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of a (proper) prime ideal
DEFINITION 6.1. precone (i)
Let
A
is a subset ~ of x,yc~
(ii)
x ~
(iii)
i~;
(iv)
xy~
for all
iff
of
{a~A I a/p e0} A
for each pair
and an order ~ on
be a commutative ring with unit. A
implies
2
P
This is done by giving
A/p. A prime
with the following properties: x+y~; x~A;
(xe~ A yc~) V (  X ~ ^ yc~).
Conversely, each prime precone ~ of A determines a pair
0
$<< which ~<< is a chain of
as
described above, by setting:
0 ~ a/p Notation.
iff
aE~
Given a prime precone ~ of A
real closure of the fraction field of by ~,
for
and by
~
aEA.
we shall denote by
k(~) the
A/~n_~ with the order determined
: A ~ k(~) the corresponding canonical map.
0
107
We shall now worry about the topology of the real spectrum, The topology of the prime spectrum, {D(a)
I a~A},
Spec
SpecR(A).
(A), is generated by the base
where D(a) = {P e Spec(A)
is the set of all It is natural,
P
such that
then,
a/p # 0.
to give SpecR(A)
H(a) of prime precones strictly positive
I a~P}
~ attributing
sign.
the topology generated by the sets a definite
sign to
n (a), say
Since the family of such sets is not closed
under intersection we are obliged to consider
finite sequences
of a's.
We are now ready for the formal definition of the real spectrum: DEFINITION spectrum of
6.2.
Let
A
be a commutative
A, SpecR(A) ,
ring with unit.
is the set of all prime precones
The real of
A
with
the topology generated by the sets H(a I ..... a n ) = { ~ S p e c R ( A ) for all finite sequences
I a I ~ ~^...A
an~}
a I ..... anEA.
Before studying the basic properties
of the real spectrum let us compute
a few examples. EXAMPLE 6.3. and
SpecR(k)
If
A = k
is a field,
EXAMPLE 6.4. the real line
The real spectrum of
in computing
irreducible say
sign.
are
r c~
Now,
{0}
~
D
ring of
4.6) are relevant
or of the form
(Q) with
,( Xr)
each ordering of
{0}
(smaller)
has to be
~r c S p e c R ( ~ [ X ] ) .
The
I F(r) ~0}.
gives rise to one point of the real spectrum for
~[X]. X
Q
is
is
We know (cf. Brumfiel
that the orderings
that the element
then
(Q)
~ ~, which has only one order.
gives rise to one point
prime precone
Thr prime ideal
is real closed,
~[X] l
~r = {F ~ [ X ]
X larger
the coordinate
(cf. Definition
~[X]
Since
Q = Xr.
corresponding

~[X],
{0}
above.
Q, the changing sign criterion 4.18 tells us that
real iff changes
[72; §I.5])
k, mentioned
the real spectrum.
Since the prime ideals of
Thus, each
of
~.
Note that only real prime ideals
linear,
the only prime ideal is
is the space of orderings
defines
in
of ~;
~[X]
[65; §7.5]
or Dickmann
are determined by the cut
these are as follows:
than every element of
~,
which gives rise to the
108
following
prime precones:
~+~=
{FE~EX]
I There is
a c~
such that
Fe0
on
(a,=)},
~. ~ =
{F ~ [ X 3
I There is
b c~
such that
F e0
on
( ~ ,b)}.
For each
r c~,
r < X < (r, ~ ) precones
X is infinitesimally
(( ~ ,r) < X < r,
larger
(smaller)
respectively).
than r, i.e.
The corresponding
are:
~r+ = { F ~ [ X ]
I There
is
E >0
such that
Fe0
on
(r,r +~)},
er = { F ~ [ X 3
I There is
E >0
such that
Fe0
on
(rg,r)}.
Observe ~r + ~ ~r
prime
that the only inclusion and
~r ! ~r
for
relations
r ~.
between
these precones
are
Thus, we can draw the picture
of
SpecR( ~[X]):
r
r
As a useful ~[X].
comparison,
Here the points
ing to each polynomial nomial {0}.
X 2 + aX + b,
r+
s
s
s+
let us draw a picture are the prime ideals Xr,
with
for a, b ~ ,
The latter is contained
tion holds otherwise.
OR
r ~;
4b < 0;
in all the others,
The picture
/'
~ (o)
one correspondto each poly
and the prime but no inclusion
is this:
(o,b) ..," • •
/ t/
~[X]:
one corresponding
a2
• • .// /////• . #~i•. I///°
"
of the prime spectrum of of
##
ideal rela
109
Note that in both the prime and the real spectrum, the set theoretic inclusion
~ c B
means that B is in the closure of the singleton
(in geometric jargon of B );
{~}
: B is an specialization of ~ , and ~ a generizaltion
this is easily checked from the definition of the respective
topologies.
In particular,
is dense in Spec ( ~ [ X ] ) . point of the variety
~
{~}
= Spec ( ~[X]),i.eo
the point
{0}
Geometers say that the ideal (0) is a generic (apparently this notion goes back to the Italian
school of 19th century geometers). The real spectrum of
~[X] does not have a generic point.
r , r , which together with ed as a sort of
r, form the closure of
"generic points" of the halflines
{r}
But the points are interpret
(r, + ~ ) and ( ~ ,r)
respectively. Exercise.
Compute
EXAMPLE 6.5.
SpecR(~[X](x_r)) , for
r cA.
In order to give an inkling on how the real spectrum
reflects geometric properties of algebraic varieties,
let us look at an
example having more "singularