constructed in LUKASIEWICZ /1929/ and in LUKASIEWICZTARSKI /1930/.
~
Protothetic based on equivalence.Of course, protothetic as a formal system can be based on different propositional connectives as primitives. That based on equivalence is most popular. Lesniewski adopted equivalence as the only primitive in the consequence of an earlier discovery by A.Tarski. In 1922 Tarski, at that time a Lesniewski's doctoral student, discovered the possibility of defining conjunction in terms of equivalence and quantifiers. Notice that the first order propositional logic based on equivalence as the only primitive connective, i.e., the logic of pure equivalence with quantifiers binding only propositional variables gives nothing essentially new. In other words, quantifiers over propositional variables make it possible to define two propositional constants /1/ and /2/
(p)p,
the falsity or the "zero" element,
(p) p ;;. (p)p,
or
(p) (p:p)
the truth or the "one" element,
and four functions of one propositional argument /1/
p,
/2/
(p)p;p,
the assertion of p,
/3/ and /4/
p=p,
the negation of p,
the verum of p,
p .. [(p)(p;p)]
but nothing more.
the falsum of p
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200
The situation changes when we go one step further and pass to what might be called the second order propositional logic based on equivalence, admitting also quantification over functions of propositional variables. Then sixteen functions of two propositional arguments are definable. It suffices, of course, to define conjunction;that can be done in many ways, e.g., p"q _
(f) {p =
pl\q'"
(f){
[f(P)
'" f(q)]}
or [f(P)
;; f(P"'P)] ;: q}
The first definition of conjunction is due to A.Tarski who published it in his doctoral thesis /Cf. TARSKI /1923a/, /1923b/ and /1924//. For Lesniewski it was a welcome discovery because equivalence constituted for him the most intuitive primitive term. However, Lesniewski was not the only man attaching to equivalence a rather great importance. Mention, for instance, F.Ramsey. In his "Foundations of mathematics" Ramsey wrote these sentences: "The preceding and other considerations led Wittgenstein to the view that mathematics does not consist of tautologies, but of what he called "equations", for which I should prefer to substitute "identities" .... /It/ is interesting to see whether a theory of mathematics could not be constructed with identities for its foundations. I have spent a lot of time developing such a theory, and found that it was faced with what seemed to me unsuperable difficulties" /Cf. RAMSEY /1925/,p.350/. Stanislaw Lesniewski was seemingly the first who realized this programm independently and successfully. Protothetic makes it possible to formulate within the system itself of many theorems which, with a different formalization of the propositional logic, are considered as belonging to its metasystem, e.g., The laws of
~eneralization,
for every propositional category.
/1/ The law of generalization for the lowest propositional category: Cf)[(p) f(p) ;: f(O)l\f(l)J where 0 and 1 stand for (p)p and 0;0 respectively, /2/ The law of generalization for the second propositional category, i.e., for functions of one argument: (A) ~ (f) [A(f) '" A(as) " A(neg) II A (ver) " A (falsum)Jj where as, neg, ver and falsum are the functions of one propositional variable: assertion, negation, verum and falsity respectively. The laws of generalization for functions of two propositional variables as well as for higher propositional categories can be formulated analogously. The laws of extensionality, for every propositional category. /1/ (p,q) { (p ,;:; s) '" (f) [f(p) ;. f(q)Jj /2/ (f,g) { (p) [f(P) ;: g(p)J = CAJ [A (f) ~ A(g)]} and so on for higher propositional categories. Deductive formalization of protothetic. Lesniewski constructed several axiomatizations for protothetic based on equivalence as the sole
WORK AND INFLUENCE OF LEsNIEWSKI
201
primitive. For illustration we quote one such an axiomatization. Axioms: /PA1/ (p,q,r) (p ;. q) =. [(r =: q) =. (p =: r)J This is a single axiom of the theory of pure equivalence based on detachment and substitution rules.
1
l
/PA2/ (P,f){ f(p) ={fCP == 0) =' (q)[f(p) ;; f(q)JH Rules of inference: /PR1/ Detachment rule for equivalence, /PR2/ Distribution of general quantifiers: From
(x 1, ... ,x
n)
(A;: B)
infer (x 1, ..• ,x i) [(x i + 1,· .. ,X n)A;: (x i + 1, ... ,X n)B] where x ••. ,x 1, n tegory, /PR3/
are variables of arbitrary propositional ca
Protothetical rule of definition: (x l' ... , x n) [ A (x l' •.. , x n)
=: B (x l' ... , x n )
J
where
A(X ••• ,x is a propositional formula with x •.. ,x 1, 1, n) n the only free variables of any propositional category and B is a new propositional constant distinct of any sign in the same propositional category,
/PR4/
Protothetical rule of extensionality, for all propositional categories /except for the lowest propositional category which is guaranted in the axioms/: (A,B) {(x 1""'xn) [A(X 1, ... ,x n) ;:: B(x1, ...
/PR5/
,xn~==(C)[C(A)
.,
c(~f
Substitution rule, which does not permit to replace complex expressions by other expressions but only to replace free variables representing propositional variables or functions of any propositional category by constants or variables of the same propositional category.
Several versions of protothetic based on equivalence are axiomatized by a single axiom. To give an example. /PA2/ (p,ql{C p:q):. (fJ{f{P,f[P'(S)s]} =: (r)[f(q,r) == (q == PJJ1} This axiom had been found as a result of a series of successive simplifications made by Lesniewski, Tarski, Wajsberg, Sobocinski and others, and it is the shortest one known as yet /Cf. SOBOCInSKI /1934//. Protothetic based on implication. Protothetic can be based on primitives other than equivalence. A system based on implication as the only primitive was developed mainly by A.Tarski. He defined the negation of p as p~(p)p and adopted the following axiom (q,f)i[Cp) (p~)J =l){f[(p) p]==) f(ql1} and reformulated the rules of inference accordingly. Protothetic as ~ generalization of the ordinary propositional logic. Protothetic contains all the theorems of the ordinary twovalued propositional logic which was proved by A.Tarski in 1922 /Cf. TARSKI /1923a/, /1923b/ and /1924//. Twovaluedness of protothetic. As a formal system protothetic is com
202
J. SURMA
plete with respect to the twoelement Boolean algebra, this algebra being considered together with all the Boolean functions stated explicitly. That makes protothetical theorems rather easy to predict. Protothetic allows to express within its own language the method of verification by the wellknown zeroone matrices. The usual manner of proving theorems resembles the ordinary verification by the zeroone matrices and i t runs as follows: first we prove a theorem in question for the constant zero, then for the constant one and finally we apply a law of generalization. As a formal system protothetic is deductively consistent, decidable and deductively complete, i.e., for every closed propositional formula A either A or the negation of A is provable. In the case of the underlying logic being twovalued protothetic has been extensively examined. On the other hand, manyvalued protothetics are not so fully investigated. One paper dealing with the problem belongs to T.Scharle /ef. SCHARLE /1971//. Scharle states without proof that finitelyvalued protothetic can be constructed with the completeness property in the following sense: For every closed propositional formula A one of J.(A) is provable, where the functions J. are such that they take desIgnated values only in case the argument has value i. This statement can be proved in many ways, e.g., by using the finitely generated trees technique which has been given in SURMA /1976/. Definitions and extensionality in protothetic. Lesniewski's formalization of protothetic includes the rule of definition allowing to extend at will the variety of propositional categories. So protothetic is presented not as a completed system but as a system developing in the course of time. This property expresses Lesniewski's very conception of the deductive system. The set of wellformed propositional formulas in not actually given as a whole. Systems are being continually enriched by newly defined terms. All definitions are formal theorems of the system in question and these definitions can be creative. Only the translatability or eliminability condition is satisfied by the defined terms. The equivalence sign ~ is just used, not mentioned, i.e., it is of linguistic, not metalinguistic nature. Another peculiarity of protothetic closely related to the role played by definitions is that it includes the extensionality as an inference rule, not as the law of extensionality. By this the formal system of protothetic is finitely axiomatized. The construction of a constant of a certain propositional category automatically supplies the rule of extensionality for that category. remark concerning the theory of propositional semantic category. In order to state precisely the conditions to be satisfied by an expression to be a new theorem Lesniewski worked out a general theory of semantic categories. The theory formally resembles the simple theory of types but, in its intuitive credentials, has more in common with the traditional "parts of speech" or with Husserl's "Bedeutungskategorien". Unlike the authors of "Principia Mathematica" Lesniewski understood logical types as kinds of expressions and not of non1inguistic entities.
~
Let s denote the basic category of propositions. As it is generally known this basic category is common to all languages. All other propositional categories are obtainable by finite application of the following rule: If~, £o"."£n are propositional categories, then
WORK AND INFLUENCE OF LEsNIEWSKI ~/~o"'~n
203
is a propositional category. Every sign, i.e., constant or
variable belongs to one and only one semantic category. Protothetic contains variables, constants and functions of all categories for which the index can be constructed by using only the symbol s. Notice that the variables and constants in the language of protothetic are not provided with category indices so the belonging of symbol to a particular category follows from the part it plays in a formula under consideration, and two formulas are of the same category if they occur in brackets of the same shape or if they precede similar brackets. In Lesniewski's theory of categories there is the following rule of contraction: Any cate~ory of the form {•.• C(a/b '" n b ) 0 b ) •.. n b )  0
contracts into the category a. This rule represents the fact that when a function of the category alb .•. b is applied successively to n propositional formulas of the cat~gorI2s b , .•• ,b respectively, then a complex formula is formed which is oIOthe ca~egory ~. Notice
that quantifiers are treated by Lesniewski as pseudosigns, as signs of no category at all or in other words, as categorially open signs. No doubt, this is a deficiency of the discussed theory. The deficiency has often been mentioned in the literature /See for example TARSKI /1933/ or PRIOR /1961//.
remark concerning Henkin's theory of propositional ~ . In 1963 L.Henkin published his formulation of the theory of propositional types based on equality and functional abstraction as primitives /See HENKIN /1963//. It was improved by P.Andrews later on /Compare ANDREWS /1963//. This theory was a modification of A.Church's formulation of the simple theory of types /Cf. CHURCH /1940//. In 1975 Henkin converted the ideas of his 1963 paper into a formulation based on equality and relational abstraction as more familiar than the functional abstraction expressed in terms of Church's lambdanotation /Cf. HENKIN /1975//.
~
Henkin's theory of propositional types can be considered as a version of protothetic though it is not identical with Lesniewski's protothetic. In Lesniewski's there are variables for each propositional category, each variable can be bound by quantifiers that are of no category. Lesniewski adopts in his system only one equivalence sign denoting the usual connective of equivalence. Also Lesniewski's protothetic includes a special rule of definition which allows the introduction of new symbols as names. of arbitrary elements of any propositional category. Henkin also constructs a theory with a distinct set of variables for each propositional type but he adopts as many equality symbols as to have one for each propositional type. Then he introduces relational abstraction as /another/ primitive of no propositional type. He starts with names for only a relatively few elements of the hierarchy of propositional types and constructs new names by means of relational abstraction. He proves that in this way each element of a given propositional type has a name in the system, and he defines quantifiers in terms of equality and relational abstraction. The possibility of defining quantifiers is essentially due to Quine /Cf. QUINE /1956//. Henkin gives a deductive formalization of his theory and shows that it is complete in the sense that A is provable in this formalization if it has the value "true" for every assignement of values to its free variables under a given interpretation.
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Later on A.Grzegorczyk showed that every equality symbol in Henkin's theory is definable by means of relational abstraction and of the equality symbol for the lowest propositional type, i.e., the equality symbol denoting the usual equivalence /Cf. GRZEGORCZYK /1964//. Notice that this Grzegorczyk's reduction does not finitize the axiom system of Henkin's theory because we need the rule or denumerable many axioms having the form of extensionality. Henkin gives his theory of propositional types an elegant semantic interpretation. His interest for the theory was caused by the problem of constructing nonstandard models for the full theory of types as described in HENKIN /1950/. He hoped for insight into the totality of models for the full theory of types through his study of all models of the much simpler theory of propositional types where each type is finite. Such an argument indicates another methodological advantage of protothetic. Similar arguments are widely known. Many problems of ordinary and higher order predicate logic can be reduced to questions about propositional logic as for instance in Herbrand's theorem. remark concerning pure equivalence. Lesniewski initiated investigations into the equivalential fragment of the ordinary propositional logic. He was the first to axiomatize the pure equivalence logic. In LEsNIEWSKI /1929c/ he stated that the axioms
~
/A1/
[(p .. r)
'"
and /A2/
[p;; (q
c.
(q
r)J
c.
p)]
.= (r c.
=[(p;;.
q).=
q)
rJ
together with substitution and detachment for equivalence form a complete system. To prove this Lesniewski formulated the so called now Lesniewski's decidability criterion for equivalence according to which any pure equivalence can be deduced from /A1/ and /A2/ if every propositional variable occurs in it an even number of times. A grouptheoretic proof of Lesniewski's decidability criterion is given in STONE /1937/. A detailed discussion of the method Lesniewski used in his proof is given in SURMA /1973a/. A Henkinstyle completeness argument for the pure equivalence can be found in SURMA /1973b/. A study of the equivalent fragment of the intuitionistic propositional logic by means of prooftheoretic methods was given in TAX /1973/. An algebraic reformulation of Tax's result can be found in KABZINSKIWROnSKI /1975/. ONTOLOGY Deductive formalization of ontol~. Lesniewski constructed a formal system of ontology as based on the axioms and rules of protothetic. Ontology includes the following additional axiom: /OA1/
(a,b)[aEb
=l~c)(c£a){c)(c£a =} CEb) ,,(c,d)(c£a
x d t a ==>C2d)]
and the following rules of inference: /OR1/
(a,x1, ... ,xn){[ataAA(x1, ... ,xnl] .=[a€.B(x1, ... ,xnlJ} where B is the constant being defined and A is a propositional formula in which x ... ,x and possibly a are the only 1, n free variables of arbitrary semantic categories. This is the rule of ontological definition.
WORK AND INFLUENCE OF LEsNIEWSKI /OR2/
(A,BH(a,xl,··.,xJ[a~A(xl,···,xn)~ ==
(C )[c
where x
(A)
B(X 1,·.·,xn
205
lJ 
S C(B)]}
.•• ,x
are variables of arbitrary semantic category. 1, n This is the rule of ontological extensionality.
The axiom /OAI/ is the so called long axiom of ontology and according to Lesniewski it gives the precise meaning of the copuls "is" in its existential sense. This axiom can be equivalently substituted by the following one: /OA2/
(a,b)[a£b .. (~c)(aE.c" Cfb)J
called the short axiom of ontology. According to the intended intuitive meaning the rule /ORI/ of ontological definition is the ordinary definition of a set by indicating the unit sets it contains. While the rule /OR2/ of ontological extensionality corresponds to the theorem stating that sets composed of the same unit sets are identical. Notice that it was Lesniewski who gave the first exact description of the rules /ORI/ and /OR2/ which was an achievement of a historical value. Observe also that the above formalization of ontology is proved to be consistent. This particular result was obtained by Z.Kruszewski as early as 1925 /Cf. KRUSZEWSKI /1925/ / by interpreting the ontological e p s.j Lon as the conjunction connective. A rematk on definitions. Lesniewski did a pioneer work in applying new approach to the theory of definition. He rejected the view that definitions are merely conventional abbreviations accepted exclusively for economy and space saving in the presentation of a formal system. According to him definitions are an act of choosing the complex and important ideas that are to be the object of study.
a
He emphasised the fact that definitions require special formative rules and he was the first to give such rules. For him definitions are formal theorems of a system under consideration and they are an important way of introducing new semantic categories. AS to the methodological role of definitions they are allowed to be creative, i.e., in the formalized systems there are theorems that do not contain defined terms and yet cannot be proved without these definitions. Lesniewski affirmed his belief in the admissibility of creative definitions under an influence of J.Lukasiewicz who discussed the problem of creative definitions in his lectures summarized then in LUKASIEWICZ /192829a/ and /192829b/. The fact that definitions are to be treated as theorems of formal systems and can be creative has not been widely recognized in the foundational studies. Nevertheless there are authors sharing this Lesniewski's view /Cf. for example MORSE /1965//. Recently several interesting papers have been published concerning reative definitions. One of them is MYHILL /1953/ where the author constructs a system of arithmetic with infinitely many creative definitions. Another paper is RICKEY /1975/ where axiomatizations of the propositional logic containing a single creative definition, a finite number of creative definitions, and also an infinite number of creative definitions are given. RICKEY /1975/ gives also the proof
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of the Lindenbaum theorem stating that the propositional calculus containing p=9p as a theorem has no creative definitions. However, there still remains more work to be done in the problem of creative definitions. Ontology and set theory. Ontology is a kind of the simple theory of types combined with the theory of a certain relation between objects synbolized "e" which is to abbreviate the copula "is" as it is used in Polish or in Latin. However, unlike Russell who refered to entitities rather than to symbols of entities, though his position was not free of certain hesitation, Lesniewski would prefer to say that "a is b", in symbols, "atb" is true if "a" is a name or more precisely "a" is a unit or singular name and the predicate "b" must apply to "a". This might suggest that ontology embodies what can be called distributive interpretation of names as symbols for sets. Is this the case will be seen from below. Observe first that according to Lesniewski the relation "£" covers all the cases like these: III Plato is wise where "is" stands for classmembership of the ordinary set theory, 121 The dog is an animal where "is" stands for classinclusion, and 131 Socrates is the husband of Xantipa where "is" stands for setidentity. These examples show that Lesniewski tried to single out Ibut not to confusel some common properties of class membership, inclusion and identity. That, of course, does not imply there is no distinction between the three relations, and it was Frege who as the first logician pointed out the necessity of making the distinction. The fact that the epsilon of ontology differs from that of the ordinary set theory can be further illustrated by the two following theorems of ontology
III
(a,bIC aj b ~ ae.a)
expressing the semireflexivity gical epsilon, and
121
(a,b,c) (aEb
A
Ibut
not reflexivityl of the ontolo
b ac ~ a sc]
expressing the transitivity of the epsilon. Notice that the ontological epsilon is not reflexive as the formul.a
131
(aJ(aea) is not true in ontology; it is even independent of ontology.
Taking this into account some authors ICf., for example, FRAENKELBARHILLELLEVY 11973/, p.2031 consider Lesniewski's ontology not as a variant of set theory in the sense of ZF but rather as a rival of set theory in the field of the foundations of mathematics. However, no matter how important a rival of set theory ontology is, or could be, this is a question still very difficult to be answered. The only certain thing is that the standard arguments leading to the logical antinomies cannot be reproduced in ontology: some of the counterparts of these arguments fail to comply with Lesniewski's theory of semantic categories while others require certain steps which are not viable in ontology. Readers interested in more details concerning Lesniewski's unpublished materials on antinomies are advised to consult SOBOCInSKI 119501.
WORK AND INFLUENCE OF LEsNIEWSKI
207
Ontology and the axiom of choice. Ontology has no axiom of choice, and the theorems provable in ZF with the aid of this axiom, in ontology appear as consequences of conditional sentences which antecedents contain the conditions corresponding to the axiom of choice. Lesniewski decided his ontology to be uncommited to the axiom of choice as its acceptance is equivalent to the wellordering principle and the hierarchy of semantic categories would require then the axiom of choice for every category. but nevertheless Lesniewski studied the contents of the axiom and even established that it is effectively equivalent to each of the following transfinite arithmetics formulas:
/1/ /2/
m·n m+n
m
or
m·n
m
or
m+n
n
=
n
for every transfinite m and n /Cf. for this KURATOWSKIMOSTOWSKI /1968/, p.297/. Recently several axiomatic extensions of Lesniewski's ontology have been examined. For instance, J.Canty added to ontology an axiom of infinity in order to derive from it Peano's arithmetic, and he showed that Goedel's incompleteness theorem is applicable to this extension /See CANTY /1969a/ and /1969b//. Ch.Davis discussed formulations of the axiom of choice that can be expressed in ontology, one of the axioms having the following very elegant form
(j nCa,g) [g(a) ~ g(f(a))] /Cf. DAVIS /1975//. J.Kowalski established that the axiom of choice, KuratowskiZorn's lemma and the wellordering principle are equivalent in ontology /Cf. KOWALSKI /1975//. However, this interesting area has not been fully treated yet. Proper interpretation of ontology in set theory. Each of the following statements is independent from ontology: /1/ There are no individuals, /2/ There is only one individual, /3/ There are more than one individual, /4/ There are infinitely many individuals, and each of them can be used to extend ontology. Lesniewski himself, due to a strong philosophical belief, carefully avoided the postulating the existence of anything, and he put empty and nonempty names into the same semantic category. This raises the question of a proper interpretation of ontology in terms of the usual theory of sets. In set theory when giving a standard interpretation to a formal system one must specify a nonempty domain of discourse which is assumed to coincide with the range of the variables of the theory. In the usual set theory to be means to be the value of a variable. In Lesniewski's the domain of discourse may be arbitrary, even empty, and it would be nonempty only if ontology contained exclusively singular names. On the other hand, ontology includes the usual first order logic and therefore the classical principle of inference to which ontology adheres is that the particularly quantified statements follow from the universally quantified ones, in symbols, (x)A =*l~ x)A which implies that the range of the variables cannot be identified with the domain of discourse. For the discussion of some further peculiarity of ontology see PRIOR /1965/. There were many discussions on how to interpret ontology in set theory. The most persistent idea which was presented by many authors L's that the Lesniewskian formula a b is interpreted in set theory as:
e.
J. SURMA
208
[a} is contained in b Under this interpretation the axiom of ontology is a theorem of set theory and the ontological rules of inference are valid as certain settheoretic conditions. In accordance with this interpretation the range of the variables can be assumed to be the power set of the domain of discourse. The elements of the power set assigned to the elements of the category of names are called sometime the extensions of the names. Using this terminology it is clear that if there are only n objects in the domain then there are 2 n extensions to be assigned to the names. If, on the contrary, the Jomain is infinite then there are more than denumerably many extensions. To end the section remark that only a very little has been done concerning model theory of for ontology. Some general modeltheoretic aspects of ontology in the style of Henkin's "Completeness in the theory of types" /Cf. HENKIN /1950// have recently been discussed in BOUDREAUX /1976/, CANTY /1976/ and RICKEY /1976a/. Ontology as ~ Boolean algebra. Some authors like, for instance, A.Grzegorczyk /Cf.GRZEGORCZYK /1955// point out a formal resemblance between ontology and the theory of complete atomic Boolean algebras with constants and functions of an arbitrary high type. From this point of view expressions of the form "a E a" should be read "a is an atom" though Ledniewski himself would prefer to read this as "a is an object". The zero element of Boolean algebra is defined in ontology as follows a EO", (a t
a)~ ..... (a E
a)
and the epsilon of ontology can be defined in Boolean algebra as follows a £ b := (a is an atom)/'{b is an element containing a) In Boolean algebraic terminology the rule /ORI/ of ontological definition is the definition of an element of Boolean algebra by indicating its atoms while the rule /OR2/ of ontological extensionality says that elements of Boolean algebra composed of the same atoms are identical. A remark on quantifiers. Adopt the following definition of the empty na~
/1/ (a) [(a f. a)" "'(a E a) '" a t J\.] and observe that the following is a theorem of ontology
/2/
",,(I'. E1\.)
hence by the familiar rule of the existential generalization /3/
~(al(a€a)
Now it is evident that quantifiers as used in ontology cannot be interpreted referentially or objectually, i.e., with variables ranging over objects refered to by names. If they could the formula /3/ would mean that every object is not identical with itself which is false in set theory. Again ontology is·to be a formal system without any ontological commitment and therefore quantifiers need have nothing to do with existence. On the other hand, the substitutional interpretation of Lesniewski's quantifiers also cannot be unconditionally accepted. Remember that a universally quantified sentence is considered as true if from the fact that for any name when substituted for the variable in question it follows that the ope~ sentence after the quantifier comes out as true. First of all, to avoid the trivial case there had to be an infinite
WORK AND INFLUENCE OF LEsNIEWSKI
209
number of names. Secondly, the names need not to be unique or singular. Thirdly, the rule /ORI/ of ontological definition when applied it extends the language, and once the definition is asserted, quantification involving variables of the semantic category in question is permitted. But in Lesniewski's ontology sets, either empty, unit or not, are recognized as purely linguistic entities serving as extensions in the range of quantification, and not in the domain of discourse. The definitions of new linguistic expressions increase only the "logical power" of the system. They do not produce any change in the domain. So the following view on Lesniewskian quantifiers should be accepted: a universally quantified sentence is said to be true if, when any extension is substituted for the variable in question, the open sentence after the quantifier comes out as true For the further discussion of the Lesniewskian quantifiers consult KUENGCANTY /1970/. comparison of Lesniewski'~ theory of semantic categories with other versions of ~ theory. In Lesniewski's system symbols "a" and "b" in the expression "a E b" belong to the same semantic category, namely, to the category n of names. In Russell's theory of types they denote sets as belonging to different types. In Lesniewski's the category of names includes not only singular names but also empty and shared or general names.
~
This property of Lesniewski's theory of semantic categories makes it more suitable for practical deduction than other versions of type theory are, e.g., Tarski's theory of classes as described in TARSKI /1933/ or that of Church's /Cf. CHURCH /1940//. In Church's we encounter only functions of one argument, i.e., only types of the form alb, and functions of more arguments are constructed by superpositions, e.g., to the category a/bb in Lesniewski's corresponds the type (~/£)/£ in Church's. In TarskI's we have variables of type ~, and variables of type sin denoting unary relations of type a where a is either n or sin. Forinstance, to the category s/nnn inLesniewski's corresponds the type s/(s/(s/n)) in Tarski's. TheIanguage of ontology seems to bethe mostappropriate when we employ but the lowest possible types or categories. From this point of view Tarski's language is less convenient. As we have already shown Lesniewski's system contains creative definitions. Now a system of the simple theory of types allowing definitions may be converted equivalently to a system not containing them but containing the so called axiom of definability lor the axiom of subsets or the Comprehension schema/ instead. Lesniewski himself investigated some anologies of the axiom and called them pseudodefinitions. Clearly systems without definitions are easier to be described and therefore more convenient in metalogical research. Application of the theory of semantic categories to Linguistics.Lesniewski's hierarchy of semantic categories has become a fruitful starting point in developing modern linguistics. The hierarchy begins with two basic categories, the category s of sentences, and the category n of names, and it consists of infinitely many complex categories, resembling "parts of speech" in ordinary traditional grammar. Lesniewski himself invented it to prevent logic and mathematics of antinomies.It remained almost unnoticed outside mathematical logic until 1935 when an important K.Ajdukiewicz's paper "Die Syntaktische Konnexitaet" appeared /Cf. AJDUKIEWICZ /1935//. Ajdukiewicz derived his conception of syntactic connection from Les
210
J.
SURMA
niewski's conception of semantic categories. He introduced a special kind of notation of complex categories, the so called Ajdukiewicz's fractions. Ajdukiewicz's paper revealed an interesting interpretation and application of the theory of semantic categories in the field of formal linguistics though neither Lesniewski nor Ajdukiewicz were directly interested in the problem of the mechanical determination of syntactic structure of expressions of natural languages. Their theory of semantic categories became a popular and influential version of what is called today categorial grammars or categorially based grammars. The importance of the theory of semantic categories has been especially proved by Y.BarHillel /Cf. BARHILLEL /1950//, H.Hiz /Cf. HIZ /1961//, D.Lewis /Cf. LEWIS /1972//, M.Cresswell /Cf. CRESSWELL /1973// and others. In this connection compare also MACHOVER /1966/. BarHillel slightly modified the LesniewskiAjdukiewicz theory and he extended it in three ways: /1/ assignment of more than one category to the words was allowed, /2/ a new kind of operator category was introduced, the operators act upon arguments at their immediate left: if a and b are categories then ab, read, "a sub b" is a new category, /3/in thIs connection an additional rule 01 cancelation was introduced: from a and ab infer b. Lewis also presented a very interesting extension of LesniewskiAjdukiewicz's ideas. As to Cresswell he returned to lambdaoperator as a new operator instead of quantifiers. For a number of interesting problems in the formal theory of language based on the theory of semantic categories the reader may consult JAKOBSON /1961/. A detailed discussion of the work on grammatical categories done by Husserl, Lesniewski, Ajdukiewicz, BarHillel, Hiz and others encompasses also LEHRBERGER /1974/. MEREOLOGY Analysis of Russell's antinomy convinced Lesniewski that he should distinguish between the distributive and the collective interpretation of the notion of set or class. The expression "a l b" when understood distributively means "a is an element of the class of b's". If, however, we interpret it collectively it means "a is a part /proper or improper of the whole which is b". The term "part" is understood here as a "piece" of some object and therefore parts cannot be empty. As concrete agregates mereological sets /but, of course, not ontological sets/ possess no cardinal number. Once the distinction between collective and distributive sets was made it was evident for Lesniewski that the presuppositions on which Russell's antinomy was based turned out to be false. Lesniewski constructed a deductive theory called first the "general theory of sets" which he subsequently renamed mereology. The reason was that mereology and set theory make use of different notions of set and of membership, and therefore he relinquished the idea of treating mereology as a set theory. The partwhole relationship has long been examined and apart of Lesniewski's mereology several other approaches were presented. To quote only two examples. J.Woodger independently developed a system very similar to Lesniewski's /Cf. FLOYDHARRIS /1964//. The so called calculus of individuals of Leonard and Goodman was also a version of mereology apparently weaker than the lastone /Cf. LEONARDGOODMAN /1940//. deductive formalization of mereology. Mereology may be formalized in terms of many primitives. The most familiar formalization uses "pr", a rnereological elementcalled part, as a primitive where the expression
~
WORK AND INFLUENCE OF LEsNIEWSKI
211
"a E. pr b " stands for "a is (a) part of b". The axioms of mereology are as follows: /MAI/ (a,b}[a~pr(bl ' 9 b H J The axiom states that only the objects having parts are individuals. /MA2/
(a,b,c)[a epr(b) " b ~pr(cl
=> a
E. pr(cJl
Transitivity axiom. To formulate the next axioms we first introduce "Kl", a mereological class, which is defined in the following way: /DEF/
(a,K){KtKl(al; K~K ~(L)[Lca ~Lepr(KJJI\ ~ (L) {L epr(K} =7 (3 M, N) [M fa liN Epr(L)~ N tpr(M)Jl}
Now the next axioms follow. /MA3/
(a,K,L)[KeKl(al~LfKl(a) =9KeL]
The axiom states that mereological classes are unique. /MA4/
(a,K) { K E.a
=>(3 L )
[L Hl(ajJ
1
The axiom postulates the class existence. No new rules of inference are added in mereology. One of the mereological theorems is the following formula (a) [a s a ~ a e pr(al] which states that only individuals are parts of themselves. At first this theorem was accepted as an axiom but afterwards it was proved to be redundant /Cf. CLAY /1970//. The definition of the notion of mereological class as given above is somewhat cumbersome and it might have been expected to be simplified. As a matter of fact a conjecture was made that it could be substituted by the following more intuitive definition. (a,K) { K EKl(a) ~ K £K II (L) [a E preLl '" K E pr(LJ]} But as R.Clay proved /Cf. CLAY /1966// it is weaker than the definition /DEF/ above. The above system of mereology is consistent which can be proved by many ways. One way is to supply it with an appropriately chosen model An interesting mode~ for mereology was given in CLAY /1968/. The domain of it consists of the set of all real numbers whose decimal expansions contain only zeros and ones with the exception of O. The relation "a is a part of b" means here that every position where a has a one in its decimal expansion, b has it too. As to the real number system it is introduced by Clay axiomatically into ontology. In the above axiomatization Kl(aJ is essentially the least upper bound of the a's. This remark was made by Cz.Lej~wski in 1960 /Cf. for this RICKEY /1976b//. The same was also studied in KUBInSKI /1971/. Many other axiomatizations of mereology have been given. For account see LEJEWSKI /1954/, /1955/ and /1962/. Extensions of mereology. The literature concerning mereology is very rich and it is richer than that on other Lesniewski's systems. It includes, among others, two long works by Lesniewski himself, LEsNIEWSKI /1916/ and /19271931/. One of the best introductions to mereology is SOBOCInSKI /1954/. An extension of mereology providing a basis for developing the arithmetic of natural numbers is given in S~UPEC KI /1958/. Other extensions of mereology are known under the names of
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J. SURMA
atomistic and atomless mereologies respectively. Atomistic mereology is a result of adjoing to mereology the new axiom stating that every object is either an atom or a mereological class constructed from these atoms which are the mereological elements of this object. Atomless mereology, on the other hand, consists of mereological axioms enriched by the new axiom statying that no atom exists in the field of mereology. For account see SOBOCInSKI /1971/ where it is shown, among others, that the definitions of atoms given by Schroeder and Tarski /Cf. TARSKI /1935// are equivalent in mereology. Applications of mereology. First application of mereology goes back to 1929 when A.Tarski used it to form a foundation for the geometry of solids. He proved that axioms for Euclidean geometry may be given in mereology with the primitive "solid" added if solids were interpreted as open /solid/ spheres. Points were defined by Tarski as distributive las characterized in ontology/ classes of concentric spheres introduced by means of rather sofisticated definitions /Cf. TARSKI /1929//. These definitions have been simplified afterwards in JAsKOWSKI /1948/. Tarski's idea has been extensively developed by T.Sullivan who obtained similar results for affine geometry where solids are interpreted as parallelepipeds /Cf. SULLIVAN /1971// and for projective geometry where solids are interpreted as convex quadilaterals /Cf. SULLIVAN /1972/ and also /1973//. Mereology seems to serve not only in the geometry of timespace solids, but also in formalization of empirical sciences like geometry of events, some chapters of theoretical biology /Cf. WOODGER /1937//, phonology /Cf. BATOG /1967// and others. On relation of mereology to complete Boolean algebra. In TARSKI /1935/, Footnote 1, one may find: "The extended" /i.e., complete/ "Boolean algebra is closely related to the deductive theory developed by Stanislaw Lesniewski and called by him mereology. / ... /. The relation of the part to the whole, which can be regarded as the only primitive notion of mereology, is the correlate of Booleanalgebraic inclusion". This is then detailly discussed in GRZEGORCZYK /1955/ where the following formal statement can be found: "The models of mereology and the models of complete Boolean algebra with zero deleted are identical". This statement has been subjected to a strong criticism in CLAY /1974/. Clay states that Grzegorczyk's proof is faulty and that the system Grzegorczyk describes as mereology is in fact not Lesniewski's mereology.Lesniewski's description of mereology was presented in colloquial Polish since it was published during his presymbolic period. The Lesniewski's expression "a is an ingredient of b" that is "a is a part of b or a is identical' with b" is symbolized in GRZEGORCZYK /1955/ as "a ingr b", i.e., by considering "part" as a binary relation and by eliminating ontological epsilon from mereologyaltogetheL Clay insists, on the contrary, on symbolizing the Lesniewski's expression as "a E pr(b) ". The next Clay's criticism of Grzegorczyk says that the later describes ontology as an elementary theory and in fact proves that just this elementary ontology shares the same models with complete Boolean algebras with zero deleted. CONCLUDING REMARKS Stanislaw Lesniewski's efforts to solve the problem of antinomies resulted in the construction of what he called the New System of the foundations of mathematics, distinguished by originality, comprehensiveness and elegance, a pioneering achievement of the 20ties. Lesniewski played a considerable role during the period of elabora
WORK AND INFLUENCE OF LEsNIEWSKI
213
rating modern tendencies of mathematical logic and the foundations ofmathematics. He was the forerunner and originator of many ideas now incorporated into logical and foundational textbooks. But Lesniewski's writings are done in a highly condensed and difficult style, most cumbersome in practice, his famous terminological explanations are hardly intelligible. He invented a special symbolism, the so called wheel and spoke notation the use of which was an additional factor determining his isolation on the international scene. This is why Lesniewski's systems have not been so popular as they deserved. Another reason is that general trends of logical research had meanwhile drifted away from "system building" to metalogical investigations mostly of first order languages. All this is quite unfortunate but the fact remains that Lesniewski·systems are not generally accepted as a tool in the foundational practice, and they" are not the systems a mathematician in the street makes use of. But on the other hand, Lesniewski'Swork has greatly influenced the very phflosophy of logic and of the foundational studies. In this field Lesniewski had worked out an original point of view he had called the "intuitionistic formalism" which he characterized by these sentences: "I might take this opportunity to point out that. for many months I have devoted a considerable expenditure of systematic work towards the formalization of these systems / ••• / through a clear formulation of their directives using a number of the auxiliary terms whose meaning I fixed in the terminological explanations / •.. /. Having no predilection for various "mathematical games" that consist in writing out according to one or another conventional rule various more or less picturesque formulae which need not be meaningful, or even  as some of the "mathematical gamers" might prefer  which should necessarily be meaningless, I would not have taken the trouble to systematize and to often check quite scrupulously the directives of my system, had I not imputed to its theorems a certain specific and completely determined sense, in virtue of which its axioms, definitions, and final directives / .•. / have for me an irresistably intuitive validity" and further "I know no method more effective for acquainting the reader with my logical intuitions tban the method of formalizing any deductive theory to be set forth. /Compare for this LEsNIEWSKI /1929/, p.78/.
214
J. SURMA
REFERENCES Abbreviations: A~ Acta Universitatis Wratislaviensis, Wroc!aw, Poland, CL Collectanea Logica, Warsaw, Poland. This journal and its first inaugural volume destroyed with its Warsaw printing house during the World War I, CR Comptes Rendus des Seances de la Societe des Sciences et des Lettres de Varsovie, Classe III, FM Fundamenta Mathematicae, JCS Journal of Computing Systems, JSL Journal of Symbolic Logic, NDJFL Notre Dame Journal of Formal LogiC, NorthHolland Publishing Company, NH PF Przeglqd Filozoficzny /Philosophical Review/, poland, PSASA Polish Society of Arts and Sciences Abroad, London, RF Ruch Filozoficzny /Philosophical Movement/, Poland, Studia Logica, Poland SL AJDUKIEWICZ K. 1935 Die Syntaktische Konnexitaet. Studia Philosophica,1/1935/,127. ANDREWS P. 1963 A reduction of the axioms for the theory of propositional types. FM,52/1963/,345350. BARHILLEL Y. 1950 On syntactic categories. JSL,15/1950/,116. BATOG T. 1967 The axiomatic method in phonology. Routledge and Kegan Paul, London, 1967. BOUDREAUX J. 1976 Settheoretic models for Lesniewski's logical systems. XXllnd Conference on the History of Logic,59 July, 1976. Jagiellonian University and Polish Academy of Sciences. Krak6w,1976. pp.25 /Abstract/. CANTY J.T. 1969a The numerical epsilon. NDJFL,10/1969/,4763. 1969b Lesniewski's terminological explanations as recursive concepts. NDJFL,10/1969/,337363. 1976 The proper interpretation of ontology. XXllnd Conference on the History of Logic,59 JUly,1976. Jagiellonian Univ. and Polish Academy of Sciences. Krak6w, 1976,pp. 68 /Abstract/. CHURCH A. 1940 A formulation of the simple theory of types. JSL,5/1940/,5668. 1956 Introduction to mathematical logic. Vol.I. Princeton University Press, 1956. CHWISTEK L. 1923 The theory of constructive types.
WORK AND INFLUENCE OF LEsNIEWSKI
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Annales de la Societe Polonaise de Mathematique,2/1923/,948. CLAY R. 1966 On the definition of mereological class. NDJFL,7/1966/,359360. 1968 The consistency of Lesniewski's mereology relative to the real numbers. JSL,33/1968/,251257. 1970 The dependence of a mereological axiom. NDJFL,11/1970/,471472. 1974 Relation of Lesniewski's mereology to Boolean algebra. JSL,39/1974/,638648. COUTURAT L. 1905 L'algebre de la logique. Paris, 1905. CRESSWELL M.J. 1973 Logics and Languages. Methuen, London, 1973. DAVIS Ch.C., Jr. 1975 An investigation concerning the HilbertSierpinski logical form of the axiom of choice. NDJFL,16/1975/,1451B4. FLOYD W.F. and HARRIS F.T.C./Ed./ 1964 Joseph Henry Woodger, Curriculum Vitae; Form and strategy in Science, studies dedicated to Joseph Henry Woodger on the occasion of his Seventieth Birthday. D.Reidel, 1964. FRAENKEL A.A., BARHILLEL Y. and LEVY A. 1973 Foundations of set theory. NH,1973. GOODELL J.D. 1952 The foundations of computing machinery. JCS,1/1952/,113, Part 11,1/1953/,86110. GRZEGORCZYK A. 1955 The systems of Lesniewski in relation to contemporary logical research. SL,3/1955/,7795. 1964 A note on the theory of propositional types. FM,54/1964/,2729. HENKIN L. 1950 Completeness in the theory of types. JSL,15/1950/,Bl91. 1963 A theory of propositional types. FM,52/1963/,323334. 1975 Identity as a logical primitive. Philosophia.Philosophical Quartely of Israel,5/1975/,3145. HI~
1961
H.
Congrammaticality, Batteries of Transformations and Grammatical Categories. In: JAKOBSON /1961/,pp.4350.
216
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JAKOBSON R./Ed./ 1961 Structure of language and its mathematical aspects. Proc.12thSymp.Appl.Math.Providence, R.I.,Amer.Math.Soc.,1961. JAsKOWSKI S. 1948 Une modification des definitions fondamentales de la geometrie de corps M.A.Tarski. Annales de la Societe Polonaise de Math.,21/1948/,298301. KABZInSKI J.K. and WROnSKI A. 1975 On equivalential algebras. Proc.1975 Intern.Smp.on MultipleValued Logic,Indiana Univ., Bloomington. Indiana, May 1316,1975,pp.419428. KOTARBInSKI T. 1929 Elementy teorii poznania, logiki formalnej i metodologii nauk. Lvov, 1929. Translated into English as "Gnosiology  the scientific approach to the theory of knowledge", London, 1966. 1957 Wyklady z dziej6w logiki. £6dz, 1957. 1965 Garstka wspomnien 0 Stanislawie Lesniewskim. RF,24/1964/,155163. KOWALSKI J.G. 1975 Lesniewski's ontology extended with the axiom of choice. ph.D. Thesis under B.Sobocinski. Univ.of Notre Dame,Indiana, USA, /unpublished/. KUBInSKI T. 1971 A report on investigations concerning mereology. AUW,139/1971/,4768. KUENG G. and CANTY J.T. 1970 Substitutional quantification and Lesniewskian quantifiers. Theoria,36/1970/,165182. KURATOWSKI K. and MOSTOWSKI A. 1968 Set theory. NH, 1968. KRUSZEWSKI Z. 1925 Ontologia bez aksjomat6w. PF,28/1925/,136 /Abstract/. LEHRBERGER J. 1974 Functor analysis of natural language. Mouton, 1974. LEJEWSKI Cz. 1954 A contribution to Lesniewski's mereology. PSASA,5/1954/,4350. 1967 A note on a problem concerning the axiomatic foundations of mereology. NDJFL,4/1962/,135139. 1955 A new axiom for mereology. PSASA,6/1955/,6570. LEONARD H.S. and GOODMAN N. 1940 The calculus of individuals and its uses. JSL,5/1940/,4555.
WORK AND INFLUENCE OF LEsNIEWSKI
217
LEsNIEWSKI S. 1911 Przyczynek do ana1izy zdan egzystencjalnych /A contribution to the analysis of existential propositions/. PF,14/1911/,329345. 1912 Pr6ba dowodu ontologicznej zasady sprzecznosci /An attempt to prove the ontological principle of contradiction/. PF,15/1912/,202226. 1913a Logiceskia razsuzdenia /Logical studies/. St. Petersburg, 1913, 83 p. 1913b Czy prawda jest tylko wieczna czy tez i wieczna i odwieczna? /Is truth only eternal or both eternal and temporal?/. Nowe Tory,18/1913/,493528. 1913c Krytyka logicznej zasady wylqczonego srodka /A critique of the logical principle of the excluded middle/. PF,16/1913/,315352. 1914a Czy klasa klas, nie podporzqdkowanych sobie, jest podporzqdko~ wana sobie? /Is the class of classes not subordinate to themselves subordinate to itself?/. PF,17/1914/,6375. 1914b Teoria mnogosci na "podstawach Filozoficznych" Benedykta Bornsteina /The theory of sets on the "philosophical foundations" of Benedykt Bornstein/. PF,17/1914/,488507. 1916 Podstawy og6lnej teo~ii mnogosci. I. /Foundations of the general theory of sets. 1/. Works of the Polish research group in Moscow, Mathematics and Science division, No 2. Printed by A.P.Poplawski. Moscow, 1916, 42p. 19271931 0 podstawach matematyki /Foundations of mathematics/. PF,30/1927/,164206; 31/1928/,261291; 32/1929/,60101; 33/1930/,77105; 34/1931/,142170. 1927a Introduction. PF,30/1927/,164169. 1927b Section I. PF,30/1927/,169181. 1927c Section II. PF,30/1927/,182189. 1927d Section III. PF,30/1927/,190206. 1928 Section IV. PF,31/1928/,261291. 1929 Section V. PF,32/1929/,60101. 1930a Section VI. PF,33/1930/,7781. 1930b Section VII. PF,33/1930/,8286. 1930c Section VIII. PF,33/1930/,8790. 1930d Section JX. PF,33/1930/,90105. 1931a Section X. PF,34/1931/,142153. 1931b Section XI. PF,34/1931/,153170. 1929a Ueber Funktionen deren Felder Gruppen mit Ruecksicht auf diese Funktionen sind. FM,13/1929/,319332. 1929b Ueber Funktionen,' deren Felder Abelsche Gruppen in bezug diese Funktionen sind. FM,14/1929/,249251. 1929c Grundzuege eines neuen Systems der Grundlagen der Mathematik. §1§11. FM,14/1929/,l81. 1930 Ueber die Grundlagen der Ontologie. CR,23/1930/,lll132. 1931 Ueber Definitionen in der sogenannten Theorie de~ Deduktion. CR,24/1931/,289309. An English version of this paper has been published in McCALL /1967/. 1938a Einleitende Bemerkungen zur Fortzetzung meiner Mitteilung u.d.T. "Grundzuege eines neuen Systems der Grundlagen der Mathematik.
218
J. SURMA An
English version of this paper has been published in McCALL
/1967/. 1938b Grundzuege eines neuen Systemsder Grundlagen der Mathematik. §12. CL,1/1938/,61144. LEWIS D. 1972 General semantics. Semantics of natural language, ed.by Davidson and Horman. D.Reidel, 1972. LUSCHEI E.C. 1962 The logical systems of Lesniewski. NH,1962, 361p. l.UKASIEWICZ J. 1910 0 zasadzie sprzecznosci u Arystotelesa. Studium krytyczne. Krak6w,1910. 1929 Elementy logiki matematycznej. Warszawa, 1929. English translation as "Elements of mathematical logic" edited by Pergamon Press, 1963. 19281929a Rola definicji w systemach dedukcyjnych. RF,11/19281929/,164 /Abstract/. 19281929b 0 definicjach w teorii dedukcji. RF,11/19281929/,177178 /Abstract/. l.UKASIEWICZ J. and TARSKI A. 1930 Untersuchungen ueber den Aussagenkalkuel. CR,23/1930/,3050. MACHOVER M. 1966 Contextual determinancy in Lesniewski's grammar. SL,19/1966/,4757. MARTIN N.M. 1953 On completeness of decision element sets. JCS,1/1953/,150154. McCALL S. /Ed./ 1967 Polish Logic. Oxford, At the Clarendon Press, 1967. MORSE A.P. 1965 A theory of sets. Academic Press, 1965, 130p. MOSTOWSKI A. 1948 Logika matematyczna. WarszawaWroclaw, 1948. MYHILL J. 1953 Arithmetic with creative definitions by induction. JSL,18/1953/,115118. NEUMANN J. von 1927 Zur Hilbertschen Beweistheorie. Mathematische Zeitschrift,26/1927/,146. 1931 Bemerkungen zu den Ausfuerungen von Herrn St.Lesniewski ueber meine Arbeit "Zur Hilbertschen Beweistheorie". FM,17/1931/,331334. PRIOR A. 1965 Existence in Lesniewski and Russell. Formal systems and recursive functions, ed. by J.Crossley and M.Dummett. NH, 1965. 1971 Objects of thought. Oxford, At the Clarendon Press, 1971. QUINE W. van Orman 1956 Unification of universes in set theory.
WORK AND INFLUENCE OF LEsNIEWSKI
219
JSL,21/1956/ RAMSEY F.P. 1925 Foundations of mathematics. Proc.of the London Math.Soc.,ser.2,25/1925/,338384. RICKEY V.F. 1972 An Annotated Lesniewski Bibliography. Department of Mathematics,Bowling Green State University, Bowling Green, Ohio, USA. Preliminary version 1972. Supplement I 1976. 1975 Creative definitions in propositional calculi. NDJFL,16/1975/,273294. 1976a Model theory for Lesniewski's ontology. XXIInd Conference on the History of Logic, 59 July, 1976. Jagiellonian University and Polish Academy of Sciences, Krak6w 1976, p.24 /Abstract/. 1976b A survey of Lesniewski's logic. Department of Mathematics, Bowling Green State University, Bowling Green, Ohio, USA, June 1976. SCHARLE T.W. 1971 Completeness of manyvalued protothetic. JSL,36/1971/,363364 /Abstract/. SINISI V.F. 1966 Lesniewski's analysis of Whitehead's theory of events. NDJFL,7/1966/,323327. SLUPECKI J. 1953 St.Lesniewski's protothetic. SL,1/1953/,44112. 1955 St.Lesniewski's calculus of names. SL,3/1955/,773. 1958 Towards a generalized mereology of Lesniewski. SL,8/1958/,131163. SOBOCInSKI B. 1934 0 kolejnych uproszczeniach aksjomatyki "ontologii" proLSt.Lesniewskiego. Fragmenty Filozoficzne, /1934/,143160. 1950 L'analyse de l'antinomie Russellienne par Lesniewski. Methodos,1/1950/,94107; 2/1950/,220228; 3/1950/,308316; 2/1950/,237257. 1954 Studies in Lesniewski's mereology. PSASA,5/1954/,3443. 1956 In memoriam Jan Lukasiewicz. Philosophical Studies, Maynooth, Ireland,6/1956/,349. 1971 Atomistic mereology. NDJFL,12/1971/,89103. SULLIVAN T.F. 1971 Affine geometry having a solid as primitive. NDJFL,12/1971/,161. 1972
The name solid as primitive in projective geometry. NDJFL,13/1972/,9597. 1973 The geometry of solids in Hilbert spaces. NDJFL,14/1973/,575580. SURMA S.J. 1973a A survey of the results and methods of investigations of the equivalential propositional calculus. Studies in the history of mathematical logic, ed.by S.J.Surma. Ossolineum, Wroclaw 1973,pp.3361. 1973b A uniform method of proof the completeness theorem for the equivalential propositional calculus and for some of its extensions. Studies in the history of mathematical logic, ed.by S.J.Surma.
220
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Ossolineum, Wroclaw 1973,pp.6379. An algorithm for axiomatizing every finite logic. Computer science and multiplevalued logic, ed.by D.Rine. NH,1976,pp.137143. STONE M.H. 1937 Note on formal logic. American Journal of Mathematics,59/1937/,506514. TARSKI A. 1923a a wyrazie pierwotnym logistyki. PF,26/1923/,6889. 1923b Sur Ie terme primitif de la logistique. FM,4/1923/,196200. 1924 Sur les truthfunctions au sens de MM.Russell et Whitehead. FM,5/1924/,5974. 1929 Les fondements de la geometrie des corps. Ksi~ga Pamiqtkowa Pierwszego Polskiego Zjazdu Matematycznego, Krak6w 1929, pp.2933. 1933 Poj~cie prawdy w j~zykach naUk dedukcyjnych. Travaux de la Societe des Sciences et des Lettres de Varsovie, Classe III,34/1933/,116p. German translation as Der Wahrheitsbegriff in der formalisierten Sprachen published in Studia philosophica,1/1936/,261405. 1935 Zur Grundlegung der Booleschen Algebra. I. FM,24/1935/,177198. 1956 Logic, semantics, metamathematics. Papers from 1923 to 1938, translated by J.H.Woodger. Oxford, 1956, 469p. TAX R.E. 1973 On the intuitionistic equivalential calculus. NDJFL,14/1973/,448456. WOODGER J.H. 1935 The axiomatic method in biology. Cambridge University press, 1937, 174p. 1976
R. Gandy, M. Hyland (Eds.), LOGIC COLLOQUIUM 76
© NorthHolland Publishing Company (1977)
INFERENCE WITHOUT AXIOM OR PARADOXES
George Temple (Oxford)
Theories of Implication The study of formal, propositional logic has known three great periods the Greek, the Mediaeval Scholastic and the Modern, which are, respectively, commonly associated with the names of their reputed founders:
Philo of Megara, Abelard and Frege.
In each period a number of different theories of implication have been advanced of which the most important are (1)
the theory of material implication;
(2)
the theory of incompatibility, and
(3)
the theory of inclusion.
All three of these theories are attributed to the logicians of the Greek school of Megara in the treatise by Sextus Empiricus ("Outlines of Pyrrhonism", Book ii, 110112, ca. A.D. 200).
Philo is credited with the
theory of material implication, according to which, a propositionp always implies a proposition
~
unless
p
is true and
~
is false.
An unnamed
Stoic, perhaps Chrysippus, is said to have introduced the notion that implies~
if
/J
is incompatible with the negation of
~.
have not been identified are said to define the implication that
'1
is virtually included in
p
p
And aome who
P
~ ~
to mean
Among the Scholastic logicians all three views are expounded at great length and with considerable acuity in the works formerly attributed to Duns Scotus and especially in William of Ockham. In modern times the theory of material implication is to be found in a casual remark by C S Pierce (Collected Papers, vol. III, para. 4413, first
221
GEORGE
222
TEt~PLE
published in The Monist, vol. VII, 1896) accepted by Russell (l~he Principles of Mathematics", 1903, para. 16) and formally expounded by Wittgenstein and Post.
The theory of incompatibility was advanced by
C I Lewis but sUbsequently withdrawn.
The theory of inclusion
reappears in the work of Pierce, and can be discerned in the writings of H McColl. The purpose of this note is to show that a careful analysis of these three theories shows that they are not merely mutually compatible, but essentially the same, the superficial differences exhibiting only a shift of emphasis.
Philonian Implication We do not possess any of the original works of the Megarian logicians and therefore do not know for certain how they formulated the theory of material implication, but it seems indubitable that they initiated the study of unanalyzed propositions, which were classified as either "True" or "False" accordingly as they corresponded or did not correspond with reality. This unique scheme of valuation was fatal to their theory of inference. Philo of Megara (ca. 300 B.C.) recognised three varieties of valid inference, viZ. from a true antecedent to a true consequent, from a false antecedent to a false consequent, and from a false antecedent to a true consequent.
This is undoUbtedly a complete classification, but it is
difficult to believe that it was accepted as a definition of inference. I cannot believe that any Greek politician, barrister or tradesman can ever have sought to persuade his adversary, his jUdge or his client that a false proposition implies any proposition (true or false), and that a true proposition is implied by any proposition (true or false). In fact what is called "Phil on ian" implication is completely ineffective as a definition, and the Megarian logicians used in its place various schemes of inference Which we should undoubtedly recognise today as completely satisfactory and sometimes of surprising subtlety. They also used the unique valUation of propositions as true or false to characterise disjunction and conjunction, but there is no evidence that they were under the illusion that they had prOVided formal definitions of these connecti ves.
223
INFERENCE WITHOUT AXIOM OR PARADOXES We owe so much to the historical researches of Jan Lukasiewicz (especially the paper "On the History of the Logic of Propositions", first published in 1934, Selected Works, 1970, translation by S McCall,
pp. 197217), and I M Bochenski ("Formal Logic", 1956, Eng. trans. by Ivo Thomas, A History of Formal Logic, 1961) that it seems ungracious to criticise their accounts of Megarian logic, especiallY as they are followed by so many writers in generously giving Philo the credit for the introduction of "truth tables".
But in fact the theory of truth tables
depends entirely on a system of the multiple valuation of propositions, and cannot, be constructed from the system of unique valuation, by which each and every proposition is ineluctably labelled as "true" or "false". It was perhaps an obscure appreciation of this fact (or was it just ignorance of the Greek logicians?) that guided the mediaeval logicians to ignore the theory of Philonian inference and to replace it by a theory of incompatibility  which we discuss below.
Scholastic Conseguence The voluminous researches of mediaeval logicians are remarkable for their fertility and subtlety, and for their complete independence, and indeed, ignorance of the work of the Stoic and Megarian philosophers. Conjunction and disjunction were loosely described in terms of the truth or falsity of their component propositions, but the problem of inference, now called "consequence", was minutely examined, not in isolation but in relation to the close1y related concepts of incompatibility, compossibility and necessity.
To summarise an extensive literature, a proposition
said to be a consequence of a proposition incompatible wi th
~
is
p , if the negation of'f. is
p .
This doctrine is essentially the same as that given by C I Lewis ("Implication and the Algebra of Logic",
!:!!.!!.2,
21, 1912, pp. 522531), and the
scholastics draw the same conclusions, viz. that an impossible proposition implies every other proposition and that a necessary proposition follows from any proposition. But whereas Lewis recoiled from these consequences, the Scholastics accepted them with enthusiasm. Neither the Scholastics nor Lewis gave any definition of incompatibility,
GEORGE TEMPLE
224
which was accepted as a primary concept.
NeoPhilonian Inference Although almost all the nineteenth century logicians, with the exception of C S Pierce (ccri ected Works, vol. III, "The Regenerated Logic", The ~,
vol. VII, 1896, pp. 1940, paras. 441443), were ignorant of the
Stoic and Megaric school, they unknowingly adopted the prime principle of that distinguished body, namely that any proposition is either true or false.
Now "truth" and "falsity" have no place in formal logic, nor,
strictly speaking has the concept of absolute "proof".
Formal logic is
concerned only with the inference from one proposition? to another proposition
~
, notwithstanding that it is equally concerned with
inferences from the negation of
~
to the negation of
p.
The gradual
realisation of this essential of formal logic is one main characteristics of the "modern" school of logicians.
Henceforward logic is concerned
with syntax rather than with semantics. It might appear that this exclusion of the concepts of truth and falsity would be fatal to the theory of Philonian implication, but this theory can be amended by replacing the concepts "true" and "false" by "deemed to be true" and "deemed to be false". Frege begins his "Begriffshrift" in 1879 by a neoPhilonian definition but it is significant that instead of saying "A is true" or "A is false" he says "A is affirmed"
or "A is denied".
Even Bertrand Russell who wrote as a convinced Philonian in his Principles of Mathematics seems to have accepted the neoPhilonian doctrine in Appendix C to the second edition of Principia Mathematica (vol. I, p. 661, 1925) where he writes When we say truth or falsehood is, for logic, the essential characteristic of propositions, we must not be misunderstood.
It does not matter, for mathematical logic,
what constitutes truth or falsehood;
all that matters is
that they divide propositions into two classes according to certain rules. I could wish that the language of Wittgenstein in his "Tractus Logico 
225
INFERENCE WITHOUT AXIOM OR PARADOXES
Philosophicus" (1921) was more transparent, but since he speaks of "truthpossibilities" I think he is fairly counted as a neoPhilonian.
McColl and Inclusion The transitive character of the relation of implication (viz. that and 'Y.)o
implies P ~;z
'1
)
p"'"
'r
must have encouraged many logicians to define
implication as a species of "inclusion" by analogy with the theory of sets of points.
However, I have singled out Hugh McColl ("The Calculus of
Equivalent Statements and Integration Limits", Proc. Lond. Math. Soc., vol. IX (1871), pp. 920, 177186, vol. X (1878), pp. 1628, vol. XI (1880) pp. 11321, XXVIII (1897), pp. 55579, vol. XXIX (1897) pp. 98109, Note, vol. XXX (1898), pp. 32032.
"Symbolic Logic" (1906) ) as the representa
tive of this school of thought because of two other characteristics of this remarkable and neglected worker:
firstly McColl must rank as an independent
creator of the propositional calculus, for he rediscovered the algebra of symbolic logic in complete ignorance of the work of Boole, Jevons and Venn and long before Pierce.
Secondly, he is to my knowledge the only logician
who has ever used his logic to solve any problems of practical utility to the working mathematician.
In fact McColl devised his system of symbolic
logic expressly "to determine the new limits of integration when we change the order of integration or the variables in a multiple integral". Considered in its full generality this is an intricate problem not discussed in any of the great treatises of analysis, but solved completely and algebraically by McColl, without any appeal to geometric visualisation. McColl bases his theory on the undefined relation of the "equivalence" of two proposi tions,
P and if , which he writes as p ~ '/ , and takes to be a
symmetric and transitive relation.
The conjunction (P'f) and disjunction
(pU,/ ), called by McColl "compound" and "indeterminate" statements are loosely defined by impli~it truthtables, and the implication Leibnitzian relation for inc lusion, f'" P
f
4~
by the
McColl's verbal definition in
Rule 2 leaves something to be desired, but his algebraic formulation is impeccable, and includes the fundamental relations, f., =
where
pi denotes
If'l)';:
I:> (1 Ui') the negation of /,
•
U{PCf/'
/>/9
J
(
1
1
1
)
GEORGE TEMPLE
226 Simple and Multiple Valuation
The Philonian theory of material implication implies that any proposition is either true or false, and hence gives rise to the so called paradoxes that a false proposition implies all other propositions, and that a true proposition is implied by all other propositions.
This theory may be
called the theory of "simple valuation", since each proposition is given one and only one value, viz. either true or false. Now it is easily shown that the simple valuation is irrelevant in formal logic.
Consider for example one form of the argument "reductio ad
absurdam".
Here we are given a proposition
proposition '1'
/>
and required to deduce a
The argument requires that we can show that
the negation of q
are incompatible and that" must imply
't.
'1
is "true" and
l'
and 'fj j,
is "false".
1/
and
The truth or falsity of
or 'fl are irrelevant, for, if we insist on saying that have proved that
to
, together imply if ' whence we deduce that
p, 9
p is "true", then we
But the argument proceeds
from the "false" proposition, p,/I to the "true" proposition "t.
However,
according to the theory of material implication, no argument is necessary, for a false proposition implies any proposition and and
1>1'
would imply
't
q I.
This example makes nonsense of the theory of simple valuation as a basis for a theory of implication, but it also shows how the latter can be revised. In the "reductio ad absurdam", and in mathematical arguments generally there is no question of the "truth" or "falsity" of the premises or conelusions.
All that is needed is a statement of which propositions are
asserted and which are denied.
!>Jq,
proposi tions , or f> ~ 'i, .... ,
l. J
',1
I f we are considering a collection of
we can begin by asserting
..
or "', 'I " 'T, 'T', ~,. " ... '\.,
~
1/ /'>',
bination of ously /~ and
jo, 9, 7.,
.,. J
or !, '1, ... " .. , or indeed any comexcept that we must not assert simul tane
/>' , or 'f and.,' or '\. and
""l ',
To indicate which propositions are asserted we can introduce a "valuation function"
reI) and while
, such that '«ft): ( ~O,)",
'"C(f') t
0
r((o ,)
if if
,.
is asserted
/
for all propositions
I
is denied
p
INFERENCE WITHOUT AXIOM OR PARADOXES
227
and their negations Such a valuation function represents a simple valuation of the propositions taken as premises and provides a syntactic alternative to the unacceptable semantic concepts of "truth" and "falsity". (There is no need to introduce the arithmetic concepts of unity and zero, which can be replaced by elements
W
and
of a two element Boolean
~
algebra. ) The criterion for material implication can be expressed in terms of a valuation function in the form,
P
implies
~
if
"C:((» r{'i) ,
"7:"((»
which yields at once the alternatives either //:""{p) . 0 or "U) =: I and
'T{q) =
t ,
But this form of the criterion is just as useless as the verbal form so long as we are restricted to one and only one truth function, i.e. to a simple valuation of propositions. But there is no reason for such a restriction.
A truth function merely
provides an arbitrary classification of propositions into two exclusive classes:
those which are asserted and those which are denied.
In fact the
real significance of the familiar "truth tables" introduced by Wittgenstein and Post is that they display the four possible truth functions from a pair
"
of proposi ti ons ,
"<:j ((» 1i,.(
L
"
!»
'fj (j» 'l""
,
and '(
«
c;
to the two element algebra of 0 and 1, viz.
,
I .
",(If)
G. ('1)
=:.
0
"1 (q)
z:
0
'l;,. ('1)
s:
o·I (
)
.
O.
The simultaneous use of all four truth functions provides a system of "multiple" valuation and gives an effective definition of implication, in the form,
p implies'j if
rr: C/»
for each truth function 'tI, "L' "), 7.,.
(In Wittgenstein's exposition
these different valuations represent the different "truthpossibilities" of the propositions
p
and
1.)
In modern expositions, these valuations are
called "realizations". The use of truth tables, or the equivalent truth functions
",,0) provides
effective definitions of the logical connectives and validates the usual axioms of equivalence and implication, but it becomes excessively
GEORGE TEMPLE
228
complicated when more than three propositions are involved.
It therefore
seems well worth while to construct a simpler system of multiple valuation.
The Hierarchy of Propositions The systematic use of truth functions is crucially dependent on the recognition that propositions form an inductive hierarchy, in the sense that there is a primary or basic collection of propositions from which all other propositions are formed by negation, conjunction and disjunction, or by Sheffer's connective.
Once the truth function
~
has been specified
for the basic set, and suitable truthfunctional definitions have been given for the connectives, the truth function propositions.
is then specified for all
~
This is one of the great contributions of Wittgenstein, Post
and Lukasiewicz.
The first formal proof that the extension of a truth
function from primary propositions to all other propositions can be effected in a consistent manner is due to Bernays. It occurred to me that, if we consider the collection of basic propositions
PI> r.',
f>,.)
fl,', ...
then each truth function
""
is the "characteristic
function" or "indicator" of a certain collection of basic propositions, namely those for which
'n..(a)
=. /
~)
and that these collections poss eaa a
number of useful properties:(1)
each collection is selfconsistent, in the sense that if then '(~ CV) ~ 0;
(2)
each collection is complete, in the sense that if
9
proposition not included in the collection, then
c,;: (q)
(3)
any pair of collections are incompatiblp., in the sense that there is always some
p
(~L~) ~ /
is any z..
0;
such that if /> belongs to one such
collection, then (./ belongs to the other. I therefore propose to study these collections and call them "atoms", as they turn out to be the ultimate irreducible units of which the propositional calculus is composed.
Wittgenstein called the basic
propositions "atomic", but this seems inappropriate since, if
i'
and "l
are any basic propositions, then r > (f>{J'l) (p V'll) My theory of inference depends on a study of collections of atoms, which I call "clusters", and I therefore proceed to introduce this theory in an informal and suggestive manner, reserving a rigorous and formal treatment for a sUbsequent paper.
229
INFERENCE WITHOUT AXIOM OR PARADOXES The Theory of "Clusters" In the light of the preceding brief survey of the main theories of
inference it is possible to present a unified theory which harmonises the different definitions and, incidentally, eliminates all logical axioms. I propose to define inference as a species of inclusion, and equivalence as mutual inference.
I define inclusion in terms of compatibility and I
define compatibility by recurrence.
Thus I establish a mapping of a
propositional calculus into a Boolean algebra. The primary elements of a propositional calculus are unanalysed propositions (the "atomic" propositions of Wittgenstein).
These basic propositions have
no structure and hence cannot be treated as if they were sets, with obvious To make the basic propositions
definitions of union and intersection.
amenable to mathematical investigation they must therefore be provided with a structure, or rather, a superstructure. We therefore begin with basic propositions
p,/, ",/,...
PI) />"...
and their negations
(The suffixes 1, 2, • • . are mere indices and have no
numerical significance.)
We are in no position as yet to define
conjunction and disjunction but we can define a complete collection of compatible basic propositions
C as
a collection
(ql' '<,., '"
is either f>", or
where each '<,< to
C
for all k
1>
{ PI)
t; / and
/>>< or />i< / belongs
where either
Thus if the basic propositions are
A.') 'fFP:..?
/>,/ )
the complete consistent collections are ((./,p,,)) (p,,?,') 0,/) Pa, ) , It is convenient and appropriate to call these complete consistent collections "atoms" for they will form the irreducible, or irresoluable units from which we shall build our algebra. From these atoms we next construct certain "clusters". atoms which "adheres" to a basi.c proposition the atoms which include
PI) the clusters are
;'1/)
?tt (or 1>.. , ;,./
p,,/).
CPu P,) , (PI! b,./) ci«, P... /) CP/J P,.) )
(p" ",)
CPt PI"
(Pl.' ;"/)
I I
(t>,/, f?/)
PtC
(or
The "cluster" of
p" / )
consists of all
Thus, if the basic propositions are adhering to adhering to
h, /,,',
adhering to f> ... J adhering to ["~f.
GEORGE
230
TEI~PLE
These clusters form the superstructure which enables us to make the transition to a species of set theory and to Boolean algebra. We shall denote the cluster adhering to a basic proposition
P.e
~/
PK
(or
p" /)
by
), and henceforward we shall define / pI' a basic proposition to be a couple (I>KI p,. ) or (/>1<1 k).
the associated capital letter
(or
We can now define the conjunction ab of two basic propositions, a and b, as the triad consisting of (1)
a
(2)
b
(3)
the intersection of the clusters A and B.
This latter is defined as the collection of atoms which include both a and b, and will be written as AB. Similarly we define the disjunction o.U i> of two basic propositions, a and b, as the triad consisting of (1)
a
(2)
b
(3)
the union of the clusters A and B.
This latter is defined as the collection of atoms which include either a or b. or both a and
b,
and will be written as A V 3.
Clearly we can proceed to define propositions in general by recurrence, with the obvious definition that the negation
LS~ 5) of a proposition
is the couple consisting of (1) S'
(2)
the complement S ' of the cluster ~... )
the latter being the collection of all atoms which do not include
s.
We thus succeed in mapping all propositions into an algebra and by a Bourbakian abuse of language we shall speak of the proposition
P ,where
is the collection of atoms specifying the proposition in question. for the basic set 13 defined above we identify {(PI}P.J)
(I'/If',') />2 with the cluster
fll>. with PI P,
or
t;
/>, or
P
Thus
with the cluster t; or
{O.,
P,), ((>'' !>,'}
and
{(Put.)]·
The crucial step is to define two propositions P and Q to be equivalent if they consist of the same collection of atoms and then we write P"'Q. with an obvious extension of our notation for disjunction, (p~)1' U(PQ~ P'Q, [>'4J') for the collections
(/>''1/»)
form the complement of
(/,',9),
((';'1).
(P',,;')
Thus,
INFERENCE WITHOUT AXIOM OR PARADOXES
231
It is an easy exercise to verify that the propositions so defined form a Boolean algebra, in which conjunction and disjunction are commutative, Moreover each proposition is double
associative and distributive. idempotent in the sense that
P?
P }
s:
'?U'P '"
P.
Moreover there is a unique null proposition ~ which contains no atoms, and a unique universal proposition
pp
(
V
which contains all atoms, such that
v.
1\
s;
It is easily verified that the conjunction of all the propositions, say ~1,aL'
...
, in an atom consists of all these propositions themselves
together with the element of the Boolean algebra which is the intersection of the clusters
A" A",...
Hence each basic proposition is expressed in
the "disjunctive normal form".
For example p, is mapped into the
collection of atoms, say A,13/,_ each of which includes
P,
Each atom
corresponds to a conjunction of propositions consistent with
f,
and the
whole collection corresponds to a disjunction of these conjunctions.
A Synoptic View of Theories of Inference The theory of clusters enables us to show that there is essentially only one possible definition of inference, although there
ar~
a number of different
ways in which it can be expressed. Any two propositions,
f
and ~
,are represented by two clusters P
each of which consists of atoms.
and ~
There is only one possible transitive
relation from p to 'I , viz. 'P , the cluster of atoms which adhere to />
is
included in Q, the cluster of atoms which adhere to 1 (or, of course, vice versa!).
Therefore we
P =
~
the implication (>
~
'i
to mean that
PQ
This definition implies that P and that .r:(p) ~ 'l:'(I')r{?).
=
("y
Hence our definition agrees with the definition in terms of multiple valuation, and is clearly the same as McColl's definition by inclusion. The defect of the definition of "strict implication" is that Lewis gave no definition of "incompatibility". following definitions,
To remedy this defect we propose the
GEORGE TEMPLE
232
f> '" !\ ,
P is a "contradiction", if empty c.Lus t ers of atoms;
thus!'r
1. e. if it is represented by the
,=
(2)
p and q are "incompatible" if f>1 = 1\;
(3)
p is a "tautology" if f:'1
j
PI
are incompatible;
, Le. if p is represented by the cluster of
f'~V
all atoms.
11
thus p and
Then the definition of strict implication is that P~'Y if r i:> /\. To show that these definitions are equivalent, we note that =: ;\, pl\ 'i' ::then 1'1' =: if p '" I''/p(,!Uq') f> V = then and i f 1',' : /\ P =
p,
(p,)
u({"I' )
s:
o«: U 1\
:;
ri
As regards the "paradoxes" of implication, it is true that, if p is any proposition, then
Il
z
Il,o
whence P
whence
and
~
V'.
Thus there is a unique null proposition, viz. the contradiction
A,
which
implies any proposition, and there is a unique tautological proposition
V'"
(f>lJ;.')
which is implied by any proposition.
right and proper.
This however is only
It seems that Lewis recoiled from these truisms
because his system did not reveal that all contradictions are "equivalent" to
A and all tautologies to
~
All the usual axioms of inference can now be established as theorems.
For
example to establish the syllogism, we express the inferences ;. ... If and
1'" L in r »>
the forms P =
(>'1
and '1 '" 11. whence
!' '"
f>li'): (1'1h :: f> ~ and
An Assessment of Cluster Theory The advantages of the cluster theory are (1)
It provides effective definitions of the logical connectives, and especially of implication and equivalence, and
(2)
It eliminates all the usual axioms of equivalence and inference, and the usual rules of inference.
There is a further advantage which has not been exploited in this paper. Here we have taken the basic propositions to be undefined elements, except that the collection of basic propositions is "polarised", in the sense that, if it contains an element p it also contains a complementary element ~ {
INFERENCE WITHOUT AXIOM OR PARADOXES and
(p'/ ~ P.
233
Now the cluster theory can also be developed for an
"unpolarised" basic set, in which case the concept of negation is absent and we obtain "posi ti ve" logic.
Or the cluster theory can be developed
for a "layered" basic set. in the sense that if it contains an element p it also contains elements
!' I)
P
~
cr")
I
~
p,
in which case we obtain a three valued logic. be developed in the same way.
Clearly nvalued logic can
R. Gandy, M. Hyland (Eds.), LOGIC COLLOQUIUM 76 © NorthHolland Publishing Company (1977)
LEOPOLD LOWENHEIM: LIFE, WORK, AND EARLY INFLUENCE Christian Thiel Philosophisches Institut RWTH Aachen, Germany
Speaking at a logic colloquium in 1976, I may presume my audience is well acquainted with the fact that any firstorder theory which has a model at all has also a denumerable model. There is abundant information on this result, on the background of the equivalent t h e cr c m of Lowenheim and on its extensions clustering into something like a theorem of Lo w e n h e i m  Skolem  Tarski  Godel  Malcev  Takeuti  Vaught etc .. But very little seems to be known about the discoverer of the basic fact. I do not mean to claim this as a singluar case in the history of modern logic. I am pretty sure that one would meet with similar problems if he tried to write the biography of Heinrich Behmann, Adolf Lindenbaum, or Emil Leon Post, even though people who have talked to or corresponded with them or who were their colleagues are still amongst us. Out of pity for the despair to Which a future historian of logic might be driven, I am going to present some material on the life of Leopold Lowenheim as well as on his writings and  though scantily  on their reception with the experts. I hasten to add that my pertinent in~uir ies are by no means finished. As, on the other hand, I do not want to bore you with already published material, I will first give a duly shortened summary of my report of 1975 1 , and then produce the outcome of some more recent investigations. Finally, I will try to scan the scene in mathematical logic mainly between Lowenheim 1915 and Skolem 1920/1922, in hope of ending up with a fair judgement on Lowenheim's influence or his noneffectiveness during the blossoming of his research and publishing activity. Throughout, I will take care to indicate what I consider the more promising directions of further historical advances. Even the first steps have proved to be entangling: the available information on Lowenheim is not only scarce, it is also partially wrong. Till very recently, the only reliable biographical source was Poggendorff's Biographisches Handworterbuch which in volume 5, covering the time between 1904 and 1922, contains the few data submitted by Lowenheim himself. 2 But this' volume appeared in 1926, and from then on one had to rcly on circumstantial evidence like the date of receipt by the JSL of Lowenheim's 1939 article Einkleidung der Mathematik in Schroderschen Relativkalkul, signed "Berlin  Lankwi tz". For subsequent years, all biographical data seemed to have simply vanished. Among the most prominent reference works, Gillispie's Dictionary of Scientific Biography does not have an entry on Lowenheim in volume 8 where it should alphabetically occur,3 nor does Meschkowski's MathematikerLexikon;" likewise, the Britannica of 1963 and 1968 5 and the German encyclopedias of Brockhaus 6 and Meyer 7 skip Lowenheim. We may disregard the fact that Zischka's Allgemeines GelehrtenLexikon 8 ignores him, too, for it does not mention Frege, Godel, Skolem, Tarski, or Zermelo either; but the gap in Kurschner's Deutscher GelehrtenKalender 9 is certainly deplorable.
235
236
CHRISTIAN THI EL
No wonder that people came to think that Lowenheim, especially as his surname clearly indicated some Jewish descendance, must have fallen victim to the Nazis and possibly have met his death through them like Adolf Lindenbaum." So, Lowenheim's name in the index of Fraenkel and BarHillel's Foundations of Set Theory of 1958 11 came to be supplemented by the data "1878  c.1940", the authors believing Lo w e n h e i m to be among the victims of the Third Reich and taking the date of Lowenheim's 1939 article as the greatest lower bound. Apparently, it was this information that was taken over in Hunter's Metalogic of 1911. 12 Now, Fraenkel and BarHillel had been cautious enough to signalize the precariousness of their second date by the prefix "circa". Others did not take such care, and so today we find the ostensibly established data "Leopold Lowenheim (18181940)" in the 1965, the 1966 and the 1912 edition of the Kleine En z g k l.opd d i e / Mathematik, 13 the "source" presumably being the index of BarHillel and Fraenkel  the latter of whom, incidentally, had himself been erroneously reported to have died at Jerusalem in Lense's book of 1949. ~ Even in 1914, this lack of data can be seen in the "Reminiscences of Logicians" at the Summer Research Institute on Algebra and Logic at Monash University at Clayton, Australia. 15 Actually, Lowenheim survived the years of darkness and was able to resume his position in Berlin in 1946, teaching up to the 72 n d year of his life. He died in Berlin on the 5th of May, 1957. We do not know very well (but see below) how Lowenheim found his way to the algebra of logic to which he devoted all of his published articles and abstracts (in the sense of Selbstanzeige), and most of his reviews and notices. It is tru~ that his father, Dr.Louis (or Ludwig) Lowenheim had been a teacher of mathematics at the polytechnic (Gewerbeschule) at Krefeld, but it seems he did not engage in mathematical research or publications, much less in such an exotic field as algebra of logic. Moreover, in 1881, he took his wife, a writer who is said to have acquired some renown already under her maiden name Elisabeth Rohn, and their threeyear old son Leopold first to Naples in Italy and then to Berlin where he lived as a private scholar (Privatgelehrter). There he prepared a comprehensive work on Democritus' influence on modern science which he hoped would gain him a teaching posi tion at the HumboldtUni versi tat. 16 When he died in 1894, Leopold was sixteen years old and still attended the Gymnasium. Having taken his final examinations in 1896, he matriculated in mathematics and the sciences at the HumboldtUniversitiit, where he enthusiastically studied synthetic geometry with Hermann Amandus Schwarz, and at the Technische Hochschule in Berlin~Charlottenburg where he may have been a student of Emil Lampe, to whom I will return in a moment. Lowenheim finished his studies in 1900'and began the normal career of a German Gymnasiallehrer, teaching as Oberlehrer at the JahnRealgymnasium in Berlin for 15 years from 1904 onwards, interrupted only by WW I when he was sent to France, Hungary and Serbia as a soldier, and then being appointed Studienrat in 1919 and (according to Johannes Teichert, v.infra) Professor soon afterwards. Only in passing will I mention his biographically most important decision to join the Anthroposophische Gesellschaft in 1924 and his marriage to Johanna Teichert, a widow to whose son Johannes Teichert we owe the bulk of biographical data on Lowenheim. I will also skip over the circumstances of Lowenheim's dismissal as "25% nonAryan" in 1934, his penurious life as a teacher of eurythmics and, at the anthroposophical School for Eurythmics in Berlin, an instructor in geometry, as well as the death of his wife in 1937 and the years after the war  ~ll these events have been extensively documented in my abovementioned paper of 1975. I will, however, take the opportunity to refer to further biographical data relating to Lowenheim's research
LEOPOLD LOWENHEIM
237
to which I now turn  with a warning that I will not be able to go very deep into the subject here. I am aiming rather at an informative survey than at a modernized systematic exposition or even at an evaluation which, in my opinion, desirable as it certainly is, will not be available without further longterm and thorough study. For my purposes, I will look at Lowenheim's writings systematically rather than chronologically. I will say something on 1. the 9 published articles (19081946), 2. the hitherto unnoticed selfwritten abstracts (19131919), 3. the reviews and notices, most of them unnoted so far (19091923), only mentioning the unpublished writings listed in the bibliography, i.e.those on geometry (except the paper "Logic of simplicity or logic of plenitude? "). Let me begin with a remark on the background of the published articles. I noted above that we do not know exactly how Lowenheim came to occupy himself with algebra of logic, although we have Lowenheim's autobiographical note stating that, shortly after taking up his teaching, he "became acquainted with the calculus of logic through reviews and from Schroder's books"17. But this sufficed to involve Lowenheim so deep in the subject matter as to reorganize it, extend it in different directions, discover his "development theorem" and his wellknown results on the satisfiability of formulae in firstorder logic with equality, on the decidability of its monadic fragment, and the reduction of the decision problem to firstorder logic with binary predicators only. I have not succeeded in finding anyone in the mathematics department of the HumboldtUniversitat or of the Technische Hochschule BerlinCharlottenburg between 1896 and 1900 who had shown interest, let alone had offered courses in algebra of logic or related topics (as, e.g., Frege did at Jena). Being fully aware of a certain danger of oversimplification, I dare say that in Germany, algebra of logic waS almost exclusively a matter for Studienrate. Ernst Schroder, though having had his Habilitation at the Eidgenossische Polytechnikum at Zurich, had been teaching in Gymnasien from about 1865 to 1874 when he got a call to the Technische Hochschule Darmstadt, two years later changing to the Technische Hochschule Karlsruhe where he taught for the rest of his life. Eugen MUller, the editor of the later parts of Schroder's Vorle~ungen Uber die Algebra der Logik and of the Abriss der Algebra der Logik, did his editorial work in addition to his obligations as professor at the Realgymnasium at Constance, and had pUblished two earlier treatises on the algebra of logic as a supplement to the annual report of the Grandducal Gymnasium at Tauberbischofsheim in 1899/1900 and 1901. Alwin Korselt, another contributor to the algebra of logic, though better known for his defense of Hilbert against Frege in their controversy on the foundations of geometry, taught exclusively at secondary schools, eventually at the Realgymnasium at Plauen in Saxony. So Lowenheim, who by the way corresponded with both Muller and Korselt and discussed his 1910 paper with them before pUblication,~ fits well into this tradition. His first article "Uber das Aufl0sungsproblem im Zogischen Klassenkalkul" appeared in the Sitzungsberichte der Berliner Mathematischen Gesellschaft, having been presented  perhaps more extensively and in slightly different form  to the members of the society on the 29th of April and on the 27th of May in 1908, under the title
"Bericht Qber die Aufl0sung von Gleichungen und die Elimination in der Algebra der Logik". ~ Lowenheim is listed as a member of the so
ciety (which was founded in 1901)H since 1906, and I have a suspicion
238
CHRISTIAN THI EL
that the aforementioned professor Emil Lampe, who edited the Jahrbuch Uber die Fortschritte der Mathematik "in cooperation with the Berlin
Mathematical Society"21 and was one of its prominent and active members, can be held responsible not only for Lowenheim's entry into that circle but also for Lowenheim's later contributions to the Jahrbuch, these suddenly coming to an end after volume 46.1 for 1918, the year of Lampe's death. u In a certain sense, this year signals at the same time the end of Lowenheim's publishing activity, for of the two later articles of 1939/1940 and 1946, the second one is only a revised version of a much earlier paper as I will show later on. First, I will sketch the content and meaning of the published articles of Lowenheim.
Item 1, Uber das AufZosungsprobZem im Zogischen KZassenkaZkuZ (1908) takes up some investigations of Schroder and Muller on the solvability of equations in the calculus of classes. Lowenheim proves several theorems stating necessary and sufficient conditions for the solvability of particular symmetric equations. An assertion of Schroder's concerning the symmetry of a certain solution is shown to be false. Result (4a) later came to be known as the "general development theorem" ("Das Lowenheimsche allgemeine Entwicklungstheorem") and constitutes § 127 of Schroder's Abriss der AZgebra der Logik. 2 3 Maybe item 2, Uber die AufZosung von GZeiehungen im Zogisehen GebietekalkuZ (1910), also belonged to the lecture given before the Berlin Mathematical Society in 1908. "Domains" (Gebiete) are the subsets of a given fixed set M (the universe); Schroder's Abriss is presupposed to be known. Making USe of the work of earlier authors, including Jevons and Johnson, three methods for the solution of equations in the cRlculus of domains are explained. Some theorems of Korselt and Muller on disjunctive (i.e., disjoint and complementary) systems are connected together and a more general form of the development theorem is formulated and proved to hold. Theorems 14' and 14" state that with the help of the development theorem, not only formulae of the calculus, but also metatheorems on deducibility and satisfiability can be proved  a feature later generally claimed by Lowenheim as an advantage of the PeirceSchroder system of the algebra of logic over its competitors. Item 3, Uber Transformationen im GebietekaZkUZ (1913), is the continuation of item 2 which the reader is presumed to be familiar with. The PeirceSchroder calculus of relatives is extended to a calculus of matrices of domains, of which the basic idea is as follows. In Peirce and Schroder, a binary relative, i.e. a binary relation, is determined iff for every ordered pair of elements from the field of the relation it is determined whether the relation holds of this pair, or not. Arranging the pairs in a matrix, (al ,a2)
(a 1, a 3 )
(a2 , a2 )
(a2 , a 3)
(a 3 ,a2 )
(a3,a3)
... ...
)
a relation R will be determined by putting "1" in place of each pair of elements standing in the relation R, and putting "0" otherwise. T~e PeirceSchroder calculus of relatives may be viewed as a technlque of calculating with square matrices of ones and zeros Lowenheim states that all formulae valid in this PeirceSchrod~r cal
LEOPOLD LOWENHEIM
239
culus remain valid for arbitrary domains instead of 1 and 0, since, thanks to the verification theorem the validity of an equivalence or implication in X l , ' " ,x n is e q u i v a l e n t to its coming out true under all valuations of the domains Xl, ••• 'X n by 0 and 1. L6wenheim sets about to develop a theory for his generalized calculus of matrices of domains, the calculus of domains being included as a special caSe since every system of domains al,'" ,am and every single domain b can be written as
, resp ..
and
The verification theorem is formulated so as to hold for the extended calculus. As matrices can be used to represent transformations, they are put to use for solving the inversion problem for transformations of a given normal form F =
... ,
aN ]
x r , . •.
'X
n
into another given one G
the ai and b· being the coefficients of the occurring ntuples of simple or negat~d variables as in
[a,
b,
C,
d ]
xy
=
a xy + b xy +
C
xy + d xy
.
As a byproduct of the skilful exploitation of the verification theorem and of L6wenheim's development theorem, together with his USe of disjunctive systems and reduction, a considerable shortening and simplification of proofs is attained. Finally, L6wenheim proudly demonstrates, in § 3, how many theorems from Whitehead's Memoir on the Algebra of Symbolic Logic~ can now not only be established by much shorter proofs but also be essentially extended. The most important result, according to L6wenheim himself, is a metatheorem stating that, under certain precautions in the case of cardinality statements, every theorem about transformations of arbitrary objects is va1id if only its validity can be shown for transformations of domains (p.261). L6wenheim hopes "to turn the thoughts of this treatise to rich profit in the calculus of relatives" (p.212), but postpones the harvest to a later publication; except that a generalization of a theorem proved in the paper is communicated, without proof, at the end. (1913), Potenzen im Relativkalkul und Potenzen allgemeiner endlicher Transformationen [1J, an a c q u a i n t a n c e v i t h the previous paper being assumed. This proof was given in a second article of the same year
More important than the rather special theorem is the method employed. In the last paper, L6wenheim had already interpreted certain special elementary matrices as permutations, and implicitly suggested a generalization for elementary matricesin general. Still pushing on, he now further generalizes the concept of permutation by a notation for relatives where just the places of the Ones in each column are listed, o indicating that the column does not contain any 1 at all. E.g. ,25
CHRISTIAN THIEL
240
(i
0 0
1 0
1 1 1
0 0 0 0
0
)
[ 2 4
.......
3 ,
o ,
1 2 3 ] .
'"
Now, this relative can be written as a generalized permutation
(~; ~ ~)

representing the replacement of the number 1 by the pair 2 4 , the

number 2 by 3, the number 3 by nothing (!), and the number 4 by the triple 1 23 A calculus for this new symbolism for relatives is then established, Lowenheim claiming that in this way the theory of permutations is being embedded in the theory of relatives and thus extended to a general theory of transformations so far unheard of in mathematics, and likely to enrich mathematics proper by a plenitude of new theorems. To support this claim, the theorem mentioned is finally interpreted as a theorem in number theory. I am inclined to asSume that Lowenheim's lecture to the Berliner Mathematische Gesellschaft on November 12th, 1912, titled "The calculus of relatives in the algebra of logic as a generalization of the theory of permutations" (Der ReZativkaZkUZ in der Algebra der Logik alB Verallgemeinerung der Permutationslehre) anticipated the considerations of [7J Another continuation of the transformation paper of 1913 is contained in Lowenheim's paper Uber eine Erweiterung des GebietekalkUls, iael.c he auch die qeunhn l i ch e Algebra umfasst [16J, published in the Archiv fUr SyBtematische PhiZosophie for 1915. The principal idea is another extension of the calculus of domains of which the ordinary calculus of domains as well as the ordinary theory of numbers (transfinite ones not necessarily excluded) and elementary algebra appear as special cases. Following Whitehead, Lowenheim considers the elementary symmetric functions of n domains a 1,··. ,an' 81 8
8
8 All 8
v
2
3
n
I
a
K
K
L
a
KfA
L KfAfll
K
a
A
a K a A all
a 1 a 2· .. an
with v > n are defined to be 0, and the ordered system
(8 1,8
8 8 n + 1' ' ' ' ) is put in correspondence with a socalled 2,,,,, n, superdomain (Ubergebiet) a = a 1 ++ a 2 t+ ••• ++ an ("++" read "superplus"). As each domain such as a 1 corresponda to the system of elementary functions 81
8
v
a1 ' 0 for V > 1
LEOPOLD LOWENHEIM
241
Liiwenheim identifies a1 with this superdomain (a1, 0, 0, ... ), thereby (with suitable definitions of equality, addition etc.) embedding the calculus of domains in the calculus of superdomains. 26 Having proved a selection of theorems in the latter, Liiwenheim proposes to visualize the concept of superdomain by "a certain similarity" with Riemann surfaces. The other paper of 1915, llb e i MogZiahkeiten im ReZativkaZkUZ [13J, is the one containing the famous theorem. As an English translation by BauerMengelberg is accessible in van Heijenoort's Source Book, with an illuminating commentary by the editor, and is perhaps Liiwenheim's best known paper, I do not plan to analyse the proof here. However, I feel somebody should some day devote himself to this task, and also try to find out whether the law of infinite conjunction, a proof of which is missing in Liiwenheim's paper, was not considered evident by all the writers on the algebra of logic, while the necessity of a proof for it may have escaped them just because of the symbolism favoured by the PeirceSchriider tradition Which sometimes, one might say, thrived upon extrapolations from the finite to the infinite. Theorem 1, opening § 1 of the paper, states that not every equation in the language of firstorder logic with equality can be expressed in the calculus of relatives, i.e. with symbols for relations only individual variables or constants (except 0 and 1) lacking. This result of Korselt's, which was emended by Tarski in 1941, is furnished with the outline of a complete proof and supplemented by a discussion of some examples of firstorder equations valid or invalid only in certain finite individual domains. For Lowenheim, this leads up to the consideration of firstorder equations which are valid in every finite individual domain but not valid generally, i.e. also in infinite individual domains; Liiwenheim calls them "fleeing equations". By a rather complicated argument, he shows that for every equation of this type there are values of its predicate letters such that the equation is not valid in any denumerably infinite domain: F finitely valid II
F valid
... F 1't o  v a l i d
or, classically equivalent, F satisfiable
... F finitely satisfiable
v F
~osatisfiable,
which is the formulation more familiar today. Liiwenheim does not pay as much attention to this theorem in itself as we all feel it deserves; his first thought is that, as an application, all independence problems concerning axiom systems for the calculus of domains can in principle be decided in a denumerable universe. Liiwenheim reports to have shown this for the axiom system of MUller, the results being ready for publication. Likewise seemingly underestimated by Liiwenheim is his next result that amounts to the decidability of the monadic fragment of firstorder logic with equality. Only the third important discovery receives Liiwenheim's wholehearted attention. This is, as the heading of § 4 announces, the "reduction of the higher calculus of relatives to the binary", which turns upon theorem 6 stating that every relative or firstorder equation is equivalent to a binary one. As Liiwen
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CHRISTIAN THIEL
heim is convinced that "every theorem of mathematics, or culus that can be invented, can be written as a relative (p.463l, it follows that "one can decide the truth of an mathematical proposition provided one can decide whether lative equation is identicallY satisfied or not" (ibid. l. this subject to turn to L5wenheim's paper of 1919.
of any calequation" arbitrary a binary reI leave
Simply titled Geb1~etsdeterminanten [27J, it parallels the paper on powers in the calculus of relatives in being another and, moreover, the proper continuation of the paper on transformations [5J. It is interesting to note that the articles so connected,
Solution in CD (1910) Transformations in CD (1913) Powers in CD (1913)
Determinants in CD (1919)
are precisely the writings mentioned by Lo w e n h e i m in 1940 ([36J, p . 1 l as those which aimed at rendering parts of the logical calculus smoother and more flexible, but which had passed unnoticed. As for the lastmentioned paper of 1919, this can be easily accounted for: the subject matter, determinants for systems of linear equations in the calculus of domains, must have appeared quite esoteric to potential readers, in spite of close analogies to properties of the usual determinants of linear algebra which, for their part, would hardly have excited the readers of the Mathematisehe Annalen in 1919. So it is not hard to explain how parts of the paper also went unnoticed, in which theorems on transformations are shown to be valid far beyond the calculus of domains  namely, for very general systems of functions, such as those on arbitrary domains, not necessarily everywhere defined, and possibly manyvalued.
Einkleidung der Mathematik in Sehrodersehen Relativkalkul [36} is that paper which I consider was composed last among the published writings of Lowenheim; it was published in the JSL in 1940 in German. In Lowenheim's opinion, the abandonment of the PeirceSchroder calculus in favour of the PeanoRussell system has doubtless led to tremendously important results, but it meant, at the same time, the renunciation of harmony and beauty in mathematics which had promoted a kind of certainty of instinct for topics significant and fruitful for mathematics proper. Above all, however, the Schroder calculus avoids the settheoretic antinomies since the concept of a set of sets is not legitimate in that system. Yet, all mathematics, including set theory, can be formulated within the Schroder system or, to use Lowenheim's unintentionally amusing expression, all mathematics can be "verschrodert", i.e. "schroderized". Referring to his result about the reducibility of ternary and higher relatives to binary ones, mathematice appears to Lowenheim like a building with three levels, level 1 comprised of quantifierfree or firstorder propositions, level 2 consisting of all propositions with at least one quantifier over unary relatives, i.e. monadic predicates, and level 3 embracing the remaining propositions with quantifiers over binary relatives. Russell's logicisation of mathematics is in principle accepted; the problem is how to transpose it into the Schroder system or rather, into the modified and simplified system of Lowenheim. The question as to sete of sets is considered answered by 11correlating with each
LEOpoLD LOWENHEIM
243
set M which is an element of another set M, an element m "representing" M, and studying, instead of the set M, the set M' of the representa+ives of the elements of M. For this purpose, a relative E is introduced such that M is represented by miff g(E i m = Mi)' Writing this as 1\ x.E(x,m)++xE: M., the affinity to the axiom of subsets or Aussonderung is obvious. Of course, not every subset of the universe has a representative (otherwise we would end up with Russell's antinomy). So, dealing with a particular domain of problems, one has to determine a suitable set of all representatives needed. It matters much to Lowenheim that in introducing his representatives of sets, his purpose was neither the axiomatization of set theory nor the es~ape from its antinomies, but solely the logicisation of mathematics by "s c h r o d e r i z a t i o n !". ~s examples of how this task could be performed, Lowenheim treats Cantor's diagonal procedure, some simple calculi like that of domains and of relatives, and a more sophisticated type of calculus. As Cantor's set theory happily survives in Schroder's system, "one need not let oneself be expelled from the paradise of Cantorian set theory by intuitionism", and intuitionism as a universal demand seems to Lowenheim to be a useless restriction. On the other hand, the concept of intuitionistic provability has also i's legitimate place in the Schroder calculus, and, as expressed by Lowenheim's beautiful formulation, "we need not let ourselves be expelled from the paradise of intuitionism either" (p.2). Lowenheim's last published paper appeared in 1946 in an English translation by w.V.Quine with the title On making indirect proofs direct [37J. As Quine says in his introduction, i L is a contribution to "elementary practical points of mathematical methodology" (p.125). Lowenheim discusses several types of proof by reductio ad absurdum and claims that every such indirect proof can be turned into a direct one by either (1) the method of complete contraposition, (2) the method of incomplete contraposition, or (3) the method of generalization. Actual proofs from mathematical texts are taken as examples. The manuscript of this paper found its way to the editors of Scripta Mathematica after quite an odyssey. Documenting this, there are two letters of Lowenheim to Bernays and one letter in the other direction which Professor Bern~ys was kind enough to let me read and quote. In his letter of May 25th, 1937, Lowenheim agrees ~o have his "treatise on indirect proof" published, and adds a word on its origin: "I lectured before the B~rlin Mathematical Society on November 28th 1917, and shortly afterwards the treatise was cast in its final form by adding the critical discussion on Hessenberg, a matter pointed out to me after the lecture."27 With his letter o'f March 15th, 1939, Bernays returns the manuscript which he had recommended for publication to the Mathematische Annalen without success; he finds it "very satisfying t'at Mr.Tarski will bring the treatise out". I assume that Lowenheim had himself asked for the manuscript back in order to send it to Tarski for Fundamenta Mathematicae, which would fit well with an autobiographical remark of Lowenheim's, saying: "Shortly before the Second World War, I published a paper on logical calculus in an American journal. Of another one destined for Warsaw, I had even read the proofs; whether the war has prevented its publication, I do not know." In April 1946, Professor Ginsburg told Professor Fraenkel that he had received "a German manuscript by Lowenheim"u some years ago, but could not find a translator. Fraenkel, who was to lecture at Harvard soon afterwards, offered to talk the matter over with Quine. In a letter
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CHRISTIAN THIEL
of August 7th, 1965, Profe~sor Quine remembers that he got the manuscript from Fraenkel "with the request that I find some student of minG who might translate it. Knowing of no such student, I did it myself." So much ings, I new. Up or with
for the published treatises. On the rest of Lowenheim's writmust of necessity be briefer. There is, however, something to now, nobody seems to have met with either the abstracts the short reviews contributed by Lowenheim to the Jahrbuch Uber die Fortschri&te der Mathematik which I mentioned before. T would not call their discovery sensational in any respect, all the more as their length, or better shortness, ranges from 11 to 23 lines only, but they certainly enable us to form a considerably better picture of Lowenheim's fields of interest and competence. From the abstracts I have drawn information on Lowenheim's particular ranking of his own diverse results and, indirectly, on the selfappreciation of his work in general. The paper of 1908 on the solvability problem had been pUblicized by a substanceless sevenline review by Lampe, and the related 1910 article by twelve lines from the same reviewer, in the Jahrbuch. But there are abstracts written by Lowenheim on the transformation paper [5] of 1913 as well as of its continuations [7J of 1913 and ~7J of 1919, the latter being somehow misplaced in the volume for 1916 to 1918. Equally due, I presume, to the circumstances of the time is the appearance of two abstracts of the famous 1915 paper on possibilities in the calculus of relatives, both written by Lowenheim, one prematurely in the Jahrbuch volume for 1913 (published, however, in 1918), and a somewhat longer second one in the volume for 191415. Among Lowenheim's reviews and notices of works and papers by other authors, those in the Archiv der Mathematik und Physik were known before and should be separated from the necessarily shorter ones in the Jahrbuch. The reviews in the Archiv on Parts I and II of Schroder's Abriss der Algebra der Logik are very carefully and thoroughly written; Lowenheim's remarks on the FregeHilbert controversy over the axiomatical approach show his sense for philosophically important key subjects in mathematical and logical research. (Of course, they also mirror Lowenheim's simultaneous correspondence with Frege to be talked ab~ut in a minute.) Essentially, Lowenheim welcomes the Abriss as making a good advance, but he finds fault with the omission of Schroder's clear separation of the calculus of domains from the calculus of propositions which leads to a faulty conception of disjunction and negation depending on a theory of "validity points" for propositions in time. Secondly, Schroder's axiomatics for the calculus of domains is shown to suffer from a serious weakness, sum and product being defined only for two domains at a time. Thus Schroder is "unable to reach infinite products and sums without a f a L'l.a c y " (p.72). Even if the definitions are expanded, "the rules of calculation for infinite sums of propositions have to be proved", and this must be done, since "already in the general calculus of domains the extension of axioms and proofs to infinite products and sums is absolutely necessary, otherwise falling short of the most important applications" (ibid.). Spicy annotations, in face of the modern critique of Lowenheim's later proof of his famous theorem. H The third review in the Archiv is on Padoa's "La logique dfiductive dans sa derniere phase de developpement", where Some interesting remarks are made on the importance of a perspicuous notation in logic and mathematics and On the comparison of Peano's symbolism with that of the Schroder school (with which, of course, Lowenheim sides).
LEOPOLD LOWENHEIM
245
The twenty short reviews and notices in the Jahrbuch are mainly concerned with works on logic and foundations and their history, the only real exception being J.Pfluger's book on the beauty of simple geometrical figures (Die Formenschonheit einfacher geometrisaher Gebilde), showing an early interest on L6wenheim's side for related questions. Of the authors reviewed, I mention only BuraliForti, Hugo Dingler, Philip E.B. Jourdain, Norbert Wiener, and Henry M.Sheffer. A look at the unnoticed writings is a suitable startingpoint for my concluding reflection on Lowenheim's recognition or neglect by his contemporaries. A first conclusion can perhaps be drawn from the fact that Lowenheim's papers, it seems to me, had to be reviewed by himself in the Jahrbuch as nobody else was willing or even able to do it  in contrast to other journals, abstracts written by the author are not very numerous in the Jahrbuch. Reviews by others came remarkably late: only the 1940 paper on schr6derization got a review in the Jahrbuch by Karl Schr6ter, and a rather saucy one; but then Curry's review of the same paper in the Mathematical Reviews, the successful competitor to the Jahrbuch since 1940, is, though more Objective, not very favourable either. The same can be said of the four lines devoted by Heyting, in the same review journal, to Lowenheim's paper on indirect proof in 1947. Admittedly, the two last papers do not in fact .count among his strongest; but then the earlier ones should have enjoyed a louder echo while at first receiving almost none. Prior to Skolem's paper of 1920, only Eugen Muller quoted L6wenheim's development theorem in the second part of Schr6der's Abriss in 1910. No reference to L6wenheim or his theorem can be found between 1915 and 1922 in the Mathematische Annalen (except in L6wenheim's oWn paper of 1919), in the Jahresberichte der Deutsahen MathematikerVereinigung, in the Mathematische Zeitschrift and of course not in Crelle's Journal which did not publish a single paper on foundations or set theory within those years. The first edition of Fraenkel's Einleitung in die Mengenlehre in 1919 does not mention Lowenheim at all, nor does the second edition of 1923 which has a footnote on p.187 that "Skolem's weighty lecture became accessible to the author only in the course of the printing of the book". No earlier than in Zehn Vorlesungen tiber die Grundlegung der Mengenlehre of 1927 does L6wenheim's name appear in any writing of Fraenkel. There is still some'unclearness regarding C.I..Lewis' Survey of Symbolic Logia (1918, abridged second edition 1966), the bibliography of which
lists all five papers of L6wenheim up to 1915, except the one on possibilities in the calculus of relatives  four of them decorated with an asterisk as "most important contributions to symbolic logic" (p. 307; see p.315 of the 1966 edition); yet none of them, nor even L6wenheim's name, seems fo appear in the text. The breakthrough of the theorem, if not of its discoverer, seems to have come more than five years after Skolem's continuation of L6wenheim's research. Skolem's paper on the axioms of the calculus of classes 30 published in 1919 but signed May 1917, shows that Skolem had read L6wenheim's paper of 1915 well and refers to L6wenheim's result on the nonexistence of fleeing equations in the monadic predicate calculus as following from a more general theorem proved in this new paper of Skolem who, as we know, worked quite in the Schroder tradition and even with the notation of that school. I take it as an indication of the neglect of L6wenheim personally that Skolem's papers were not reviewed by L6wenheim in the Jahrbuah  whatever the reason for his disappearance from the staff of contributors may have been.
246
CHRISTIAN THIEL
A last question remains: Why did Lowenheim withdraw from further research on the algebra of logic and, as his unpublished writings prove, shift to comprehensive but rather elementary expositions of special fields of geometry? Did Lo ve n h e i ra resign from logical research in bitterness? Let me first state that his oftmentioned isolation was rather an isolation of the whole PeirceSchroder tradition. Schroder had died in 1902, Peirce in 1914, Mliller lived to 1935, and Korselt, after his retirement in 1924, to 1947, but neither of them to my knowledge published anything on the calculus of logic after 1910. And Skolem, as the only and, geographically at least, isolated researcher in the movement, could not singly change the course of history. On the contrary, he was the one to meet with open hostility from Zermelo when the latter in 1931 made a fierce assault on "skolemism, the doctrine that every mathematical theory, the theory of sets not exempted, is realizable in a denumerable model", and severely criticized the "very odd consequences" of what he baptized "the finitist prejudice".n Personally, I would not view Lowenheim as so isolated as he is sometimes pictured. We must not forget that he loosened his contacts with logicians by turning to other fields of intellectual life as a consequence of what he called his spiritual turn. As I mentioned before, he had been in contact with Mliller, Korselt, and Zermelo during the blooming of his logical research, he received visits by Bernays before 1931 and by Scholz and Tarski about 1936, and he was still interested in publishing then. Most fascinating is his exchange of 20 letters with Frege in which he is reported by Bachmann and Scholz to have convinced Frege that an uncontestable foundation of formal arithmetic was possible. The correspondence must be considered completely lost, our knowledge being restricted to the general topic, the dates of most of the letters (19081910), a remark on this correspondence in Lowenheim's first letter to Bernays, and some circumstantial evidence to be found in a volume containing what is left of Frege's collected scientific correspondence.~ Did Lowenheim have unpublished manuscripts He did. Many of them were on geometry, one reason for his work in this field being Rudolf Steiner's inclusion of synthetic geometry in the mathematics curriculum for Waldorf schools. The mathematics teachers of the Waldorf mother school at Stuttgart asked Lowenheim to elaborate parts of synthetic geometry for teaching purposes as he was known to have been very proficient in synthetic geometry since the days of his study with H.A.Schwarz. In the bombing of Berlin on August 23rd, 1943, Lowenheim's flat at BerlinLankwltz was completely destroyed. Among other things, nearly 1200 geometrical drawings and models and two book cases with many unpublished scientific and literary manuscripts were burnt to ashes. We do not even know exactly what these manuscripts were about. It would be a rash conclusion to interpret Lowenheim's tiny remark of 1940  "as long as I had time for 10gistics"33to mean that he had given up his logical work altogether; it may just have meant that he wanted to, or had to, devote the greater part of his time to other things. In 1915,1939 and 1940, Lo w e n h e i.m indicated that he still kept unpublished manuscripts on logic. And in autobiographical notes, he does not conceal a justifiable pride in his logical work. Let me come to an end with two quotations from these reflections which are kindly given to me in letters from Johannes Teichert: "I have a lively interest in logic and have pointed out new paths for science in the field of the calculus of logic which had been founded
LEOPOLD LOWENHEIM
247
by Leibniz but which had come to a deadlock. ( ... ) On an outing, a somewhat grotes~ue landscape stimulated my fantasy and I had an insight that the thoughts I had already developed in the calculus of domains might lead me to make a breach in the calculus of relatives. Now I could find no rest till the idea was completely proved, this giving me still a host of troubles. (oo.) This breach has been scarcely noticed, but all the more some other breaches which I made through my treatise 'On possibilities in the calculus of relatives' were noticed. This became the foundation of the modern calculus of relati v e s , But just in this paper I did not take much pride, for the point had only been to ask the right ~uestions while the proofs could be found easily without ingenuity and imagination. As I was then the only one working seriously in the field, I could gain a respected name among experts in a rather cheap way. My first paper on the calculusof relatives which did not find much attention had cost me much more intellectual exertion. The later works have remained unpublished and, with the eiception of a single one, fallen victim to the ~lames." In a letter to his stepson during the last war, he says: "The loss of the entire calculus of logic is the most painful thing for me." Thus I gather he was still serious about his work in logic. As there is no reasonable hope of enriching our knowledge of Lowenheim's writings extensionally by the discovery of hitherto unknown manuscripts or published writings, historians of modern logic interested in Lowenheim would be well advised to concentrate on clarifying and penetrating his available writings, connecting his ideas up with earlier and contemporary work, and finding the key concepts and methods in Lowenheim which were to become pregnant for the development of presentday lOgic and foundational research. There will be plenty of work to do. NOTES
2
3 4 5 6 7 8 9
Christian Thiel, Leben und Werk Leopold Lowenheims (18781957). Teil I: Biographisches und Bibliographisches. Jahresberiohte der Deutsohen MathematikerVereinigung (subse~uently abbreviated "JDMV") 77 (1975) 19. On p.5, line 13, for "Reylgymnasium" read "Realgymnasium"; on p.7, line 16, for "1909" read "1896". CLalso note 16 below. Poggendorff's Bipgraphischliterarisches Handworterbuch fUr Mathematik, Astronomie, Physik, Chemie und verwandte Wissenschaftsgeb i e t e , Band V: 19041922. II.Abt.: LZ. Berlin 1926, p.759. The latest edition (under the title "Poggendorffs Biographischliterarisches Handworterbuch der exakten Naturwissenschaften") , covering the years from 1932 to 1953, does not mention Lowenheim any longer (cf. Bd.Vlla, Teil 3: LR: Berlin 1959). Charles Coulston Gillispie (ed.), Dictionary of Scientific Biography, vol. VIII (LaneMacqueur), New York 1973. Herbert Meschkowski, MathematikerLexikon, Mannheim/ZUrich 1964, 2.erw.Aufl. Mannheim/Wien/ZUrich 1973. Encyclopaedia Britannica, 1963 edition and 1968 edition. Brockhaus Enzyklopadie, Wiesbaden 1966ff; Elfter Band (LMah)1970. Meyers Enzyklopadisches Lexikon, Mannheim/Wien/ZUrich 1971ff; Bd. 15 (LetMeh) 1975. Gert A. Zischka, Allgemeines GelehrtenLexikon. Biographisches Handworterbuch zur Geschichte der Wissenschaften, Stuttgart 1961. KUrschners Deutscher GelehrtenKalender. Erster Jg., 1925. Berlin/ Leipzig 1925.  2.Jg. (auf das Jahr 1926), Berlin/Leipzig 1926. 3.Ausgabe 1928/29.  4.Ausgabe 1931.  5.Ausgabe 1935.  6.Ausgabe 2.Band, Berlin 1941.  7.Ausgabe Berlin 1950.  8.Ausgabe Berlin
248
10
CHRISTIAN THIEL 1954.  9.Ausgabe Berlin 1961.  10.Ausgabe Berlin 1966 (Lowenheim not mentioned in the necrology of the two lastmentioned editions). Cf. J.H.Woodger's preface to Alfred Tarski: Logic, Semantics, Metamathematics. Papers from 1923 to 1938 (Oxford 1956), p i x . According to Abraham A. Fraenkel, Lebenskreise. Aus den Erinnerungen eines judischen Mathematikers, Stuttgart 1967, p.205, Lindenbaum lived from 1904 to 1941, according to Meschkowski (op.cit.) from 1905 to 1942. A.Fraenkel and Y.BarHillel, Foundations of Set Theory, Amsterdam 1958, p.409. Geoffrey Hunter, Metalogic. An Introduction to the Metatheory of Standard First Order Logic. London etc. 1971, p.189. Kleine En z y k Lo p a d i e / Mathematik, hrsg. v. W.Gellert, H.Klistner, M.Hellwich und H.Kastner, Leipzig und Basel 1965, Zurich/Wien 1966, and Frankfurt a.M./Zurich 1972, p.785. Josef Lense, Vom Wesen der Mathematik und ihren Grundlagen, Munchen 1949, p.67. Reminiscences of Logicians, edited by J.N.Crossley, in J.N.Crossley (ed.), Algebra and Logic. Papers from the 1974 Summer Research Institute of the Australian Mathematical Society, Monash University, Australia, Berlin/Heidelberg/New York 1975 (Lecture Notes in Mathematics no.450), p.28. Louis L6wenheim, Die Wissenschaft Demokrits und ihr EinfluB auf die moderne Naturwissenschaft, hrsg. v. Leopold L6wenheim, Berlin 1914. I have not yet been able to find a copy of this book, but a 48 page paper (ending on page 48 in the middle of a sentence, the text presumably being nothing else than page 1 to 48 of the book) appeared, under the same title, as "Beilage zu Heft 4 des Archivs fur Geschichte der Philosophie, Band XXVI" in 1913. The title "Uber den EinfluB Demokrits auf Galilei", quoted on p.5 of my 1975 paper, is incorrect, this being the title of a paper by Louis L6wenheim published in Archiv fur Geschichte der Philosophie I (1894) 230268. Copies of autobiographical notes by Leopold L6wenheim, including all passages quoted in this paper, were kindly provided by Mr. Johannes Teichert, with a letter of 14 September, 1976. The notes are undated, but all written after 1945. Cf.[3J, p.171. cr . JDMV 17 (1908) 2.Abt., p.62. Cf. JDMV 11 (1902) p.71. I t seems L6wenheim lectured before the Society for the first time on October 30, 1907, this lecture (titled "Einfuhrung in den logischen K'I a s s e n k a Lk ii L'"] being the only one preceding his lecture of 1908. Cf.the title pages, e.g: of the volume quoted in note 19. The Jahrbuch uber die Fortschritte der Mathematik was the immediate predecessor of the Mathematical Reviews and the zentralblatt fur Mathematik and played a comparable role. Emil Lampe died on September 4th, 1918. See Arthur Korn's obituary of E.L. in JDMV.!2 (19141915, pub1.1922) pp.IVII. Lo w e n h e i m is still listed as a member of the Berlin Mathematical Society in 1918, but the publication of the Archiv der Mathematik und Physik, to which the Proceedings of the Berlin Mathematical Society were appended, had to be discontinued in 1920. :n Pt.2, pp.756f in the 1966 joint reprint of Vorlesungen andAbriss. Special cases of the development theorem were mentioned on p.746f, op.cit .. Lowenheim [lJ reports in 1908 that Zermelo claimed to have known the general theorem "for a long time", but it seems it had never been published before. A.N.Whit~head, Memoir on the Algebra of Symbolic Logic, American Journal of Mathematics ~ (1901) 139165 and 297316. i
11 12 13 14 15
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18 19 20
21
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LEOPOLD LOWENHEIM 25 26
27
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30
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33
249
In column 4 of the full matrix of the original paper, p.65, the of row 2 and the 1 of row 4 should be interchanged. Dr.Albert C. Lewis has pointed out to me that a comparison of Lowenheim's construction of superdomains with some ideas of Hermann Grassmann might be promising. I would like to take this opportunity to state my indebtedness and express my gratitude to Dr. Lewis for many helpful and stimulating discussions of the present paper as well as for a large number of more appropriate formulations suggested by Dr. Lewis for ~he final version of the text. Whereas the lecture of November 28, 1917, is documented by the Sitzungsberiehte (151st session), I am puzzled by the reports of the Berlin Mathematical Society in the JDMV where two lectures by Lowenheim under the title "tiber die Yerwandlung eines indirekten' Beweises in einen direkten" are listed: beside that on November 28th, 1917, an earlier one in a doubtful session on July 27, 1917, undiscoverable in the Sitzungsberiohte where June 27, ·1917, is given as the date of the 149th session and October 31, 1917, as the date of the 150th session. The only explanation I can think of is an error of the JDMV concerning the month, Lowenheim's lecture possibly having been scheduled for the session of June 27, 1917, but finally being postponed because of Lowenheim's absence (who at that time served in the army, might have announced his coming, and then been unable to appear). Professor Fraenkel in a letter to the present author of 20 August 1965. See, however, Hao Wang's "digression on Lowenheim 1915" in A Survey of Skolem's Work in Logio, in Th.Skolem, Selected Works in Logic, ed.by J.E.Fenstad, OsloBergenTromso 1970, pp.1752 (the digression on pp.2729). Untersuchungen tiber die Axiome des Klassenkalklils und tiber Produktations und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen. Reprinted in Th.Skolem, Selected WorkS in Logic, op.cit. in n.29, pp.67101. Ernst Zermelo, tiber Stufen der Quantifikation und die Logik des Unendlichen, JDMV 41 (1932) 2.Abt., pp.8588, quotations from p.85. Gottlob Frege, wissenschaftlicher Briefwechsel. Herausgegeben, bearbeitet, eingeleitet und mit Anmerkungen versehen von Gottfried Gabriel, Hans Hermes, Friedrich Kambartel, Christian Thiel, Albert ·Yeraart. Ijamburg 1976. (XXIX. Frege  Lo v e n h e i.m on pp.157161. ) [36], p.7.
o
BIBLIOGRAPHY OF LEOPOLD LOWENHEIM
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5
tiber das Auflosungsproblem im logischen Klassenkalkul. Sitzungsberichte der Berliner Mathematischen Gesellsohaft 7 (1908) 8994.
Review of Ernst Schroder, Abriss der Algebra der L~gik, Erster Teil [Elementarlehre. Leipzig 1909J. Arohiv der Mathematik und Physik, III.Reihe, 15 (1909) 194195. tiber die Auflosung VOn Gleichungen im logischen Gebietekalkul. Mathematische Annalen 68 (1910) 169207. Rezension von Ernst Schroder, Abriss der Algebra der Logik, Zweiter Teil [Aussagentheorie, Funktionen, 'Gleichungen lind Ungleichungen. Leipzig 1910J. Archiv der Mathematik und Physik, III. Reihe, 17 (1911) 7173. tiber Tr~sformationen im Gebietekalklil. Mathematische Annalen 11 (1913) 245272.
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21
22 23
CHRISTIAN THIEL Abstract of [5], Jahrbuch uber die Fortschritte der Mathematik 44 (1913, publ. 1918) 11. Potenzen im Relativkalkul und Potenzen allgemeiner endlicher Transformationen. Sitzungsberichte der Berliner Mathematischen Gesellschaft 12 (1~13) 6571. Abstract of [7T, Jahrbuch uber die Fortschritte der Mathematik 44 (1913, publ.1918) 17. Review of Alessandro Padoa, La logique deductive dans sa dernie~ re phase de developpement. Avec une preface de Giuseppe Peano rParis 1912], Archiv der Mathematik und Physik, III.Reihe, £1 (1913) 360361. Review of Alfonso Del Re, Sulla indipendenza dei postulati dell' Algebra della Logica[Rendiconti dell'Accademia delle Scienze Fisiche e Matematiche, Napoli, (3) 11 (1911) 450458], Jahrbuch uber die Fortschritte der Mathematik 42 (1911, publ.1914) 19. Review of E.Stamm, Beitrag zur Algebrader Logik [Monatshefte fur Mathematik und Physik 22 (1911) 131149J, Jahrbuch uber die Fortschritte der Mathematik 42 (1911) 19. Review of Balakram, The Gene~l Equation in the Algebra of Logic [Journal of the Indian Mathematical Society 2 (1911) 213218], Jahrbuch uber die Fortschritte der Mathematik 42 (1911, publ. 1914) 80. tiber M6glichkeiten im RelativkalkUI. Mathematische Annalen 76 (1915) 441410. English translation by Stefan BauerMengelberg: On possibilities in the calculus of relatives, with introduction by the editor, in Jean van Heijenoort (ed.), From Frege to G6del. A Source Book in Mathematical Logic, 18791931, Cambridge, Mass. 1967, pp.228251. (First) Abstract of [13]. Jahrbuch uber die Fortschritte der Mathematik 44 (1913, publ.1918) 78. (Second) Abstract of [13], Jahrbuch uber die Fortschritte der Mathematik 45 (191415, publ.1922) 108109. tiber eine Erweiterung des Gebietekalkuls, welche auch die gew6hnliche Algebra umfasst. Archiv fur systematische Philosophie ~ (1915) 131148. Review of P.E.B.Jourdain, The development of the theories of mathematical logic and the principles of mathematics [QuarterlY Journal of Pure and Applied Mathematics ~ (1912) 219314J, Jahrbuch uber die Fortschritte der Mathematik ~ (1912, publ.1915) 97. Review of Alessandro Padoa, Frequenza, Previsione, Probabilita [Atti della Reale Accademia delle Scienze di Torino ~ (1912) 878886], Jahrbuch uber die Fortschritte der Mathematik ~ (1912, publ.1915) 98. Review of Cesare Burali~orti, Gli enti astratti definiti come enti relativi ad un campo di nozioni [Atti della Reale Accademia dei Lincei (Roma)(5) £12 (1912) 677682J, Jahrbuch uber die Fortschritte der Mathematik 43 (1912, publ.1915) 9899. Joint review of W.H.Young, On a proof of a theorem on overlapping intervals [Messenger (2) .!±£. (191213) 113118J and Arthur Schoenflies, Entgegnung [Messenger (2) .!±£. (191213) 119121], Jahrbuch uber die Fortschritte der Mathematik 43 (1912, publ.1915) 112. Review of Theophil Henry Hildebrandt,~ contribution to the foundations of Frechet's Calcul fonctionnel [American Journal of Mathematics ~ (1912) 237290], Jahrbuch Uber die Fortschritte der Mathematik 43 (1912, publ.1915) 113. Notice of G~e Vries, Calculus Rationum [Amst.Ak.Versl. 20 (1912) 10571073 and £1 (1912) 11831199], Jahrbuch Uber die Fortschritte der Mathematik 43 (1912, publ.1915) 227. Review of. Eugenio Maccaferri, Le Definizioni per Astrazione e la Classe di Russell [Rendiconti del Circolo Matematico di Palermo
LEOPOLD LOWENHEIM
251
12 24
25
26
27 28 29
30
31 32 33
35
36 37 38
39 40 41 42 43
(1913) 165171J, Jahrbuch fiber die Fortschritte der Mathematik 44 (1913, publ.1918) 75.
Review of P.E.B.Jourdain, The development of the theories of mathematical logic and the principles of mathematics [Quarterly Journal of Pure and Applied Mathematics ~ (1913) 113128J, Jahrbuch fiber die Fortschritte der Mathematik ~ (1913, publ.1918) 7576. Review of Henry M. Sheffer, A set of five independent postulates for Boolean algebras, with applications to logical constants [AMS Trans. II (1913) 481488], Jahrbuah fiber die Fortschritte der Mathematik ~ (1913, publ.1918) 76. Review of P.A.MacMahon, The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects [American Journal of Mathematics 12 (1913) 281322J, Jahrbuah fiber die Fortschritte der Mathematik 44 (1913, publ.1918) 7677. Gebietsdeterminanten. Mathematische AnnaZen 79 (1919) 223236. Abstract of [27], Jahrbuah fiber die Fortsahritte der Mathematik 46.1 (191618, publ.1923!24) 93. Review of Hugo Dingler, Das Prinzip der logischen Unabhangigkeit in der Mathematik, zugleich als Einfuhrung in die Axiomatik [Munchen 1915], Jahrbuch fiber die Fortschritte der Mathematik ~ (191415, publ.1922) 100101. Review of Cesare BuraliForti, I numeri reali definiti come operatori per le grandezze [Atti della Reale Accademia dei Lincei (Romal 24 1 (1915) 489496L Jahrbuo n fiber die Por t e ch r i bt:e de» Mathematik 45 (191415, publ.1922) 100. Review ofcino Poli, Paradossi logici [Atti della Reale Accademia dei Lincei (Roma) ~2 (1914) 6265], Jahrbuch fiber die Fortsahritte de r Mathematik 45 (191415, publ.1922) 107. Review of J.Pfluge~ Die Formenschonheit einfacher geometrischer Gebilde [Stuttgart 1915], Jahrbuch fiber die Fortschritte der Mathematik 45 (191415, publ.1922) 118. Notice of~orbert Wiener, Certain Formal Invariances in Boolean Algebras [AMS Trans. ~ (1917) 6572J Jahrbuch fiber die Fortsahritte der Mathematik ~ (191618, publ.1923!24) 92. Notice of Percy John Daniell, The Modular Difference of Classes [BUlletin AMS gj (1917) 446450] Jahrbuch fiber die Fortschritte de r Mathematik 46.1 (191618, publ.1923!24) 92. Notice of Lagne~Logique des propositions [Atti della Reale Accademia delle Scienze di Torino 21 (1918) 428444] Jahrbuch fiber die Fortschritte der Mathematik 46.1 (191618, publ.1923!24) 92. Einkleidung der Mathematik in Schr6derschen Relativkalkul. Journa l: of SymboZia Logic 5 (1940) 115. On making indirect proofs direct. Translated from the German manuscript by w.V.Quine. Scripta Mathematica ~ no.2 (1946) 125139. Die logarithmische Spirale. Eine ausfuhrliche geometrische Darstellung ihres Wesens und ihrer GesetzmaBigkeiten. Mimeographed (Vol.VI of the series "Geometrie als eine Bildersprache" edited by Johannes Teichert), BerlinCharlottenburg 1957. 169 pp. with a separate FigurenMappe consisting of 145 plates with 204 fig .. FuBpunktkurven und Spiegelkurven. Unp ub l i e he d MS., 30 pp., 38 fig. Allgemeine Satze uber Zykloidalen und Trochoidalen. UnpubZished MS., 99 pp., 101 fig .. Rosenkurven und ihre Conchoiden (Eurythmiewellen). UnpubZished MS., 67 pp., 65 fig .. Pascalsche Schnecken. UnpubZished MS., 154 pp., 176 fig .. JacobSteinerKurven und Astroiden. UnpubZished MS., 141 pp., 210 fig ..
252 44 45
CHRISTIAN THIEL Die fiinf regelmiiBigen (Platonischen) Kiirper. UnpubZished MS., 206 pp., 470 fig •. Einfachheitslogik oder Reichtumslogik ? Unpublished MS., 29 pp. of the first version; some pages of an (incomplete) revised ver
sion.
R. Gandy, M. Hyland (Eds.), LOGIC COLLOQUIUM 76 © NorthHolland Publishing' Company (1977)
RELATIVELY HOMOGENEOUS STRUCTURES David M. Clark
and
Peter H. Krauss
State University of New York New Paltz, New York 12561
~ categoricity is a model theoretic notion. An elementary (that is firstorder) tgeory is called !focategorical if its denumerable (that is countably infinite) models are all isomorphic. A structure is called ~ocategorical if its elementary theory is ~ocategorical. The following question arises immediately: What morphisms are available to establish isomorphisms between the denumerable models of T
x
+
(x)
x + y
o
y +
x.
An elementary theory 2! abelian ~ is any set of firstorder formulas of this similarity type which contains these axioms and is closed under deduction. Rosenstein [11J describes all ~ocategorical elementary theories of abelian groups, and from his description we can directly extract axiom systems for these theories. Although Rosenstein's theorem is not entirely trivial, it turns out that there are
255
256
D.M. CLARK, P.H. KRAUSS
"rather few" ~ocategorical elementary theories of abelian groups. Nevertheless, this result tells US explicitly which denumerable abelian groups can be characterized up to isomorphism by properties expressible in a firstorder language. This is a piece of information which to us appears to be interesting also from an algebraic point of view. In this paper we shall initiate the study of algebraic morphisms which can be used to establish the Hocategoricity of elementary theories. Aside from the many technical results of the more detailed investigation two general observations may be considered the most surprising outcome of this study: First, these morphisms suffice to establish all known applied ~ocategoricity results we have analysed thus far (that is Boolean algebras, linear orderings, abelian groups, rings with identity and without nonzero nilpotent elements). Although S. Shelah has constructed examples of Hocategorical theories where categoricity cannot be directly established with the help of these morphisms, we have thus far been unable to produce such an example which is "algebraically interesting", such as an elementary theory of groups or rings. Secondly, these morphisms play an important role in the structure theory of algebras which goes substantially beyond kocategoricity results. In fact, the structure theory of countable abelian torsion groups of finite Ulm type can be completely developed in terms of these morphisms. This leads us to expect similar results for other important classes of algebras, notably the class of locally finite commutative rings. Finally, we should point out that due to the special properties of these algebraic morphisms we shall be able to substantially strengthen H o categoricity results to include results on (relative) quantifier elimination for the theory. Although we would like to attract the interest of both model theorists and algebraists, we shall intuitively mo~ivate our approach from the model theoretic point of view. We recommend that the interested algebraist read the introductory section of Rosenstein [llJ and hope that he will stay with us until he can convince himself that our investigation is also interesting from his point of view. To wetten his appetite we should like to point out that our investigation led us to conjecture a refinement of the KaplanskyMackey Lemma (Kaplansky and Mackey [4J ) which is the corner stone of the celebrated structure theory for reduced abelian torsion groups. We were unable to prove this conjecture and would like to express our gratitude to Paul Hill who provided us with a proof. His result will appear as an appendix to this paper. 1.
Introduction.
Throughout this paper we shall use standard notation and terminology so that we may restrict ourselves to a few preliminary remarks. Given is a basic denumerable similarity type t determined by a countable set RI of relation symbols and a countable set Op of operation. symbols. t is called finitely based if both RI and Op are finite sets. A tstructure en. has universe lOti, and for each nary relation symbol RERl has an nary relation ROl., and for each nary operation symbol f EO Op has an nary operation f~. 01. is called an algebra if RI ~ ¢ and Ol. is called a relational structure if Op = if;. 01 is called locally finite if every nonempty finite subset of lOtI generates a finite substructure of en. 01. is called uniformly locally finite if for each positive n
forall
XEX.
If ~ is a class of tstructures then .~ is called uniformly locally finite if for each positive n
RELATIVELY HOMOGENEOUS STRUCTURES
257
every subset of I~l with at most n elements generates a substructure of OL with at most k(n) elements. Next we introduce the (finitary) firstorder language of similarity type t. Then denotes the universal (elementary) theory of ~, that is the set of all quantifier free tformulas (all tformulas) holding in all members of~. We define
o WI = Mod ThQf /IV! Em= Mod Thm (the universal (elementary) class generated by~) and h1 is called a universal (elementary) class if Wi = om (Wl = EM). We define
OL
== Qf~
if if
Ot=~
and say that Ot and ~ are universally (elementarily) equivalent respectively. A set ® of tformulas is called an (elementary) theory if @ = Th Mod®. Now let ~ be a consistent theory and let L be a set of formulas. E is called an n~ for G:> if (i) the free variables of the formulas belonging to L all occur among vo , ••• , vn l ; (ii) L is maximally consistent relative to <ED. If x "" \011 n then we say that x realizes the ntype L in ilL if x satisfies all formulas belonging to L , that is 0l.1= If ex1 for all 'f G E • If E. is an ntype for then the following observations are easy to verify: has a model in which L is realizable, and if L: is realizable in en. then en. is a model of e . n Moreover, if x E: lotl and y ~I:Cln realize r:! in O!. and £'r respectively then
e
e
((n.,
x o,···,xn_1)= (;jj., Yo' •.. 'Yn1).
In particular, the substructure of Ul generated by {xi I i< n J is isomorphic to the substructure of ,t;.. generated by {Yi I i <. n 1 by an isomorphism f such that f(xi)=Yi for all Lc n , This motivates our next definition. f is called an elementar.x local monomorphism from az. into ;e.. if for some nonempty finite X ~ 1Lll.1 , f:X ;> I and (en. ,X)X6X =. (;t; ,f(x) )XEX •
*'"
Let EL( (Jl , :;e,.) be the 'set of all elementary local monomorphisms from en. into ~ • We notice immediately that "few" types for the theory @ means ''many'' elementary local monomorphisms between models of 8>. This is the clue to H ocategoricity.
e
RyllNardzewski's Theorem Let be a complete elementary theory. Then @ is if and only if for every n<~, ~ has only finitely many ntypes.
~ocategorical
Although RyllNardzewski's Theorem is of great theoretical interest, it plays only a rather limited role in the proof of applied ~Qcategoricity results because in most cases its condition for r< categoricity iscdlfficult to verify. To describe the main tool for the progf of applied Hocategoricity results we need a few more definitions. f is called a local monomorphism from Ol into l& if for some nonempty finite X s 1Jll.I, f:X ~ 1Jti'.1 and there exists an isomorphism from the substructure of 01. generated by X onto the substructure of.& generated by f'(X) which extends f. Let LM ( en. ,itr) be the set of all local monomorphisms from Ol.. into ~. Next, let K~LM(Ol. ,lG). We say that K has the finite extension property if (L)
fGg
K
f. ¢;
(ii) for every and a EDom(g); (iii) for every
f
E
K and every
a EO I Ott
there exists
g Eo K such that
f
6
K and every
b 6 IlG1
there exists
g E' K such that
D.M. CLARK, P.H. KRAUSS
258 f6.g
and
bERan(g).
Lemma 1.1 If 01. and ~ are countable and there exists finite extension property then OJ.;;;;;e.... Proof:
Ks;,LM(Vl.,.:e)
Use a Cantor backandforth argument.
Now we can state the main tool for the proof of applied results. Theorem 1.2 equivalent: (i)
(Li )
with the
Let G> be a complete elementary theory.
is Hocategorical. For every 07.,~ E Mode,
fj)
property. (iii) For every Ol,CGfinite extension property.
Eo
EL(02,~)
~ocategoricity
Then the following are
has the finite extension
ModG there exists a
Ki:oLM( 01,£1)
with the
Proof: (i) implies (ii) by RyllNardzewski's Theorem and (iii) implies (i) by Lemma 1 .1. Anybody who studies applied T{ocategoricity results will notice that these investigations usually come in two parts. In a first part RyllNardzewski's Theorem is used to earmark certain theories as candidates for Hocategoricity. And then the second part consists of showing that these theories are in fact H 0categoricaL Now frequently this is accomplished by an algebraic construction of isomorphisms between the denumerable models of the theory. This is where Theorem 1.2 comes into play. It turns out that condition (ii) of this theorem in most cases is just as unwieldy for applications as RyllNardzewski's Theorem because usually it is quite difficult to determine the family of all elementary local monomorphisms between two structures. However, frequently it:fs possible to purely algebraically construct families of local monomorphisms between the models of a theory and then to show that these families have the finite extension property. Now an application of condition (iii) of Theorem 1.2 yields a purely algebraic proof of ~ocategoricity. In this study we shall first develop a uniform algebraic method of constructing families of local monomorphisms with the finite extension property and then we shall verify that this method can be used to establish several major applied H ocategoricity results known from the literature (namely linear orderings, Boolean algebras, abelian groups and rings with identity and no nonzero nilpotent elements). To motivate our approach we shall begin with a few remarks on the notion of an elementary local monomorphism for the benefit of the algebraist. Let us consider an elementary local monomorphism f, where (01. ,x)x~.x
== (t;. ,f(x»xcx
The problem with this notion is that it does not just mean that f is a local monomorphism from 01. into;(;., but that it entails an extremely complex description of how the substructures of 07. and;{; generated by X and f'(X) "sit inside" rJL and ;(Jrespectively. In particular, to explore the "environment" of these substructures inside en and ;{; as it is described by an elementary local monomorphism requires an unbounded search on both sides. It appears that algebraists are only willing to conduct a bounded search into the environment of a morphism. This motivates the next definition which is due to Fraisse. For any two structures rJL and JJ. we first define
if either 01. and iff. have no distinguished elements (that is elements denoted by individual constants) or the substructures of Ul. and ~ generated by their distinguished elements respectively (that is the substructures determined by the elements which are denoted by constant terms) are isomorphic. Next we define
259
RELATIVELY HOMOGENEOUS STRUCTURES 01. =r+1 ;t;if for every
a
lOll there exists bE: 1;(;,1 such that
Eo
(ill
and vice versa.
,a)=r
(~,b)
Finally we define
if Ol == t for all r<w. Now we can define the central notion of this study. f is called a local "monomorphism from 01. into ;& if for some nonempty finite Xs;. lOl.l , f:X ~I£..I and (07. ,x)x~X =0<. (,'6 ,f(x) )x~X • Let LM",,( O!,;?) clear that
be the set of all local
0(.
LM(Ol,£:) LM,) 01.
monomorphisms from cJI. into
:e.
It is
LMo(O!'~)
l:'1LM r( OL ,;e,.)
,if!:,.)
Intuitively speaking, a local ~ monomorphism is a local monomorphism describing its environment by carrying out a search ~steps deep on both sides. For finite ~ such a search often has a clearly discernible algebraic meaning and this is where the analysis of this study begins. To show that ~ monomorphisms close the gap between local monomorphisms and elementary local monomorphisms we have to review a few wellknown facts. Lemma 1. 3
Suppose
Y,= X = 1011 and (01
then
If
,x)XGX =01. (~,f(x) )X~X
(01 ,Y)y~y Proof:
f:X ..... I:& I.
=01. (~,f(y) )Y6Y
Use induction on
Corollary 1.11
If d..., P> '" co
Corollary 1. 5 I f
oi. ~
(:>
~
w
and Ol.=(l>.:t;. then then
LM(!>(tl/
llI.=,,;e...
,~)S.LM",,( 0l.,lG).
Next we define the quantifier rank of formulas in the usual fashion. (i) If 'f is atomic then rank ( 'f) = o. (ii) rankh f) = rank( r ). (iii) rank( )"vy.) = max{rank( 'f), rank(¥')}. (Iv) rank(3vn 'f ) = rank('f) + 1. For each n, r (uo, let FIDnr be the set of formulas whose free variables occur among vo' ••• ,.vn  1 and whose quantifier rank is at most r. Lemma 1.6 then for every
Suppose ~'=
x
EO
lill,n and
Fffinr' OL ~
Proof:
Use induction on r.
Proof:
Use Lemma 1.6.
Corollary 1.8
y,= 1~ln.
If
if and only if ;g
LM,)01 • £) s. EL( Ot • h)
t=
'fry]
260
D.M. CLARK, P.H. KRAUSS
In general we do not obtain equality in Corollary 1.8. we have to impose severe restrictions.
For equality to hold
~ 1. 9 Suppose t is finitely based and @ is an elementary theory whose model class is uniformly locally finite. Then for every n , r < ev there are only finitely many requivalence classes of (01..x). where O!. E Mod ® and x E.IOLl n •
Proof:
Use induction on r.
Theorem 1.10 If t is finitely based then the following are equivalent: at is uniformly locally finite (ii) 00l is uniformly locally finite (iii) EOi is uniformly locally finite.
rrr
Proof: Assume (L) and let O
'f x[y] if and only if
Since there exists finite
X ~I illl n
Ol.
(07. ,x)
=0
(&,y).
such that
t=
Y' 'r
x,
it follows at once that OaL is uniformly locally finite. assertion follows from Ol Eo E 01. ~ OOL •
The remainder of the
Lemma 1.11 Suppose t is finitely based and at is uniformly locally. finite. For every n , r < wand every x G!Ol.l n there exists lj' Eo ~r such that for all if} and all YEIiGl n ,
C6 1= 'fey] ~:
If
if and only if
(oz. ,x)
=r ( S,y)
Use induction on r.
Corollary 1.12 Suppose t is finitely based and 01. is uniformly locally finite. XE',IOtl n and YE/iGt n then the following are equivalent: (L) (Ol,x) =r (~.y) (ii) For every ~ G ~r' 01 1= :f [x] if and only i f ;& F ':fry] •
and
Proof:
Assume (ii).
'5 E Fffinr such that 01. F= 'f [x]
By Lemma 1.11 there exists
((). 1= 'f [yJ if and only
if
(Ul,x)
=r (
The converse is Lemma 1.6. It is wellknown (and obvious) that a local mapping between ~ two structures is a local monomorphism if and only if it preserves satisfaction of quantifier free formulas. Corollary 1.12 says that a local mapping between uniformly locally finite structures of finitely based similarity type is a local rmonomorphism if and only if it preserves satisfaction of all formulas of quantifier rank at most r. Then
Theorem 1.13 Suppose t is finitely based and (jt =G;)'&if and only if 01. == ~ • Proof:
Then
~
is uniformly locally finite.
Use Theorem 1.7 and Corollary 1.12.
Corollary 1.14
Suppose t is finitely based and EL(
a, t;.)
=
LM,}
at
is uniformly locally finite.
01.,
The next result tells us that we can use Corollary 1.14 in the investigation of
RELATIVELY HOMOGENEOUS STRUCTURES
261
~ocategoricity.
It is a wellknown consequence of RyllNardzewski's Theorem.
Theorem 1.15 locally finite.
If
e
is an Hocategorical theory then
Mod IE) is uniformly
Thus in the study of ~ocategorical elementary theories we may confine ourselves to theories whose model class is uniformly locally finite. Moreover. if the similarity type is finitely based, then the elementary local monomorphisms in such a model class are exactly the local evmonomorphisms. Now let us return to our discussion of H categoricity and consider a theory ~ of finitely based similarity type whose mod~l class is uniformly locally finite. In order to apply condition (iii) of Theorem 1.2 we have to find, for each m,~ E Mode, a KSLM(Dr,;&) with the finite extension property. We begin with a simple observation. Lemma 1.16 K!i:EL( 0l.,16).
Ks= LM(
If
ilL,;;&)
has the finite extension property then
Proof: It follows directly from the definition that Corollary 1.8.
K SLM",( m,05).
Now use
The characteristic feature of an "algebraic" proof of Ii categoricity is a definition of families of local monomorphisms with the finiteOextension property which does not touch upon the "elementary" quality of these monomorphisms. To give such a definition is surprisingly simple. Lemma 1.17 If extension property. X
l'
~
LMr(at,lB) = LMr +1 ( Of. , 'l6 )
then
LMr ( OZ , )6. )
has the finite
Lemma 1.18 If 0l:=ow;& then for every I' <0 and for every nonempty finite there exists a local rmonomorphism f:X"71l61.
~1~1
Proof: Suppose r <.wand X = {Xi I i <. k] , By hypothesis, We inductively define
Yi El£.I ,
x ~ tau,
where
0< card X
k<w.
Let
01. ~r+k,t;. • for each
i< k ,
(Ol, xo'" .,xi_1) =r+ki
such that
cs , yo.· ...Yi1)
Corollary 1.19 Suppose t is finitely based and ~ is a complete theory whose model class is uniformly locally finite. Then for every r
Use Theorem 1. 2, Lemma 1.17 and Corollary 1.19.
In our view the most surprising outcome of this study is the discovery that all applied }{ocategoricity results we have analyzed thus far can be uniformly established as consequences of Theorem 1.20. The details will be worked out in Sections 3 to 6, but we shall outline the pattern that will develop in this investigation. First notice. if K!i:LM( O! .~) has the finite extension property then
D.M. CLARK, P.H. KRAUSS
262 SK =
{gELM(07.,~)
I g s. f
for some
f~K
also has the finite extension property. Now, wherever people have discovered, for each Ol,~ E, Mod@, a KSLM(Ol.,~) with the finite extension property we can produce a computable I' < W such that for all at, '£y G Mod @ ,
Moreover, we shall see that this " uniformisation" will actually strengthen these categoricity results to include a result on (relative) quantifier elimination. o we conclude our rather expansive introduction with few remarks supporting claim that method "algebraic". do algebraists study local rmonomorphisms? for example, 1010monomorphism useful tool the algebraist? fortunately can answer this question an emphatic~! most striking examples thus far are in structure theory of abelian torsion groups. 1monomorphisms at core celebrated finite more generally, countable groups ulm type be completely developed terms rmonomorphisms. reader find details section 5. 2 shall refine sections 3 6 give applications. some applications require good deal technical detail. hope has motivated follow us path. 2.
Relatively Homogeneous Structures
In this section we shall investigate the hypothesis of Theorem 1.20. the main result of this investigation we introduce a definition. We set
.eo,
LM...( O!) = LMo<,.( en. EL( at) EL(at,OL) .
To state
where " ~ '" ;
Theorem 2.1 Suppose t is finitely based and ~ is a complete theory whose model class is uniformly locally finite. Then the following are equivalent: (i) For every at, 'I:r e Mod <8l there exists I' <w SUch that LM r( i.Jl,:(}) " LMr +1 ( Ol. , c6 )• (il) There exists I'
e . e ,
As a consequence of Theorem 1.20, each of the conditions (i)  (iv) implies 'riocategoricity for the theory G>. Moreover. we shall see that these conditions are equivalent to some (generalized) quantifier elimination for the theory. A structure at. is called locally rhomogeneous if LMr ( OJ.) = LM r Ti ( uz.). A locally Ohomogeneous structure is also called locally homogeneous. ~ is called relatively homogeneous if OL is locally rhomogeneous for some r<:w. Notice. if r~ sand lJl.. is locally rhomogeneous then is locally shomogeneous. Thus with increasing I' we are considering a weakening notion of homogeneity. If OL is relatively homogeneous then the homogeneity rank of az. is the smallest r<", such that OL is locally rhomogeneous. Our next observation is obvious.
o:
Lemma 2.2 Then
LMr(Ol,~)
= LMrTi(Ol.,:B) i f and only i f
LMr(cJl.,~)
= LM",(02.,1G).
Corollary 2.3 Suppose t is finitely based and Ol. is uniformly locally finite. ~( Ol., t;.) = LM (Ol., '1&) if and only if LM (Jl.,;(,.) = EL{ Ol., ;c.). r{ r Ti Proof:
Use Corollary 1.14 and Lemma 2.2.
RELATIVELY HOMOGENEOUS STRUCTURES
263
Next we slightly strengthen Lemma 1.1 and adjust its proof in the obvious fashion. Lemma 2.4 I f 01 and ;J;. are countable and K £, LM ( 01. ,;(,.) has the finite extension property then for every f G K there exists an isomorphism g: az. 7:t.:'r extending f.
en
~ 2.5 If D1.. is countable then is locally rhomogeneous if and only if every local rautomorphism of 01. can be extended to an automorphism of
az. •
Proof: If 01. is locally rhomogeneous then by Lemma 1.17, LMr(Ol.) has the fini teextension property. Thus. by Lemma 2.4, every f (';, LMr ( 02. ) can be extended to an automorphism of oz. •
¢
Lemma 2.6 If 01. and ;& are locally rhomogeneous and f. LMr( 01. ,~) = LMr +1 (01. • .:tr). Proof:
¢
Let
f. xs!Ol.l,
07. =",;(,. then
where X is finite, and suppose
f:X;'/ClerI,
where
(Ol ,x)X&X =r (eXr,f(x»XEX Consider
a61OtI.
By Lemma 1.18 there exist
(01. .x,a)xeX  r
Thus
and
g:X'jlI~1
c cl<&1
such that
(~.g(x).c)xex
(<(;. .g(x) )XEX  r (i{J. ,f(x) 'xeX '
and since i(T is locally rhomogeneous, there exists
b EO
I~I
such that
($,g(x),c)xeX r (qe.,f(x),b)x<X
It follows that
(Ol. ,x,a)Xf,X  r (,iG,f(x).b)xEX
and by symmetry,
(o/',x)xeX =r+l (~,f(x»X&X Then
Theorem 2.7 Suppose ~ and ~ are countable relatively homogeneous structures. i3i ~ t:r if and only if (Jl. ESc.> ij.
en
Proof: Suppose and ~ are locally r and shomogeneous respectively. where r~ s
2: of
Proof: Consider any uniformly locally finite i(; and x 61 JC.I n+l, where 0 < n eo, By Lemma 1.11, there exists 'f'6 Fmnr such that for all rJ; and all Yo""'Ynl E: lotl,,c 1= 'f [yo"" 'Ynl] if and only if
(16. x o,· ...xn_l) =r (.t, yo' .. ·.ynl) Similarly there exists If Eo Fmn + r t .tF1f lyo,· .. ,YnJ if and only l.f
such that for all
«::
and all
Yo"" 'Yn
EO
I.eI ,
(1(';., x o'''' ,x n) =r (
Let {u i I i ~ n ] the formula
be a set of distinct variables not occurring in 'f' or
'If and take
D.M. CLARK, P.H. KRAUSS
264
x(:tt.,x) Finally, let ~ be the set of all formulas )C(~ ) obtained in this fashion. Now suppose m. is uniformly locally finit~ aifd suppose &L E ModL:. Let x, y Eo IOtt n , where 0 < n < c.;) and (01. ,x) =r (m,y) Consider any
a EI O!.I •
Then
OZ F=
::t(Ol,x~a)
Oz.F 'tf'[xo""'~_l,aJ
Thus
,
and
01/:== 'f[Yo""'Yn1]
Ot. l= (3~)1f[Yo"" 'Yn1]
and therefore there exists
bel OL I such that
OJ. F 1f' [Yo"" 'Yn_1,b]
It follows that
(01. ,x ,a) := r (01. ,y ,b)
and by symmetry,
WI. ,x)
== r+1 (Ol. ,y)
This extablishes that OL is locally rhomogeneous. Conversely, suppose at is locally rhomogeneous and consider any Let a o"" ,an ,b o" .• ,b n_1 EO IiJl.l where
X(~x)'
and
Then
(01. ,a o ' ••• ,an) 
r (~,xo"" ,x n)
(m,bo"" ,b n_1) =  r (~,xo" •. ,x n_1)
Thus
(it ,a o"" ,a n_1) =r (01. ,b o" .• 'bn1)
and since OL is locally rhomogeneous, there exists It follows that and therefore Thus
bnE {OLI
such that
(Ot ,ao" •• ,an) =r (01. ,b o" •• ,bn)
01. l= ~[bo,··.,bn]' iJI/= (3 ~)'lf{bo,... ,bn_1]
and hence 01. F X<.:t:,x)' Corollary 2.9 Suppose t is finitely based and 19 is an elementary theory whose model class is uniformly locally finite. Then the class of locally rhomogeneolls models of @ is elementary. Proof of Theorem 2.1: Assume (iv) and let 01. E Mode be locally rhomogeneous. Consider any 'l6 E Mod <5l. Since e is complete, by Theorem 2.8, ifJ is locally rhomogeneous. This establishes (iii). Next, assume (iii) and let en. , E: Mod Gl. Since ED is complete, by Theorem 1.13, OL=uo~ (ii) follows from Lemma 2.6. The remainder of the assertion is trivial.
*
The next result is a generalization of a theorem due to Morley and Vaught [8J. Theorem 2.10 Suppose t is finitely based and at is a relatively homogeneous structure. Then en. is Hocategorical if and only if Ol. is uniformly locally finite.
RELATIVELY HOMOGENEOUS STRUCTURES
265
Proof: Suppose ~ is locally rhomogeneous and uniformly locally finite. By Theorem 1.10, EOl. is uniformly locally finite and therefore, by Theorems 1. 20 and 2.1, at is kocategorical. The converse follows from Theorem 1.15. As we pointed out before, each of the conditions (i)  (iv) of Theorem 2.1 implies Kocategoricity for the theory GY. A further analysis of these conditions leads to a notion of quantifier elimination. An elementary theory ED is said to admit rquantifier elimination if for every formula ~ whose free variables occur among v0" •• vn  l there exists 'If E: F1nzlr such that ® 1= 'f~ 'If. If @ admits Oquantifier elimination we also say that @ admits quantifier elimination. ® admits relative quantifier elimination if @ admits rquantifier elimination for some r
Proof: x~IOtI
n,
Assume @ admits rquantifier elimination. Y61eGln, where O
«JL,x)
Let
Q,2& E Mode
and
( ~.y)
Now consider any formula ~ whose free variables occur among vo, ••••vn  l such that 01. 1:= 'f [x] • By hypothesis, there exists 1.f Eo Fmnr such that ® F 'f~ 1.p • Thus 01. F "f Cx] and by Lemma 1. 6
(Ol. .x)
==
(£..y)
OZ.:G
~
Mod 0.
LM r ( Ol., d(;.) = EL( o: , e i, Let 'f o c n c «> , By Lemma 1.11
be a formula whose free variables are among vo, ....vn  l ' where Consider any Ol. fi Mod ® and any x E \il.1 n such that oz. 1== 'f [x]. there exists '4' E Fmnr such that for all «; and all Yfil.J>2l n •
?fJ.1:= ?f[y]
if and only if
(o[.x)
Let ~ be the set of all V' obtained in this fashion. that ;[ is finite. We claim that Indeed. by definition,
=r (c(;.,y). By Lemma 1.9 we may assume
G> 1= 'f ~ V:L. .
Now suppose l(;. E Mod e. y eol iCl n and for some 1f€: L ,
(Ol,x) =r (0/.
,x):=
and
(~,y) ($,y)
and therefore <8 1= 'f [y]. This establishes our claim and ® admits rquantifier elimination. We have shown that (i) and (ii) are equivalent. and (ii) and (iii) are equivalent by Corollary 2.3. Corollary 2.12 Suppose t is finitely based and e is a complete theory whose model class is uniformly locally finite. Then the following are equivalent: (i) ® admits rquantifier elimination (ii) Every model of ED is locally rhomogeneous
266
D.M. CLARK, P.H. KRAUSS (iii)
Some model of
~
is locally rhomogeneous.
Theorem 2.13 Suppose t is finitely based and @ is a complete theory which admitsrelative quantifier elimination. Then G> is Hocategorical if and only if fV has a uniformly locally finite model. Proof: Suppose 6D has a uniformly locally finite model. By Theorem 1.10, Mod G>rsuniformly locally finite. By Theorem 2.1 and Corollary 2.12, ~ is Hocategorical. Corollary 2.14 Suppose t is finitely based and G> is a complete theory. Then Ko categorical and admits rquantifier elimination i f and only if ~ has a uniformly locally finite and locally rhomogeneous model. ~ is
Proof: In one direction use Theorem 1.15 and Corollary 2.12 and in the other Theorem 1.10 and Corollary 2.12. Remarks on Quantifier Elimination: For r = 0 Theorem 2.11 may be strengthened by deleting the hypothesis that t is finitely based and Mod Gl is uniformly locally finite. An elementary theory G> is called substructure complete if for every OL, J(l.E Mod®, LM(Ol,~) = EL(Ot,lS). Theorem 2.11 then becomes a wellknown result on quantifier elimination: An elementary theory admits quantifier elimination if and only if it is substructure complete (sacks [12J). In Krauss [6] "algebraic" methods of quantifier elimination are investigated. It appears that the natural extension of this notion of quantifier elimination to formulas with bounded quantifier rank is only a useful tool for model theoretic investigations in case the similarity type is finitely based and the model class is uniformly locally finite. Theorem 1.15 shows that k categorical theories of finitely based similarity type do satisfy this prerequisi~e. Therefore applied HQcategoricity results can frequently be strengthened to include a result on (relat~ve) quantitier elimination. Remarks ~ the Similarity~: Most of our results depend on the assumption that the similarity type is finitely based. This restriction can be removed, however in this study we have no opportunity to take advantage of such a generalization. We shall indicate what is involved. Let us call a class f1.'1 of structures strictly uniformly locally finite if it is uniformly locally finite and for every n c co there exists a finitely based subtype t n of t such that for all OI.,:.e. .. m and all n n, xtClUl
ye,I.t'l
(1h ,x)
=; 0
(~,y)
if and only if
(OJ. 1
t n ,x) =: 0
(~l
t n ,y) •
Now the proof of Theorem 1.15 actually yields Theorem 2.15 locally finite.
If ®
is Hocategorical then
Mod Gl is strictly uniformly
It is a routine chore to check that all our arguments go through if we replace everywhere "uniformly locally finite of finitely based similarity type" by "strictly uniformly locally finite". Since in this study we do not give an application of this generalization we have stated our results in a somewhat less general but considerably more intelligible form. Finally we shall address ourselves to the question of the scope of our investigation. We say that an elementary theory ~ has a uniformly algebraic k ocategoricity problem i f for any model en of lED, Ol. is kocategorical if and only if OL is uniformly locally finite and relatively homogeneous. Lemma 2.16 e has a uniformly algebraic Hocategoricity problem if and only if for any countable model 01 of , 01.. is 110 categorical if and only if O!. is uniformly locally finite and relatively homogeneous. Proof:
Use Theorem 2.8 and the LoewenheimSkolem Theorem.
RELATIVELY HOMOGENEOUS STRUCTURES
267
Theorem 2.17 @ has a uniformly algebraic k categoricity problem if and only if all complete kocategorical extensions of [email protected] admit relative quantifier elimination. Proof:
Use Theorem 2.14.
In Sections 3 to 6 we shall show that the following theories have a uniformly algebraic ~ocategoricity problem: (1) Boolean algebras; (2) Linear orderings; (3) Abelian groups; (4) Rings with identity and no nonzero nilpotent elements.
H"
S. Shelah has suggested to us an example of an categorical structure with one binary relation which is nct relatively homogeneous. With his permission we shall now give this example. Let t be the similarity type determined by a binary relation symbol R and for each k and m, where 0 < k < co and k ~ m~ 2k, an mary relation symbol P For i,j<m set i'Vj if i,j
k.
pm _ kVO·· .vm_1
' ) (J.
/\ i<j
oJ
vi" v j
[n
~+lvo· •• vm 7~vo" ,vm_1
(ii) (iii)
P~vo" ,vm_1
/\ [ iNj
(iv)
RViVj /\

1\ Hj
~.., RViV j 1
RViVj /\
J') Pk2kVo"'V2k1 RViVj
It is easy to verify that Mode has the embedding and amalgamation properties. By (i). Mod @ has only countably many finite models and therefore has a denumerable
Notice that OL is a model of
homogeneous [email protected] model 0(..
Since ~ is homogeneous and. for each n
p~Ol(a) but not
Lf f
ROl(bi ,b j )
p~Ol(b). By (ii) 
• 1.+1 • ( mod n ) J==
(v ) , a satisfies
0, Rv.v.J
(3vn",v2n_1) [1. in Ol but b does not,
1.'ff
~J
J.
/\
/\ .., RViV'] ifj J
Thus
On the other hand we will show that
(OLo.a) =n1 (Oto,b) ,
r Let t* be the similarity type obtained from t by removing p for all r, where n s, l' ~ zn , and let at* = Ol.lt.... We first establish the fO~lowing auxiliary claim: Let
x,YGIOl*ll'.
such
that I.e""
Proof:
where
1'< 2n1.
If
(Q*.x)
Consider xJ;'EIOl.* I and let ={Yi 1J.~r} and
=0
Yr4101*I,
(Q.*.y)
then
(Ol.*.x) =1 WL*.y).
Define a t"'structure J:'"
D.M. CLARK, P.H. KRAUSS
268
01.*1 lx. I i~ r} : .(;* ot* I I: /"~ I i < I' 1 = Ol.*I{Yil i
.t * to atstructure .(; by defining
We notice at once that d: is a model of axioms (L)  (Hi) and, since r<2nl, is also a model of axioms (Lv ), Since 01. is homogeneous ModGJuniversal, we can obtain Yr ~10l.*1 so that "c* = Ol*I{Yil i~r}. Thus (Ol.*,x,xr)
c
=0 «/l*,Y,Yr)
and by symmetry our claim is established. Returning to the original setting we notice that (0I1~ ,a) =: (07.* ,b). It follows .!.rom our auxiliary.claim th~t (OI.*,a) =n:l (01*,b), .an~ therefore (Ol;o,a) =nl (OLo,b). Thi s establ~shes that 02 0 as not relahvely homogeneous. Th~s concludes our presentation of Shelah's example. It may be interesting to notice that Th ~o is axiomatizable. We are left with some natural questions: (1) Does a finitely axiomatizable H categorical theory with an infinite model have a relatively homogeneous model? Ig this context we recall that every Hocategorical theory of linear orderings is finitely axiomatizable (see Rosenstein [10J). (2) Does the elementary theory of groups (rings) have a uniformly algebraic ~ocategoricity problem? 3.
Relatively Homogeneous Boolean Algebras
In this section we shall consider Boolean algebras < B, /\ ,v,  >. Since the class of Boolean algebras is uniformly locally finite, we obtain at once from Theorem 2. 10 : Corollary 3.1
Every relatively homogeneous Boolean algebra is Hocategorical.
The converse of Corollary 3.1 will establish that the class of Boolean algebras has a uniformly algebraic Hocategoricity problem. Theorem 3.2
If ~ is a Boolean algebra then the following are equivalent: H categorical. (H) ~ has ~initelY many atoms. (iii) ~ is relatively homogeneous.
rr>
t:,. is
The equivalence of (i) and (ii) in Theorem 3.2 is wellknown. proof of Theorem 3.2 we need the following observation.
To complete the
Lemma 3.3 Suppose 6l.. is uniformly locally finite and KSLM(Ol.) has the finite extension property. If L'\.(01.)SSK then Ol. is locally rhomogeneous. Proof: If K has the finite extension property then so does SK. Thus, by Corollary 1.8 and Lemma 1.10,
Notice, if Ol. and ~ are Boolean algebras with n atoms and K is the set of local monomorphisms from Ol. into :e preserving the atoms, then K has the finite extension property. It follows from Theorem 1. 2 that the theory of Boolean algebras with n atoms is Hocategorical. We shall use this idea and Lemma 3.3 to complete
RELATIVELY HOMOGENEOUS STRUCTURES
269
the proof of Theorem 3.2. Let at be a Boolean algebra with atoms a •••••a 1 and let K be the set of local automorphisms of 01 permuting {ao ••••• an_~~' Ifnf EO LMn+1 ( at.) then there exists g E LM1 ( at.) such that [a •••• '~1} So Dom(g) and f s g. Since g is a local 1automorphism of tTI... g Eo K. We°have shown that LMn+1 ( Ot) s SK and by Lemma 3.3. OL is locally (n+1 )homogeneous. This completes the proof of Theorem 3.2. The proof of Theorem 3.2 reveals that a Boolean algebra with n atoms is locally (n+1)homogeneous. Although this is a very crude result. it appears that an exact determination of the homogeneity rank of a Boolean algebra with finitely many atoms is a difficult combinatorial problem. We have only been able to give a rather tight approximation which shows that the homogeneity ranks of relatively homogeneous Boolean algebras are unbounded. Then
Theorem 3.4 Let r n be the homogeneity rank of a Boolean algebra with n atoms. I' 0 = O. r 1 = 2 and for n ~ 2 log2(nl)  1 < r n < log2(nl) + 4 •
In the remainder of this section we shall prove Theorem 3.4. We shall represent a Boolean algebra with n atoms as the field = < F , n. l.J.  ) of clopen subsets of its Stone space Xn • The atoms of 'if: cgrrespl:lnd to the isolated points of X. We shall use the isolated points of Xn ¥o refer to the atoms of fn. so that we gay unambiguously talk about the atoms of a sub field of A finite subfield Ol. s.~ n is uniquely determined by its atoms which form a c~open partition of Xu' If Ol.. q ~ then Ol 50 iG i f and only i f the partition induced by £;. is a retinement of the pRrtition induced by 01.. For ME:Fn we define
f
f.
e
H(M)
=
r: (M»
•
where fo(M) is the number of isolated points in Mand if Mcontains only isolated points
f1(M)
otherwise.
Notice that fl(M) = 1 if and only if M is infinite. Let automorphisms of ~n permuting the isolated points of Xn' Lemma 3.5 If fELM(fn) (i) fE El( .£:"n)· (ii) fE (iii) f preserves H
be the set of local
~
then the following are equivalent:
sK./:
Proof:
Show that (i) implies (iii), (iii) implies (ii) and (ii) implies (i) .
6'
The determination of .LMr( n) requires a coarser indicator. define m Hm(M) <,r·"o(M),/\(M». where min{ro(M). m} • :(M)
For
ME; Fn
we
r
Lemma 3.6
There are formulas
t
nl= 1f'o[M]
fn Fri(M] Lemma 3.7 If k ~3 and such that for all M6Fn•
'!fo ' 1f1 .: Fro1 , 3 such that for all Mf' Fn i f and only i f heM)
0
f< 1 (M)
1
i f and only i f
i ~2k3 + 1 = m then there is a formula
D.M. CLARK, P.H. KRAUSS
270 fn F Proof:
If'~(MJ
We argue by induction on k.
l.f~
(biz)
If't
"lf~ /\ "'f~
If' ~
( :J
i.
if and only if r:(M)
For
k = 3
we have
m = 2
and take
w <: y v
IV ~ w<
[ciV;iz<>x]J>(3w)[(/)<w
y)( :3 z ) [ y ;i z " y < X
A
Z
<
X A
(\I w)
Ir
zJ 7 w = ~
k 3 Now consider k"'3 a~d i~ 2 + 1 = m, Case 1: i = 2 k 3 + 1 = 2 4 + (2 k4 + 1). Then i = r + s, where r = 2k4 and k4 s = 2 + 1. For MEFn, jlm(M) = i if and o!!.ly if there is an N ~M in Fn such that N contains exactly r iso~ated points and NnM contains at least s isolated points. We take
\f'1
(3u)[u~x Alf'~l(u)
where u is a new vari~b¢e. By induction, Case 2: i ~ 2k3 = 2  + 2k4• Then i in Case 1. Lemma 3.8 Let k.,,3 and m phism of ~n then f preserves ~. Proof:
x·1.,J. E Fm1 , k
k tfi
=r
= 2k 3 + 1.
Let M6/G7.land ~(M) ="i,j:>. such that for all N E Fn; fn 1= Xi,j[N]
Corollary 3.9 homogeneous.
Let
k
Fm
~ 3 and
Proof: If n s, m then H = H. H and~Lemma 3.5, is elemen~ary.
so
If'~l(ul''lx) k'
+ s, 1 ~here n , s (, 2k4 • If
f: 0!~<6
Now argue as
is a local kautomor
By Lemmas 3.6 and 3.7, there is a
if and only if
~nF Xi,j [f(M)]
By Corollary 1.12,
A
~(N) =
~(f(M» = < i,p.
m = 2k 3 + 1.
If
n s m then
tr n
is locally k
By Lemma 3.8 a local kautomorphism preserves
This provides an upper bound for the homogeneity rank of f n close to log2n. Coming from the other side we now show that for r much less than log2n, ~ n is ~ locally rhomogeneous.
t.
Lemma 3 10 Let f: at ? ;J;;. be a local automorphism of which preserves H2m where m = 2k • Then for any M(; Fn t:here exist NE Fn and all extension g of f such that g preserves Hm and g(M) = N. Proof: Let {Ai l i <: s 1 and {.Bi I i c s ] be the atoms of en and & respectively, where f(Ai) = Bi for all i< s , Then the atoms of the subalgebra of n generated by 101.1 v ~ M} are the nonempty members of
f
{AinMli<s}v {Ai n"Mli<:s1.
If we can find
Ni foBi
such that
(*) then we can take H(Ai) = H2m(Ai)
N=
Hm(Aif'l_M)=~(Ni)
{
U {N i I
= H2m(Bi)
Hm(AiI"lM)
~(Bil"lifi)
i < sl. If A. has fewer than 2m isolated points then 1can = H(Bi) and we choose Ni so that <*) holds with ~
RELATIVELY HOMOGENEOUS STRUCTURES repl~ced by Ai~M have
271
H. Suppose Ai has at least 2m isolated points. Then not both M and fewer than m isolated points. Thus we can choose Ni so that (*) holds.
Lemma 3.11 A local automorphism of fn that preserves local kautomorphism. Proof:
llm.
where m" 2k,
is a
USe induction and Lemma 3.10.
Proof of Theorem 3.4: An atQlllless Boolean algebra is locally homogeneous (Lemmil3:"'5):whereas hhas a local iautomorphism, but no local 2automorphism that fails to preServe H. By Corollary 3.9, tn is locally khomogeneous if log2(n1) + 3 ~ k. Thus r n < 10g2(n1) + 4. On the other side we must show that Cn is not locally khomogeneous provided 10g2(nl)  1 , k. Since 2 kt 1 .. 2k + (2 k+l )
= (2 k+1) + 2k
Hm,
we aan find a local automorphism f of &~n which preserves where m .. 2k, but does not preserve H. By Lemma 3.11 fELMk(&n) but, by Lemma 3.5, f$ EL(IYn)' 4.
Relatively Homogeneous Linear Orderings
In this section we shall consider linear orderings < L, < >. Since any class of relational structures is uniformly locally finite we obtain again from Theorem 2.10: Corollary 4.1
Every relatively homogeneous linear ordering is kocategorical.
Next we shall show that the class of linear orderings has a uniformly algebraic Hbcategoricity problem. Lemma 4.2 hQlllogeneous.
Every countable
~ocategorical linear
ordering is relatively
Proof: Let ~ be a denumerable Hocategorical linear ordering. By RyllNardzewski's Theorem, there are only finitely many 2types realizable in.e , and each 2type realizable in ~ is principal. For each 2type realizable in Je choose a formula which generates it, and let r be the maximum of the ranks of the generators. We claim that It, is locally rhomogeneous. Indeed, let x ,y E; ldeln, where Then for each
i < na ,
and therefore Since
ce
ue,y).
(£ ,x)
(rJ: ,Xi ,xi+1) r
(oe
(Of: ,Xi,xi+1) 
(,;e'Yi'Yi+1)'
,y 1'Y i+l)
is denumerable and riocategorical,
and it follows at once that
(oe ,xi,x i+1) (;e ,x)
~
(£

( ~ ,y ) •
'Y1'Yi+1)'
It follows from Lemma 2.16 that the class of linear orderings has a uniformly algebraic k categoricity problem. Notice that we have been able to establish this fact wi~hout solving the H categoricity problem, that is without characterizing the relatively homogeneousolinear orderings. Rosenstein [10J gives a characterization of all countable ~Qcategorical linear orderings, and it is easy to extract from his work a character1zation of all relatively homogeneous linear orderings. In order to do this we give a brief review of Rosenstein's definitions. If Ie = < L ,< > is an (irreflexive) linear ordering and ~lO. e , then ~ is called a segment of ;£ if c e l b I whenever a < c < b and a, b e I f" I. A linear
272
D.M. CLARK, P.H. KRAUSS
ordering tc = < M,< > is called a splitting of .:£ i f N is a partition of L whose parts are the universes of segments of !e and < is the induced ordering of the parts. If ~ is a finite set of linear orderings then there is (up to isomorphism) a unique linear ordering ~3r, the rational shuffle of ~, with the following property: There is a splitting n of 0& with the order type of the rationals such that each part of 'tr; (with its ordering inherited from;e ) is isomorphic to a member of 3'"" and between any two parts of ft. are parts isomorphic to each member of &". Theorem 4.3 (Rosenstein) If ~ is a countable linear ordering then ~ is H9 categorical i f and only if ;e belongs to the smallest class of linear orderings wh1ch contains a oneelement linear ordering and is closed under concatenation, rational shuffles and isomorphisms. To give a complete characterization of relatively homogeneous linear orderings we have to generalize the notion of rational shuffle. Let OL be a dense linear ordering without endpoints, let 0
If if, is a linear ordering then the following are equivalent is H categorical. (ii) ~ beloggs to the smallest class of linear orderings which contains a oneelement linear ordering and is closed under concatenation, generalized shuffles and isomorphisms. (iii) .e is relatively homogeneous.
rrT"";e
Proof: The equivalence of (i) and (iii) follows from Lemma 2.16, Corollary 4.1 and Lemma 4.2. To establish the equivalence of (i) and (ii) we refer to Rosenstein [10]. As Rosenstein points out in the "Note added in proof", it is easy to extract from his work (finite) axiom systems for all K categorical theories of linear orderings. Subsequently it is not hard to see ~hat (ii) describes the models of these theories. An exact calculation of the homogeneity rank of a relatively homogeneous linear ordering requires an effort which does not appear to be justified by its results. To reveal the complexity of the problem we inductively define a set ~ of (countable) linear orderings as follows. Let m = {< 1, < >} and let VVJ n +1 be the set of all sums and rational shuffles of finitel~ many members of ~n' Finally, let
U mn '
1'Yl= n
then the complexity of
~
Theorem 4.5 For every positive ings of complexity n are unbounded.
is defined to be the least n
n < c.v
such that
the homogeneity ranks of linear order
To prove Theorem 4.5, it suffices to show that the homogeneity ranks of finite linear orderings are unbounded. Let if., be a finite linear ordering and a e 1£1 n. Then a determines a splitting of < l;tl{ail i
we define
Rj (a),
where
(<'e ,a)
>
j < n + 1. o
( £,b).
(at: ,a) ~m (C£,b)
Next, let
a, b e l.el n, where
RELATIVELY HOMOGENEOUS STRUCTURES if for each j
Lemma 4.6
either
If tie is a finite linear ordering and (.t
then for any
card Rj(a) = card Rj(b)
~tl;el
,ai)i
'V
2kTll
there exists
bnEI.e1
(dC ,ai)i
'V
2Ll
or
273 card Rj(a),
k
a,bE:I'£:jn,
where
(.e ,bi)i
such that (dC ,bi)i
Proof: Let anE Rj(a). If card Rj(a) < 2kil  1, card Rj(b) and we choose bn so that
then
card Rj(a)
equals
< Rj (a), < '~> ~ < Rj (b) , < ,bn > • kTi If card Rj (a) ~ 2 then card R· (b);, 2kTi  1 and the result follows since 2kil _ 1 = (2 k _ 1) i 1 i (2 k _ 1). J Lemma 4.7
Let
~
be a finite linear ordering and (OC,a)
then
~2k_l
a,bel.e,n,
k
If
(~,b)
(o'e ,a) =k (oe ,b).
Proof:
Use induction on k and Lemma 4.6.
Lemma 4.8 Proof:
Let
A linear ordering with 2nil elements is not locally nhomogeneous. card ~ = 2nTl and let a and b be the 2nth and (2 nTl)st Since 2nil = (2 n  1) i 1 i 1 i (2 n  1) we have
eleme~espectively.
By Lemma 4. 7 , but clearly
ue ,a) (ce
"'2 nl
tal n
(ae
(ae
,b).
.s),
(it tal '1 (£tb).
Notice that the proof of Theorem 4.5 is now complete. 5.
Relatively Homogeneous Abelian Torsion Groups
In this section we shall consider abelian groups < A,ittO >. An abelian group is locally finite if and only if it is a torsion group, and it is uniformly locally finite if and only if it is of bounded order. Rosenstein [11J noticed that for abelian groups the converse of Theorem 1.15 holds: Theorem 5.1 bounded order.
An abelian group is Hocategorical if and only if it is of
The proof of this result requires a minute portion of the celebrated structure theory of countable abelian torsion groups. If ~ and ~ are abelain pgroups with the same Ulm invariants and K is the set of local monomorphisms from at into ~ preserving (generalized) height, then K has the finite extension property (KaplanskyMackey). It follows that any two countable abelian pgroups with the same Ulm invariants are isomorphic (Ulm). Since any two elementarily equivalent abelian pgroups of bounded order have the same Ulm invariants (Rosenstein), Theorem 5.1 follows at once. We shall use this idea to first give a quick proof of a remarkable result which implies Theorem 5.1 and establishes that the class of abelian groups has a uniformly algebraic ~ocategoricity problem.
274
D.M. CLARK, P.H. KRAUSS ~
5.2
An abelian group of bounded order is locally ihomogeneous.
We begin with the standard reduction to abelian pgroups, relative to some fixed enumeration < Pi \ i <w > of the primes. Let at be an abelian torsion group. Then ~ is a direct sum Ol. = ~ Oli' i<w
where for each i<w, VIi is the p,ecomponent of ClI.. If f is a local automorphism of 01. then there exists s < «> suc n that f is a local automorphism of 2. 01.. i<s J. Moreover, for each i< S there exists a local automorphism f i of a i such that
L ail = L fi(ai) • i< s i< s Our next observation can be verified by a routine computation. f(
Lemma 5.3 Let OL be an abelian torsion group and let f be a local automorphism of 01..(i) f is a local rautomorphism if and only if for each i< s , f. is a J. local rautomorphism. (ii) 01. is locally rhomogeneous if and only if for each locally rhomogeneous. Next, let at be an abelian pgroup, where p is a fixed prime. define the sequence of ~ subgroups of 01.
01.,(.+1 =
n pn(
mol.);
nc ce
0l.A.=
We recursively
n Ol.oI.
c0<)..
where :t. is a limit ordinal. The Ulm !lE!:. of Ol , denoted by e( Ol.), is the smallest .,(. such that at"" = Otol..+ 1 If '1:"( lJl.) = ol.. then OI. oL is called the divisible subgr'OUP of 01. and is denoted by D(0l.). The divisible subgroup of 01. is a directsum of pquasicyclic groups. Several notions of "height" playa role in the theory of abelian pgroups. The (ordinary) height is defined as follows: hf a )
but
= {eoo n if if
The generalized height is defined to conduct a "deeper" search: H(a)
={<<,<,n> i f 00
if
a~I!lld.1 but a 1 101
01.+11 ,
and
hf a ) = n in rJI..<
a€!D(OL)I
Lemma 5.4 (KaplanskyMackey [4]) Let OL be an abelian pgroup and let K be the set of local automorphisms of OL preserving H. Then K has the finite extension pr'Operty. Now we can prove Theorem 5. 2. Let 01. be an abelian group of bounded order. By Lemma 5.3, we may assume that 01. is an abelian pgroup, Let K be the set of local automorphisms of
RELATIVELY HOMOGENEOUS STRUCTURES height, defined for each
r <<..1 :
Notice that we may identify
275
en rt
if
a41
if
a",IOLrj
H1(a) = h(a),
and if '1:'«(}).) r",s<w then Hs = H. 1.:; r < w, then f ~reserves Hl (as we have pointed out before) so that the restriction f 1 of f to lOt I is a local rautomorphism of at 1 • If f is a local rautomorphism of iJL, where
f
~ 5.5 If f is a local (r+1)automorphism of and f 1 is the restriction of f to Im 11 then f 1 is a local rautomorphism of Ol •
Proof: The assertion is obvious for r = 0. Assume r > 0, subgroup of iJl , where (Ol,x)XEI~1 r+l (OL,f(X»XEIOJ' • Let
~ = ~ rv at 1
and
a<=
1000ll.
Then there exists
b
~
and OJ is a finite
lol.! such that
(OL,f(x),b>XEI~1
(Ol. ,x,a)XE 1<11 1 =r
Let g be the extension of f to the subgroup generated by I~I and a, and let &1 be the restriction of g to I (}).1 1 • By induction hypothesis, gl is a local (rl)automorphism of 01.1. In particular, bE lOll, so that 1
(cTl. , x ,a )XEo I:t:1 rl
By symmetry,
(Oll,x)XEoI£.1 Sr ~
5.6
«(J[1,f1(x),b)XGI.t1I· (0l1,f1(x»XEI&I'
Every local rautomorphism of
oz.
preserves Hr'
The assertion is obvious for r = 0,1. Assume r)o 1 and ~ is a finite where (01. ,x) =. r (OL,f(x» x <= 1'"1' x E 1111/ 0 0 Let a EI~I and distinguish two cases. Case at Im 11. By induction hypothesis, f preserves lIz.l' Since r  1 i1 1, f(a)~ I and
Proof: subgroup of
oz. ,
1
Hr(f(a»
= Hr_1(f(a»
= ~_l(a)
=
~(a).
Case 2 a Eo I at 1 1 • • Let H1 be the (11 )st height function for 0l1. By Lemmas:5, the restriction f to Hit' is a local (r1 )automorpr.sm of 0l.1 which, by induction hypothes Ls , preserves ~1' But then f(a)E; 101. I and
f:{ot
~(f(a» = H;_l (f1 (a» + < 1,0> whereweset
= ~1 (a)
{o<...,n>+
+ < 1,0)0 = ~(a),
and 00 +<1,0>=00.
The proof of the converse of Lemma 5.6 requires a refinement of the KaplanskyMackey Lemma which is due to Hill [3J. ~ 5.7 Let f be a local automorphism of OL which preserves ~, where 0< r < w. Then for each x GIlJLl there exists a local automorphism g of (]l. such that f..g, x Eo Dom(g) and g preserves ~1'
Theorem 5.8 A local automorphism of an abelian pgroup is a local rautomorphism if and only if it preserves '\.. Proof: By Lemma 5.6, a local rautomorphism preserves 1Iz.. Conversely, by Lemmas:=rand induction, a local automorphism which preserves ~ is a local rautomorphism.
D.M. CLARK, P.H. KRAUSS
276 Corollary 5.9
An abelian pgroup of Ulm type
r<<..?
is locally rhomogeneous. and let t?J be a finite
Proof: Let OL be an abelian pgroup of Ulm type r
and by symmetry
(tJl,x)XEoI~1
r =r+l
Rr = H. (iJ(
By Lemma 5.4 and Theorem 5.8
,f(x),b)x "I~f
(aL,f(x»X"lqjl
Next we shall investigate the converse of Corollary 5.9. This will result in a complete characterization of relatively homogeneous abelian pgroups and the determination of their homogeneity ranks. Lemma 5.10 Proof:
If OL is locally rhomogeneous then Vl. r is locally homogeneous.
Let ~
be a finite subgroup of OZr, where
«()Lr,x)XEol~1
0
(iJ1.r,f(X»uoJt2l1
Consider a E I Ulr,. Since f preserves Rr, by Theorem 5.8, f is a local rautomorphism of iJ1.. Since 01. is locally rhomogeneous, there exists b Eo I aLI such that (<Jl. ,x,a)x E: I !{I I
By Theorem 5.8,
b
EO
IlJI. r I
r
(01.,f(x),b)
x EI~
I'
and therefore r
(01. , x , a )Xe liljl
0
(otr,f(X),b)xcl~1
Our next task is the characterization of locally homogeneous abelian pgroups. Lemma 5.11 If at is locally homogeneous then elements of order have the same height. Proof: Let x , yEo 1In.1 have the sarne order. therefore (01 ,x) =1 (cJ1 ,y). Thus hf x ) = h(y). Lemma 5.12
Then
(0/.
.x )
(]L
with the same
=0
(01.,y)
and
If 01. is locally homogeneous then iJ1. is either divisible or reduced.
Proof: Let 01. = ;) Ee n where ~ is divisible and ~ is reduced. If ~ is nontrivial then 1R..1 ¢ ~ so that there exists XE.I3?1 with h(x)
If 01. is locally homogeneous and reduced then O!1 is bounded.
01. 1 is nontrivial and reduced then ~O} ¢ Ill'Ll! ¢ 1011 and is unbounded. If litl is also unbounded then there exist x E. 101. 11 and y~ 101 1 , such that x and y have the sarne order. Since hf x ) = 00 and h(y)
1001JO'tIi
If
Lemma 5.14
If 01. is locally homogeneous and reduced then IJl.l is trivial.
Proof: Suppose 011 is nontrivial and let x Eo 1111 11 with order p , Lemma 5.13, there exists mc co such that pm 1 is trivial. Let pmy y ~ Hn1, so that h(y) = r < '" • Since x € 101. I, there exists Z E10l./ x pm+r+l z• Now we compute o ¢ x pmy pm(pr+l z)
1
=
By
=x ,
Then such that
RELATIVELY HOMOGENEOUS STRUCTURES
277
o : px : pm+ly : pm+l(pr+l z) Thus y and pr+l z have the same order pm+l. contradicting Lemma 5.11.
However,
h(y): rand
h(pr+l z)?r + 1
Theorem 5.15 If en. is an abelian pgroup then the following are equivalent: rrr~ is locally homogeneous
(ii) ~ is either a direct sum of isomorphic cyclic groups or is a direct sum of isomorphic quasicyclic groups. (iii) Every local automorphism of en. can be extended to an automorphism of 01. Proof: Assume (i). If Ol is divisible then at is a direct sum of quasicyclic group8:'"Otherwise by Lemma 5.12, at is reduced and by Lemma 5.14, 01. 1 is trivial. We claim that Ol is bounded. Indeed, let x ~10l.1 where h(x): 0 and x has order pro If pr at is not trivial then there exists y,= IlJJl such that h(y): 0 and pry # O. Let the order of y be pS, where s~ r. Then x and psry have the same order ~r, however h(x): 0 and h(psry) <> s  r > 0, contradicting Lemma 5.11. Thus p at is trivial and our claim has been established. Since (Jl is bounded it is a direct sum of cyclic groups, and it follows easily from Lemma 5.11 that these cyclic groups are isomorphic. Assume (ii) and let f be a local automorphism of 1Jl. Let <)} be a countable direct summand of m such that f is a local automorphism of ~. If 01 is a direct sum of isomorphic cyclic groups then elements with the same order have the same height. In either case f~ considered as a local automorphism of ~ , preserves H and therefore, by Lemma 5.4, can be extended to an automorphism of ~. Since ~ is a direct summand of m, f can be extended to an automorphism of en. • Theorem 5.16 An abelian pgroup is relatively homogeneous if and only if it has finite Ulm type. Proof: Suppose t(Oi):¥w and r<w. Then 't"(lJLr): 1:(aL);pu.:>'l and therefore, by Theorem 5.15, 01r is not locally homogeneous. It follows from Lemma 5.10 that en. is not locally rhomogeneous. This proves one direction of the assertion and the other is Corollary 5.9. Theorem 5.17 Let 01 be an abelian pgroup of Ulm type r < w. Then the homogeneity rank of 01. is (i) r if 01rl is not a direct sum of isomorphic cyclic groups; (d i ) rl if OJ. rl is a direct sum of isomorphic cyclic groups. Proof: By Corollary 5.9, en. is locally rhomogeneous. Since t ( OJ.) = r , Ol.rl~ so that Olrl is not a direct sum of quasicyclic groups. Now first assume that mc r  1 is not a direct sum of isomorphic cyclic groups. By Theorem 5.15, (It rl is not locally homogeneous and therefore, by Lemma 5.10, 01 is not locally (rl)homogeneous. Ne~t assume that cn. r  1 is a dire~t sum of isomorphic cyclic groups. Since 'i:' r ) : 2, by Theorem 5.15, at r is not locally homogeneous and therefore, by Lemma 5.10, 01 is not locally (r2)homogeneous. To show that en. is actually locally (rl )homogeneous, let f be a local (rl )automor~~fsm of at. By Theorem 5.8, f preselves "rl so that the reslriction f rl of f to 1V1. I is a local automorphism of ()1.rB~ Iheorem 5.15, Ol r is locally homogeneous so that f r_ 1 preserves height in Ol  . But then f preserves ~ and therefore, by Theorem 5.8, is a local rautomorphism of Ol. This establishes that Ol. is locally (rl )homogeneous.
«(]l
In the remainder of this section we shall use Lemma 5.3 to extend our results to abelian torsion groups. Accordingly, let Ol. be an abelian torsion group and recursively define the sequence of Ulm subgroups of en.
01 0 =(Jl.;
01.""+1:
n
new
n(01"');
mAo:
nOlo(.
«
D.M. CLARK, P.H. KRAUSS
278
where ?. is a limit orjt~al. The Ulm ~ of Ol. • denoted by " such that en"' = (h ,and we aga m set D( In)
L
i<.<.:>
m into
is the smallest
= <Jl. to( (}).)
If
is a direct decomposition of induction that for every 01..
't' (Ill.)
0'/..
J.
its pcomponents, then we verify by transfinite
Ol.,,(.
L tJi."!
i<w
J.
•
Theorem 5.18 An abelian torsion group is locally homogeneous if and only if each pcornponent is either a direct sum of isomorphic cyclic groups or is a direct sum of isomorphic quasicyclic groups. Proof:
Use Lenuna 5. 3 and Theorem 5.15.
Theorem 5.19 An abelian torsion group is relatively homogeneous if and only i f it has finite Ulm type. Suppose OL i f locally rhomogeneous.
Proof:
By Lemma 5.3, for each i<w, t:( Qi)';; r + 1.
O1 i is locally rhomogeneous and therefore, by Theorem 5.17,
Thus
(JL r+l
and therefore
t: ( m )
~
at ~+1 =
=
J.
r + 1.
L
m~+2 =
i<w
J.
Olr+2
The converse is proved similarly.
Theorem 5.20 Let m be an abelian torsion group of Ulm type homogeneity rank of Ol. is (i) r if O1 r1 is not locally homogeneous; (ii) rl i f O!rl is locally homogeneous. Proof: Let P be the homogeneity rank of ill. and for each the homogeneity rank of O1. . By Lenuna 5.3, i Moreover, since
P =max{Pili<w}. 2: <Jl. ~ = ()t r = Q r+l = ~ en. r+l
i <.w
J.
i
i
r < w.
i <W
,
let
Then the
Pi be
'
for each i<w, t:(C!1i)~r and therefore, by Theorem 5.17, Pi~r. Thus?.:;. r , Now, i f 01 rl is not locally homogeneous, then by Theorem 5.18 there exists J < W such that mjl is neither a direct sum of isomorphic cyclic groups nor a direct sum of isomorphic quasicyclic groups: By Theorem 5.17, fj = r and therefore
p
r
 Finally, if Ol.rl is locally homogeneous then by Theorem 5.18, for each i<w, 1 is either a direct sum of isomorphic cyclic groups or is a direct sum of isomorphic quasicyclic groups, and th~E!fore Pi~rl, by Theorem 5.17 and Corollary 5.9 respectively. Sins:e Oi. 1 Ol.r, there exists j < W such that 0l~1 1and therefore is a direct sum of isomorphic cyclic groups. Thus by theorem ~.17, (Vi = rl arld it follows that p = r1.
<Jl.l
01:
m:
Corollary 5.21 Let Ol and
.
Proof:
Use Theorems 2.7 and 5.20.
Notice that Corollary 5.21 is a weak version of Ulm's Theorem. In fact, in this special case, Ulm's Theorem gives an analysis of G,) equivalence in terms of Ulm invariants.
RELATIVELY HOMOGENEOUS STRUCTURES 6.
Relatively Homogeneous Rings with Identity and Elements
~
279 NonZero Nilpotent
In this section we shall consider rings (R,+,,O,':> • The complete characterization of ~categorical rings is still an open problem. Macintyre and Rosenstein [7J have init~ated the study of Hocategorical rings by considering rings with identity and no nonzero nilpotent elements. We shall first indicate how this restriction leads to the more general problem of Kocategoricity in the variety generated by a quasi primal algebra. By Theorem 1.15, an Hocategorical ring has finite characteristic and therefore is a direct sum 1? i<s 1,
z
where each IR. is a definable ideal of prime power characteristic. It is wellknown that thIs reduces the problem of Kocategoricity to the case where ~ is a ring of prime power characteristic. Macintyre and Rosenstein discovered that if an K categorical ring ~ has no nonzero nilpotent elements then there exist a prime p agd a positive integer m such that ~ satisfies the following equations: px
x'y
0
y·x
Thus ~ is in the variety generated by a field with pm elements, and a finite field is a quasi primal algebra. (See Pixley [9J). Algebras in the variety generated by a quasi primal algebra can be represented by sections of sheaves over Boolean spaces (Keimel and Werner [5J). Macintyre and Rosenstein take advantage of a special case of this sheaf representation which is due to Arens and Kaplansky [1J and requires that the ring has an identity. Let X be a Boolean space with field f of clopen sets, let OL be a finite field and let n
such that
Lemma 6.1 (Arens and Kaplansky) If ~ is a countable ring with identity which belongs to the variety generated by a finite field ot , then there exists a Boolean space X with field 'IT of clopen sets, and for some n
In a Boolean space there exists a onetoone correspondence between closed sets and ideals in the field of clopen sets. For each closed Yi define
Y\
Yi={UElrll
Then is the ideal corresponding to Yi distinguished ideals (called an augmented
unYi=~J· and (fi. Y) is ~
of sets).
a field of sets with
Lemma 6.2 (Macintyre and Rosenstein) Suppose at is a finite field and X is a Boolean space with field ; of clopen sets. For each i < n , let Yi So X be closed and let iSi SOl ~be nontrivial. Then CC
O.M. CLARK, P.H. KRAUSS
280
Macintyre and Rosenstein characterize countable H categorical rings with identity and no nonzero nilpotent elements by characte~izing ~categorical augmented Boolean algebras. An augmented Boolean algebra is a structure 011= (1)'(0' Ji)i
= fXEMol
x·ISK}.
Let < H On), 1\ , v , ~ > be the subHeyting algebra generated by {(1) ,:it, J.1 i
L
i<s
12. , ~
s
where each ~i is a pcomponent of R which is isomorphic to a ring ACe( ;Y),(Ol. ,£r)] of sections or a Boolean sheaf and the augmented field of sets ( f ' Y) is Hocategorical. (iii) ~ is uniformly locally finite and relatively homogeneous. Corollary 6.5 The class of rings with identity and no nonzero nilpotent elements has a uniformly algebraic Hocategoricity problem. Proof:
Use Lemma 2.16 and Theorem 6.4.
The equivalence of (i) and (ii) in Theorem 6.4 is again due to Macintyre and Rosenstein and follows easily from Lemmas 6.1 and 6.2. To prove the remainder of Theorem 6.4 we need a substantially stronger "transfer principle" than Lemma 6.2. Theorem 6.6
Suppose ~ is a finite field and X is a Boolean space with field For each i < n, let Y. =X be closed and let
f9' of clopen sets.
trivial.
Then
RELATIVELY HOMOGENEOUS STRUCTURES
281
is relatively homogeneous. Our proof of Theorem 6.6 is rather long and requires a substantial amount of technical detail. It will be presented in the more general context of an investigation of ~ocategoricity in quasi primal varieties. Proof of Theorem 6.4 Assume (ii). Then ~ obviously is uniformly locally finite. and it is easy to see that 'R. is relatively homogeneous i f and only if each ~.1relatively is relatively homogeneous. By Theorem 6.3. each augmented field of sets (f .Y) is homogeneous and therefore. by Theorem 6.6. each ring C[(f .Y).(ot.
r
Lemma 6.7 Proof:
Use Corollary 1.12.
Let K be the set of a+l local automorphisms of Tn' which are isomorphisms between finite frames ofm. Since on is locally finite. by hypothesis m has a finite frame. Lemma 6.B
(Macintyre and Rosenstein)
For each J €Ho(m),
let PJ be the number of atoms of s =
Finally let Lemma 6.9
m
K has the finite extension property.
is locally (s
t
L.
J€ Ho(m)
t)homogeneous.
PJ .
'~/J
and let
D.M. CLARK, P.H. KRAUSS
282
Proof: Let fl; LM{s+t)(Oll). Since m is locally finite, there exists g" LMt(m) such that f<e g and the domain of g is a finite frame of m. Since r.;;t, by Lemma 6.7, geLM('ln'). By Corollary 1.12, g preserves the formulas lfJ and "i'J' for each JEo Ho(m>. Thus the range of g is also a finite frame of 11'1 and g E K. We have established that and by Lemmas 3.3 and 6.8, ·nt is locally (s + t)homogeneous. The proof of Theorem 6.3 is now complete.
References 1.
Arens, R.F. and Kaplansky, I., Topological representation of algebras, Trans. Amer. Math. Soc. 63(1948), 457481. 2. Henson, C.W., Countable homogeneous relational structures and ~ocategorical theories, J. Symb. Logic 37(1972), 494500. 3. Hill, P., Extending automorphisms on abelian pgroups, Appendix to this paper. 4. Kaplansky, I. and Mackey, G., A generalization of Ulm's Theorem, Summa Brasil. Math. 2(1951), 195202. 5. Keimel, K. and Werner, H., Stone duality for varieties generated by quasiprimal algebras, Memoirs Amer. Math. Soc. 148(1974), 5985. 6. Krauss, P.H., Quantifier elimination, Logic Conference Kiel 1974 (Proceedings), Berlin, Heidelberg, New York (Springer Verlag, 1975). 7. Macintyre, A. and Rosenstein, J.,' l4 categoricity for rings without nilpotent elements and for Boolean structurgs (Preprint). 8. Morley, M. and Vaught, R., Homogeneous universal models, Math. Scand. 11(1962), 3757. 9. Pixley, A.F., The ternary discriminator function in universal algebra, Math. Ann. 191(1971), 167180. 10. Rosenstein, J.G., Wocategoricity of linear orderings, Fund. Math. 64(1969), 15. 11. Rosenstein, J.G., Wocategoricity of groups, J. of Algebra 25(1973), 435467. 12. Sacks, G.E., Saturated ~ Theory, Reading (W.A. Benjamin, 1972).
APPENDIX EXTENDING AUTOMpRPHISMS OF ABELIAN pGROUPS Paul Hill Florida State University Tallahassee, Florida 32306 This note is in the nature of an appendix to the preceding paper [1] by D. Clark and P. Krauss. Our purpose is to show that certain automorphisms can be extended on abelian pgroups. In particular, we prove Lemma 5.7 in til, which is the reason for the inclusion of this note in the present series of papers. Let G be an abelian pgroup. As usual, we inductively define p~G for each ordinal c{, by the equations poC.+ 1G = p(p"'G) and p~G = p"'G whenever
I"
nG'
1lL<
is a limit ordinal.
By the height of an element
x ",G,
denoted by
RELATIVELY HOMOGENEOUS STRUCTURES H(x) ,
we mean an ordinal number or the symbol
{:
H(x)
00
More specifically,
•
if
XE'tG
if
x s p"'G for all
but
283
x4 p"'+1G 0/,
The symbol p"'G[p] denotes the set iXe P"'"G: px = OJ. The rudiment of the proof of the fOllowing lemma can be found in [2J and [3]. Lemma Let A and B be finite subgroups of the abelian pgroup G and let be a fixed ordinal. Suppose that 'f is an isomorphism from A onto B that satisfies the following property for each a € A: ;L
H( (PA. ):
[
r
H(a)
(a»
H( r(a» > A
whenever
H(a)
A.,
whenever
H(a) » A.. •
~
and
X"
For each ordinal cI., < A there exists p<'G CpJ having the property that H(xta) ~ oL for each a E A if and only if there exists y .. p"'GCp] having the property that H(ytb) ~ 0(, for each b € B, Proof:
For the fixed ordinal .,("
Similarly, let
let
p"'GCpJ = S
A!:.. = {a "An p"'G
ats
B: = { bE B () p"'G
bs E p<+1G
Ii
<$
p"'tl G for some for some
p"t1G[pJ ,
Set
s e S} , s .. S} ,
Observe that A:l = {a e An p"'G: pa e p"'t2G 3; if pa E p"'t 2G, we can write pa = pg where gEp"tlG, If aEAnp"'G, agEpol.G[p] • Thus we can write ga = stz, where s "S and z E polt1GCp] . Therefore, as e p'"t1 G and a E AJ:.. Since and
A:" = {a EAnp"'G
pa Epol.t2G}
B;={beBl)p"G
pbEp"'t2 G 3
The restriction of 'f is an isomorphism from A* onto B* because A!f. is a finite subgroup that maps into B~ under 'f; we are,~of course~ making strong use of the fact 0<.<:>and that 'f satisfies condition (P;>..)' We continue the proof of the lemma by defining and
S'(A) = {St; S
sta Ep",tl G for some
at.A!f.l
t s E: S
stb E: p<=<+l G for some
b e B~} •
S*(B) =
For simplicity of notation, let inclusion relations
Ao/,t1 = A 0P"'t1G
A2A~2A"'+1 and
and
Bo(.t1 = Bn
f+1G •
Observe the
B2B.;l2Bo<.+l'
Moreover, the isomorphism ~ :A~~ B£ induces an isomorphism A~/A"'t1~ B,;':/B...t1 since 'f must map Ao( t1 onto B", t1' We also notice that there are natural isomorphisms: for example, the first of the two preceding isomorphisms is induced by the correspondence s ~ a obtained from the description of the set S*(A)
=ts
E
S:
sta Epo(.+lG for some
a E A!:.}.
In view of the above isomorphisms, we conclude that S*(A) and S*(B) are both finite since A~ and Bt are. Moreover, since we have established the existence of an isomorphism
284
D.M. CLARK, P.H. KRAUSS S*(A) ~ A~/A<J.+1 r:> B:IB"'+1:>7 S*(B)
from S*(A) onto S*(B), we conclude that S = S*(A) if and only if S = S*(B) because S*(A) and St'(B) are finite. Since there exists x E p«G[p1 having the property that H(x+a) 'It 0<. for each a E A i f and only if S 7 S*(A) and since, likewise, there exists y E p"'G[pl having the property that H(y+b h 0(. for each b e B i f and only i f S 7 St'(B), the lemma is proved. Theorem Let A and B denote finite subgroups of the abelian pgroup G. Suppose that 'f is an isomorphism from A onto B that satisfies the following property for each a E:A: H( If (a» = H(a) whenever H(a)~ ),.+ 1 (P?,+1): { H( 'f (a» >;t + 1 whenever Hfa ) :> ),.+ 1 Suppose that onto ,
px s A where XE'G. Then there exists an extension of 'f from
Proof: We may, of course, assume that x f A. Furthermore, we may assume that x is proper with respect to A since A is finite; recall that x is proper with respect to A if H(x+a) <> H(x) for each a E A. In the remainder of the proof, several cases are distinguished. case 1: X" p?.+lG. Then px Eo p:>"+2G• Since 'f' satisfies (P;>..+l) on A, we know that 'I'(px) " to+ 2G. Choose WE G so that w € p:>"+1G and pw = 'f (px ) , If w If. B, we set y = wand extend the mapping 'f' to < A,x '> by mapping x onto y; more generally, we map nx+a onto ny+b where b = 1j>(a). Since y = w the extended map remains an isomorphism between and < B,y>. Observe that x+a has height ~+1 or greater exactly when a has height )"+1 or greater. Furthermore, if the height of a does not exceed ~, then x+a and y+ ~(a) have the same height because each has the same height as the element a. It follows easily that the extension of 'f satisfies property (P),.) on < A,x>. Incidentally, we do not know that the extension continues to satisfy property (P:>..t 1 ) because x and y may not have the same height even though both belong to p~ IG: for example, one could have height 'X. +1 and the other 'A +2. We have only finished the proof of Case 1 when w 4B. Now suppose that the element w chosen above does belong to B. Choose c in A so that ~(c) = w. Since 'f' satisfies (P.:I.+1 ) on A and since WE p:>"+1G, we know that H(c)~;l,. +1. Furthermore, since 'f' (p(xc» = 0 and since 'f' is monic on A, we conclude that ptxe ) = 0 and that xcEpA+1GCp1. For each aEA, it is necessary that H(xc+a) ~ H(x) because x was chosen pro£er with respecI to A and because c is also in A. Using the fact that An p;t+ G and Bn p"+ G are finite and isomor~hic under 1., we conclude p"+1 GCp J 7 Bn p>'+lGCp] in view of the fact ?+ G[pJ 7 A'l p"'+lGCpJ; £ht element xc can be used to demonstrate the latter inequality. Choose Z E P + GrpJ outside of B, and set y = w+z. Then y ~ B and YE p:41G. The extension of ';f that maps x onto y is an isomorphism from that satisfies property (P:>..) on . Case 2: x f p>'+l G• Let H(x) = «: <::, ~ • We need to distinguish two subcases. case 2A: px Ep"'+2 G; this case actually encompasses the case that p(x+a) E poL+2G for any a e A such that x+a e p"G (for we can exchange x for x+a). Since we are assuming that pxepl+2G, choose gepol+1G so that px = pg , Set u = xg and observe that H(u+a);;;o<. for each aE'A; if H(u+a):>o<. , then x+a is contained in poL+1G since g is. However, this is contrary to the fact that x is proper with respect to A and H(x) = d.. Since Dl < x, +1 and since satisfies (P~+l) on A, we can conclude from the lemma that there exists v in p"'GCP] having the property that H(v+b)~o<. for each b s B. We let y = w+v where w is chosen, as before, so that WE' p"'+1G and pw = If' (px ) , If we extend 'f by mapping x onto y, the extension is an isomorphism from < A,x:> onto that satisfies property (P A) on < A,x>. Case 2B: H(p(x+a» ~ 0<. +1 whenever x+a E p"G and a E A. Let y E po(G be such that 'j'(px) = py. First, we claim that H(y+b)";o<. for each bEB. Suppose that H(y+b)~O<+1 for some bEB. Choose a in A so that 'f(a)=b. Since YE:p"G
tB,
r
RELATIVELY HOMOGENEOUS STRUCTURES
285
and yTb E p"Tl G, we know that b ~ pcl.G• Thus a and consequently XTa are contained in pcl.G• However, p (xea ) e:p"'T2G because 'f(p(xTa» = p(YTb)G pclT2G• This is contrary to case 2B. Thus we must conclude that H(yTb)"o<. for each b e B, It follows that y has height exactly ~ and that y is proper with respect to B. The extension of 'f that maps x onto y has the desired properties, and the theorem is proved. Corollary (Lemma 5.7 in [1]) If 'f:A.,....~ B is an isomorphism between finite subgroups A and B of an abelian pgroup G and i f 'f satisfies property (P"',Hn) for each positive integer n, then for any x E G there is an extension of 'f from onto /AI pn, we know that n is the smallest nonnegative integer with the property that pn x EA. Now apply the theorem repeatedly n times; if n = 0, there is nothing to prove. In the first application of the theorem, we use the element pn1x and obtain an extension to < A,pn1x > that satisfies property (P",«Tn1)' The process is continued until we reach with an extension that satisfies property (Pw~)'
References 1. 2. 3.
D. Clark and P. Krauss, Relatively homogeneous structures. 1. Kaplansky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor. 1. Kaplansky and G. Mackey, A generalization of Ulm's theorem, Summa Brasil Math. 2(1951), 195202.
R. Gandy, M. Hyland (Eds.). LOGIC COLLOQUIUM 76 © NorthHolland Publishing Company {1977}
APPLICATIONS OF LOGIC TO THE PROBLEM OF SPLITTING ABELIAN GROUPS Paul C. Eklof* University of California, Irvine
Throughout this paper the word "group" will mean "abelian group."
We shall
always identify a cardinal with an initial ordinal and an ordinal with the set of its predecessors (so that, for example, ~l
= wI =
V
I
V < wI ) .
If A is a set
or a group, we let IAI denote the cardinality of A. 1.
THE PROBLEM OF SPLITTING GROUPS. One of the most fundamental problems in the study of the structure of abelian
groups is to determine when a group splits i.e. is the direct sum of a torsion subgroup and a torsionfree subgroup. If a group B splits  say B = B ~ B l 2, where Bl is torsion and B2 is torsionfree  then the torsion subgroup is uniquely determined: Bl is in fact the subgroup of all torsion elements of B (i.e., elements b such that nb = a for some n ~ a), called the torsion part of B and denoted T(B). The subgroup B is only determined up to isomorphism: it is isomorphic to B/T(B), 2 but by an abuse of language we shall refer to BIT (B) as the torsionfree part of B. The problem of when a group splits was studied by Baer (1936] who determined all torsion groups T with the property that B splits whenever T(B) is isomorphic to T.
These groups are in fact precisely the torsion groups which are the direct
sum of a divisible group and a group of bounded order. (torsion) cotorsion groups.
Such groups are called
The analogous problem for the torsionfree part was
solved by P. Griffith [1969J who proved that a torsionfree group J has the property that B splits whenever B/T(B) is isomorphic to J if and only if J is free. Baer considered also the general problem of when knOWledge of the isomorphism types of the torsion and torsionfree parts of B is sufficient to insure that B splits.
We shall make use of the notation of homological algebra in order to state
the problem succintly: we write "Ext (J,T) = 0" to mean that B splits whenever T(B)
~
T and B/T(B)
that Ext (J,T) = O.
~
J.
The problem then is to determine the pairs (J,T) such
In the sequel we shall refer to this as the splitting problem.
If we restrict the question to pairs (J,T) where J has fixed cardinality K we shall call this the splitting problem for cardinality K.
*Research supported by NSF MPS7406409AOl
287
PAUL C. EKLOF
288
Baer [1936] solved the splitting problem for cardinality ~O. to his criterion in this case as Baer's criterion;
We shall refer
we will not state it here, but
we do want to mention an easily derived consequence of it (see §2 of Eklof [197J). 1.1
Proposition.
For any torsion group T there is a sentence 8
that for any countable torsionfree J, J~8T
<=)
Ext (J,T)
= O.
of L such T Wl W Moreover, the
sentences B may be chosen so that if T is elementarily equivalent to T then 2 l T BTl = BT2 Remark. We have used the notion of elementary equivalence for convenience, but in fact a stronger result holds. If T and T have the property that for any l 2 prime p and any n e ill , Pn T [P] /p neL T [p ] is zero if and only if l l n n+l p T [pJ/p T [p] is zero, then BTl and B can be chosen to be the same. 2 T2 2 It follows from 1.1 that Baer's criterion is absolute: 1.2 of
Corollary.
m; let J,T
Suppose
mand
~
are models of ZFC such that
~
is an end extension
be respectively, a countable torsionfree group and a torsion group
which belong to
m.
Then m~ Ext(J,T) = 0 if and only if ~ ~ Ext(J,T) = O.
In the following sections we shall see that the analogs of 1.1 and 1.2 do not hold if the assumption of countability of J is dropped.
We shall, however,
obtain a solution to the splitting problem under the assumption of the Axiom of Constructibility (V = L). Before proceeding to these considerations let us close this section with a
reminder of a basic fact of homological algebra which will play an important
role (see for example Fuchs [1970J, for a proof). 1.3
Lemma.
If J and T are groups such that Ext (J, T)
0 and J' is a subgroup
of J, then Ext (J', T) = O. 2.
THE SPLITTT.NG PROBLEM FOR CARDINALITY
~
1.:
Our approach to the splitting problem is to proceed by induction on the cardinality of J.
The result of Baer referred to above gives a criterion for
Ext (J, T) = 0 when J is countable.
Thus, to solve the problem for pairs (J, T)
when J has cardinalitY~l we want to r~duce ourselves to the consideration of countable subgroups.
We begin with some considerations about the "small" subgroups
of an arbitrary group J. For any infinite cardinal K and any group J of cardinality K, let 8 K (J) th~ set of all subgroups of J of cardinality
denote
8 K(J ) by inclusion.
A chain in 8 K( J ) is a subset C of 8 (J ) which is totally K A chain C in 8 (J ) is called closed (or smooth) if K UX e C whenever UX e 8 K( J ) ; C is called unbounded if U C = J. If C is closed and unbounded we say C is a cub chain. A subset E of 8 ( J ) is called stationary K in 8 ( J ) if for every cub chain, C, in 8 (J ) , En C f ¢. K K ordered by inclusion.
SPLiTTING ABELIAN GROUPS
289
By Lemma 1.3 a necessary condition for Ext (J,T) = 0 is that Ext (AO,T) = 0 for all A @K(J). In general this is not a sufficient condition 08 (although it is if \J\is singular: see §3). In order to obtain a sufficient condition we define ET(J) = {ADc
tf\, (J)
: Ext (AD,T)
f 0 or
aA{ @K(J) (AO ~ Al and Ext (AI/AO,T) f O)J The following theorem is essentially proved in Eklof (197J, using ideas of Shelah (1974 J. 2.1
Theorem.
IJI = K and
Let T be a torsion group and J a torsionfree group.
t..
cf(K).
Suppose
Consider the following three conditions:
(1)
Ext (J,T) = 0;
(2)
ET(J) is not stationary in @K(J);
(3)
There is a cub chain
{Aviv
< t..} in @K(J) such that
Ext (AO,T) = 0 and Ext (Avrl/AV,T)= 0 for all V < Then
(i)
t...
Conditions (2) and (3) are equivalent and imply (1);
(ii) (V = L)
If K is a regular uncountable cardinal (i.e.,
K = A ~ ~l) andlT\ ~ K, then (1) implies (2) and (3). Remarks on the proof. of the definitions.
The equivalence of (2) and (3) is an immediate consequence The implication (3)
~
(1) involves only some elementary
homological algebra (see Theorem 1.2 of Eklof (197J).
The proof of (b) makes
use of a consequence of the assumption V = L formulated by Jensen (1972J; in fact, if (2) fails, i.e., if ET(J) is stationary, the combinatorial principal OK(ET(J»
is used to prove that (1) fails.
The implication (1) ~ (2) is not a
theorem of ZFC; in fact\ assuming Martin's Axiom plus the negation of the continuum hypothesis we can prove that there is a torsionfree group J of cardinality
~l
such that for any countable torsion group T which is not cotorsion, Ext (J,T)
=
0
but ET(J) is stationary in @K(J) (cE Theorem 3.1 of Eklof (197J). Theorem 2.1 will give problem for cardinality
~l
~s
almost immediately a solution to the splitting
(at least).
Let us, however, first introduce a formal
language in which the criterion for splitting can be expressed.
The language we
use was introduced by Shelah [1975bJ and studied by Barwise and Makkai (197J. Let L' be a first order language i.e., a collection of predicate and relation symbols.
Add to L; an arbitrary set of unary predicate symbols Ai to obtain a
language L.
If
~
and
t.. are infinite cardinals (or the symbol
~),
the formulas
of L ,( stat) are defined as in the usual definition of L,  i.e., we allow ~"
~
conjunctions over fewer than
~
~"
formulas (or any set of formulas if
~
= ~)
and
PAUL C. EKLOF
290
quantification over fewer than A variables (or any
~
of variables if A
oo) 
with the addition of one new rule of formula construction: If
is a formula of
~
L~A (stat)
then so is
stat~
We introduce the following abbreviation: ~~
is an abbreviation for ,statA'qJ
The semantics of L, (stat) is the usual one with, in addition, the following rule, ~"
~
where m is a model of cardinality K.
ml=s~tAtp
i f and only i f {AWi\<mr\<m,
A)F~}
is stationary in @K(m). Now let L' be the language of group theory.
Let T be a fixed torsion group
and consider the following sentence: ....,
s~tAo
[(Ext(AO,T) Ext(AI/AO,T)
'1
0) V
(s~tAl
(A
O
,;; Al 1\
F O»J
Using Proposition 1.1 we replace the expression "Ext(AO,T) of 0" by a formula of L we do a similar thing for "Ext(Al/AO,T) '1 0", making use of the definition Ol l Ol; of a quotient group to translate Baer's criterion into a formula of L Thus OlIOl .T can be regarded as a sentence of LOllOl(s~t) involving the predicates AO and AI; it is the sentence which makes the following true.
2.2
Theorem (V = L).
For any torsion group T there is a sentence WT of L (stat) OlIUJ
 depending only on the elementary theory of T  such that for any torsionfree group J of cardinality Proof.
~l' JpW <=> Ext(J,T) O. T This result is an almost immediate consequence of Theorem 2.1, but there
are two points to observe.
First, the substitution of the quantifier "stat AI"
in the definition of W for the quantifier ":;[ AI" in the definition of ET(J) is T harmless because of Lemma 1.3. SecQndly, we must remove the restriction that ITI ~ ~l in the statement of 2.1(ii).
Bya result of Sasaida [1956j, we may
assume that T is a direct sum of cyclic groups.
Then we claim that T = T' Ell T"
such that T' is a countable group with the property that for any countable J, Ext (J,T) = 0 if and only if Ext (J,T') = O.
To see this, one may either apply
Proposition 1.1, using the Szmielew invariants (Szmielew [1955j or Eklof  Fisher [1972J) or use the remark following Proposition 1.1.
If IJ\
= ~l
and Ext (J,T)
then Ext (J,T') = 0 since Ext (J,T) = Ext (J,T' Ell T") = Ext (J,T') Ell Ext (J,T"). Hence by Theorem 2.1(ii) ET,(J) is not stationary in @Oll(J), but by choice of T', ET,(J)
= ET(J).
Since the
This completes the proof.
la~guage
LOOK is powerful enough to describe up to isomorphism
all groups of cardinality
(Note,
= 0,
SPLITTING ABELIAN GROUPS
291
however, that this result is less constructive than 2.2 since we have not yet shown how to explicitly exhibit the sentence 2.3
~T (K~.
Theorem (V = L). For any torsion group T and regular cardinal K there is a
sentence
~T (K)
of LooK(ssat) such that for any torsionfree group J of cardinality
K, J~~T(K) ~>Ext (J,T)
= O.
A comparison of 2.3 and 1.1 shows that the given solution of the splitting problem for uncountable cardinals is more complicated than Baer's criterion
~O' in that the sentences ~T(K) involve a quantifier
for the case of cardinality s~t,
which is second order in character.
The following result suggests that this
is unavoidable. 2.4
Theorem (V = L).
Let K be a regular uncountable cardinal which is not
weakly compact and let T be a torsion group which is not cotorsion. exists torsion free groups J
and J
Then there
of cardinality K which satisfy the same
l 2 sentences of LOOK but such that Ext (Jl,T) = 0 and Ext (J
f O. 2,T) In particular, the solution of the splitting problem for cardinality
~l
cannot be expressed in the language L ~ fortiori. the criterion cannot be oowl expressed in the language LWlW(Q), where Q is the Chang (equicardinal) quantifier, which is a sub language of L (stat) (cf. BellSlamson [1969J, pp. 267ff). WIW ~ The hypotheses that K is regular and not weakly compact are essential in
Remark.
Theorem 2.4.
For regularity, see £3; for weakly compact see Theorem 1.6 of
Eklof [197J or Mekler [1976]. Using Theorem 2.1 we can obtain by induction on the splitting problem for cardinality cardinal
~W
~ntl'
n~
explicit solutions to
However, when we get to the singular
we are temporarily stymied since Theorem 2.1 does not apply.
We deal
with this case in the next section. 3.
THE SPLITTING PROBLEM FOR ARBITRARY CARDINALITY. Given a set U and a regular infinite cardinal A, let
~
(D) denote the filter
on g(D) (the set of all subsets of D) generated by all X eg(g(D»
satisfying
(i)
X is closed under unions of chains; and
(ii)
for all S e g(U), there exists HeX such that S \H\ ~
lsi
~
Hand
+ p for some cardinal p < A.
(An element of KA(U) satisfying (i) and (ii) is called a generating element.) It is easy to see that KA(D) is a Acomplete filter on g(D) and that K (D) ~ KI'(D) i f A ,;; K. If P is a property of subsets of U we shall say that P A holds for 'almost all subsets(w.r.t.K (U» if {Seg(U): S satisfies belongs A to ~ (D) (cf. Kueker [197J). In the case A = ~l we shall omit the parentheti
PJ
cal reference and simply say that P holds for almost all subsets.
For example,
if D is a group, almost all subsets of U are subgroups and if U is a direct sum of countable groups, then almost all subsets of U are direct summands of D. latter because if U=
i~IAi'
is a generating element of
where each Ai is countable. then X = ~l(U»,
~'JAi:
J
>;;
(The I}
PAUL C. E:KLOF
292 3.1
Definition.
Let J be a nonempty class of abelian groups.
We shall say
that J is an abstract notion of free if it satisfies: (0)
J is closed under isomorphism;
(1)
If A~ and A' ~ A, then A'
(2)
If
o}
{A~l~ <
€
J;
is a closed chain of elements of J
such that AV+I/A~ belongs to J for all ~ < 0, then (3)
U{A~I~
o} belongs
<
If A € J then
AlA'
€
to J;
J for almost all subsets A' of A.
FJr example, if J is the class of all free groups, then J is an abstract notion of free.
(Property (3) holds by the remarks preceeding the definition).
The following theorem was proved by Shelah [1975aJ in a more general setting. 3.2
Let J be an abstract notion of free and let A be a group of
Theorem.
singular cardinality such that all subgroups of A of smaller cardinality belong to
J.
J.
The A belongs to
Now we can return to the splitting problem. J
T
For any torsion group T, let
be the class of all torsionfree groups J such that Ext (J,T) = O.
Our aim
is to prove the following result. 3.3
Theorem (V
L).
J
is an abstract notion of free. T Theorems 3.2 and 3.3 together with 2.1 yield, by induction, a solution to the
splitting problem for arbitrary cardinality under the assumption V = L. we know that J
T
notion of free.
Now,
satisfies (0), (1) and (2) of the definition (3.1) of an abstract (Property (1) holds by 1.3 and (2) by 2.1(i»
(3) that gives us some difficulty.
It is property
In fact, it appears to be necessary to prove
Theorems 3.2 and 3.3 simultaneously by an induction on cardinality. theorems are immediate consequences of the following Theorem 3.4.
These two In the course
of proving 3.4 we shall give an account of Shelah's proof of Theorem 3.2 which we hope will serve some readers as a useful introduction to the more general arguments of the paper of Shelah [1975aJ. Recall that in the proof of 2.2 we showed that given T there is a countable T f such that for any J, Ext(J,T) 3.4
Theorem (V = L).
=0
implies Ext(J,T')
= O.
For any torsion group T there is a countable torsion
group T' (depending only on the elementary theory of T) such that for any torsionfree J, if K (I) (II)
IJI then: J €:l
T
i f and only i f J c
(Shelah)
Jr'
I f K is singular and J' e J
T
for all subgroups
J' of J with IJ'I < K, then there is a closed chain
{A)~< ef(l,)} A~ €
J
J r and
of elements of @K(J) such that
AV+l/A~
belongs to J . T
e J
r
for all V < A.
Consequently
SPLITTING ABELIAN GROUPS If J €
(III)
~T
293
and K is uncountable then J/J' €
~T
for almost all subgroups J' of J. Remark.
By definition of "almost all", 3.4 (III) implies Theorem 2.1(ii).
Hence our proof must make USe of the assumption Proof.
V
L.
The remainder of this section is devoted to the proof of Theorem 3.4.
We prove (I), (II) and (III) simultaneously by induction on K. Case l:K
Then (I) is true by 1.1 and (II) and (III) are vacuous.
= ~~
Case 2:K is a regular cardinal be as in the proof of 2.2. implies J € ~T"
~ ~l~
We shall prove (I) first.
Then for J of arbitrary cardinality, J
Now suppose J € ~T' and IJI
=
K.
Let T'
€ ~T
By Theorem 2.1(ii), J is
the union of a closed chain {Aviv < K} of subgroups of J of cardinality < K such that Av € ~T' and Avtl/A € ~T' for all v < K. By inductive hypothesis v A € ~T and ~l/Av € J for all \i < K. Hence by Theorem 2.1(i), A belongs T v to ~T' Since (II) is vacuous in Case 2, we turn to the proof of (III). First we need a definition and a lemma. {Aviv <
o} of
subgroups and H (H)
3.5
Lemma.
=
~
{v <
If J contains the union of a closed chain
J, define
01
(H n Avt l)  (H n Av)
(1)
\(H)!"; IHI
(2)
If H is the union of a closed chain
(3)
Given X € v
(H)
=
U {(H~) \~ K
WI
<
f
¢}
{H~I~ <
T}
(J) for all v < T, let Y = {H
T},
then
~ AI H€X, , v
for all v € (H>}.
Proof.
Then Y € K (J). WI Since (1) and (2) are immediate from the definitions, we proceed to the
proof of (3). We shall in fact prove that Y is a generating element of K (J). WI It is clear that Y is closed under unions of chains, so it suffices to prove that for any H? € @(J) there exists H € Y such that R b Hand IHI ~ IRol + ~O. O = nL~I\i€
Define zo
KWI(J) ~ ~+(J) and ~+(J) is \+  complete.
Zo
So there exists HI €
such th~t H ~ HI and IHII ~ IHol + \ = A. Continuing, we obtain a chain O {Bnlh€Ul} such that Hn+ l € n{X)V€(H n>}, and IHnt11 ~ IHnl + x. Let H = nltwHn Then H \;; H, IHI ~ A and by (2), H €Y. This completes the proof of the lemma. O Now we may prove (III). Suppose J €J • By (I) we may suppose that T is countable.
T
Then by Theorem 2.l(ii), J is the union of a smooth chain {Aviv < K}
of subgroups of J of cardinality < K such that for all V < K, AK+l/AK€ ~T' By Avtl/~ and "pulling back
applying the inductive hypothesis (III) to the group
to AWl" we obtain an element Xv of K (AWl) such that for all H WI Av+l/(A + H) € ~T' Define V Xv • {H ~ J IH Avt l e Xv}
n
€
Xv'
PAUL C. EKLOF
294 We claim tha t Xv E:K:llJ. 0).
Now
Xv
is clearly closed under unions of chains, so such that S ~ H
it suffices to prove that for any S e g(J) there exists HeX and IHI ~
lsi
(S nAv+l) ~ Hand
Since \I X e K(1)1 (Av+ 1) there exists HexV such that
IHI
\s
~
desired element of Xv' Y e K
V
v
+ ~O.
nAV+I1 + ~O.
If we let H ~
HU S
then H is the
This proves the claim., Now by Lemma 3.5 there exists
for all v e (H\ We claim that for { _ v Since J/H ~ U A + H/Hlv < KJ it suffices, by Theorem 2.1(i), v T• to prove that (Av+l + H)/(A + H) e.7 for all V < K; but v T (Av+l+ H)/(A + Jt) =:. AWI/Av + (H n Av+l) so i f v c (H) then (Awl + H)/ (~ + H) e.7T v by definition of Y; and if V ~ (H) then H A ~ H A and vr l v + H)/(A + H) ~ Av+l/A e .7 This completes the proof of (III) in Case 2. (A T. vt l v v Case 3: K is singular. Let us first observe that if (II) holds, then (I) OJl
Hey,
(J) such that Hey if and only if HeX
.l"H e.7
n
n
and (III) follow by the same arguments as in Case 2 except that in place of Theorem 2.1(ii) we use (II) to write J as the union of a closed chain with the desired properties.
It therefore remains only to prove (II).
Fix a torsion group T.
We know that.7 satisfies properties (0), (1), (2) T of Definition 3.1 andby inductive hypothesis  property (3) for A of cardinality
< K.
For the remainder of the proof we shall use the following terminology:
if A and B are torsionfree groups, say that A is free if A belongs to .7 T (i.e., Ext (A,T) ~ 0) and A is free over B if A + B/B belongs to .7 We shall T. say that A is ~ (over B) if every subgroup of A of cardinality < K is free (over B).
We must, therefore, prove:
closed chain {Aviv < cf(K)
J of
if J is Kfree then J is the union of a
subgroups of cardinality < K such that for all
v < cf(K), Av is free and AWl is free over A (By Theorem 2.I(i) it follows v' that J is free). Since this is proved in Shelah [1975aJ we content ourselves with a fairly detailed sketch which will make it clear that we need 3.4(111) only for cardinality < K.
The following lemmas depend upon V
~
L in that they
depend ultimately on Theorem 2.I(ii). 3.6
Lemma (V
~
L).
A of cardinality < p. cardinal > S
>;;
H,
Proof.
~O
IHI <
Let
~O
<
~ ~
p
~
K.
Let X e
~
(A).
Let S be a subset of
If P is regular and A is p+free, or if p is a limit
and A is pfree, then there exists a submodule HeX such that P and A is pfree over H.
We may assume that X is a generating element of
that p is regular and A is p+free. module H of X, if S ~ Hand jHI < and HI/H is not free.
P
~
(A).
Suppose first
If the lemma is false, then for any subthere exists HI e X such that H ~ HI' \H I
\< P
By induction  using the fact that X is closed under
unions of chains  we construct a closed chain {Hvlv <
p} of elements
cardinality < P such that for all v < p, HWI/~ is not free.
of X of
But then V~KHv
is a subgroup of A of cardinality p which is not free by 2.I(ii), contradicting the hypothesis that A is p+free.
SPLITTING ABELIAN GROUPS
295
Now suppose that p is a limit cardinal and A is pfree.
Suppose S is of
Since A is r+ free, there exists H € X such that S ~ H,
cardinality r < p.
[HI ~ r and A is r+ free over H.
We claim that A is p free over H.
B be a subgroup of A containing H such that IBI < p.
Let
Since A is pfree, B is
c, leI
free.
Hence by 3.4(111) there exists C such that H ~
free.
Then C/His free since A/H is r+ free and hence B/H is free.
~ rand B/C is
This com
pletes the proof of Lemma 3.6. We now introduce the key definition, due to Shelah.
3.7
Definition.
Let
~
be a regular cardinal.
ordinals O! the property A is
PO!(~)free
We define by induction on
over B (resp. strongly
PO!(~)free
over B).
For any A and B, A is Po(}.,)free over B (resp. strongly PO(}.,)free
over B).
We say that A is Patl(}.,)free over B (resp. strongly Patl(}.,)free
over B) if A has a P decomposition over (0)
and for all
< A.:
V
(i) (ii)
and
t'
 decomposition over B (resp. strong P (A)O! i.e., a closed chain Aviv < AJ of subgroups A satisfying:
(~)
B)O!
A is free over (JJAAv) + B, (resp. A A is free over B; v A is free over A + B whenever V 1*1
~
<
~AAv);
V;
(iii) AV+l is pO! (~)free over A + B. v If O! is a limit ordinal, we say A is PO!(A)free over B if A is B for all 3.8
~
P~(}.,)free
over
< O!.
= L).
Lemma (V
(1)
Let IAI ~ K and O! < K A is (strongly) PO!(A)free over B if and only if A + B/B is (strongly) PO!(A)free
(2)
If A is (strongly) PO!(}.,)free for any O! > 0, then A is Afree
(3)
If A is strongly PO!(A)free, then there exists X such that if HO' HI PO!(A)free over HO'
Proof.
€
€
~+(A)
X and HO ~ HI' then HI is
Since (1) is proved easily by induction and (2) follows by definition
and Lemma 1.3, we turn our attention to (3).
In fact, we prove the following
relativized version of (3).
(3')
If A is a subgroup of U and A is strongly PO!(A)free over B + C, then there exists X
€
~+(U)
HO' HI in X, if HO ~ HI then HI (H
n A) O
We assume that O! given a strong
=
~
+ (HI
n B)
such that for any
n A is
PO!(h)free over
+ C.
+1 and that (3') holds for~.
P~(~)decomposition{Aviv
< AJ of A.
Suppose that we are
Then by 3.4(111)  for
groups of cardinality
is A+ complete, there is an element X of
HO' HI in X
~+(U)
such that for any
PAUL C. EKLOF
296
(a)
A is free over (H nAv) + B + C v O (b) A is free over (H n A + A~l + B + C v O v) (c) HI n Avr l is P~(\)free over H n A + (HI n (A + B) ) + C O v v Moreover, by definition of ~+(U) we may also require that any H in X satisfies: (d)
H n (B + C)
(e)
H n (A + B + C) = (H n A + (H n B) + (H n C) v v) H (Av + B) = (H B) "
(f)
(H n B) + (H n C)
=
n
n
We claim that if HO' HI € X and HO ~ HI then {HI nAvlv < AJ is a strong Pa(\)decomposition of HI nA over (H nA) + (HI n B) + C. We must verify that O conditions (O)(iii) of Definition 3.7 are satisfied. Condition (0) is trivial. As for 3.7(i), we must prove that HI nAv is free over (H nA) + (HI nB) + C. O By (a), A is free over (HO nAv) + B + C, so by Lemma 1.3, HI n A is free over v v (H nAv) + B + C. Therefore it suffices to prove that O (HI nA)I{(H l n A) n[(HO nA) + (HI nB) +
C]J
is isomorphic to (HO
n A) I{(HI'"' A) n[ (H a r
I
A) + B +
c]}.
It suffices to prove that expressions in braces ({ }) are equal, but this is a consequence of (d).
Since the arguments for 3.4(ii) and (iii) are
similar  they use (b) + (e) and (c) + (f) respectively  we shall omit them. This completes the proof of 3.8. Now suppose that J satisfies the hypothesis of 3.4(11) i.e., J has singular cardinality K and J is Kfree.
Let A = cf(K).
We prove by induction 
using Lemma 3.8  that J is strongly P (A)free for all V < A+.
+
V
we have proved that J is strongly Pa(A)free for some a < A. in 3.8(3).
Indeed, suppose Let X be as
Using Lemma 3.6 we can construct a closed chain {Hvlv < A} of
elements of X such that for all V < A, IH

v
I<
K, H
V
€ X and J/H
"'+
1 is Kfree.
Then { Hvlv < AJ is clearly a Pa(A)decomposition of A, so A is Pa+l (A)free. Thus, in particular, we have proved that if J satisifes the hypothesis of 3.4(11), then J is PW+l (cf(K»free.
To finish the proof of 3.4(11) it
suffices to prove that any Pw(cf(K»free group of cardinality < K is free, for then a Pw(cf(K»decomposition of J will be a chain with the properties desired for 3.4(11). If A is P (A)free, IAI S K and A s cf(K) < K, we prove by induction w on (\AI,A) that A is free (where we wellorder the set of pairs (p,A) by the relation (P,A) S (p:\/) if and only if 0 < pi or p = pi and A S A'). IAI < \ then the result follows from Lemma 3.8(2). follows easily from the definition of
~O < A < IAI;
Pw~O)free
If A
= ~O'
If
the result
and Theorem 2.l(i).
Suppose
then by Lemma 3.8(3) and the fact that ~+(A) is ~lcomplete,
there is an element X of
~+(A)
such that if H HI O'
€
X and H O
~
HI' then
SPLITTING ABELIAN GROUPS
297
HI is Pw(A)free over HO. Then A is the union of a closed chain {~lv < IAI} of members of X of cardinality < IAI. For all V < IAI, H\itl/Hv is Pw(A)free and 1H\itl/HV\ is < JAI, so by induction HV+l/Hv is free. 2.1(i), A is free. There remains the case ~O < A = IAI. from the succeeding lemma.
Hence by Theorem
The result in this case will follow
First, if {Aviv < A} is a Pa(A)decomposition of
A and H is a subset of A let [H] be the least b H
n (J
= H
~
A such that
n (V~b A)
(i.e. [H] = sup (H) where (H) is defined before Lemma 3.5). 3.9 Lemma (V = L). Let A be P (A)free and suppose IAI < Ie There exists w X e ~ (A) such that if HO' HI eX, H ~ HI and IH < A, then [H is a limit O O] l\ ordinal and HI is P w(cf[HO])free over H O. Assuming the lemma for the moment, let us complete the proof of Theorem 3.4. Let A be Pw(A) free where ~O < A = JAI.
Let X be as in Lemma 3.9.
Then A is
the union of a closed chain {Hvlv < AJ of elements of X of cardinality < A. By choice of X, HV+l is Pw (cf[Hv])free over H for all V < A, so since v cf [H ~ IH < A, we have by induction that H l/H is free. Therefore, v] V v+ V by Theorem 2.1(i), A is free.
I
It remains to prove Lemma 3.9.
It suffices to prove by induction on n > 0
that if A is a subgroup of U and A is Pn(A)free over B + C, there exists X e ~ (U) such that for any HO' HI in X, if HO ~ H, and IHll < A, then [H O] is a limit ordinal and HI n A is Pnl (cf[HO])free over (H n A) + (HI n B) + C. O Let us fix a Pn (A)decomposition {Aviv < A} of A over B + C and for each V < A, l fix a P 2(A)decomposition {A I~ < A} of A\it lover AV + B + C. Let [H] nV,~ and [H]v be defined as above relative to these decompositions, i.e., [H] = least b ~
A such
that H n ( U A ) = H n <,~.A ) and [H] = least y ~A v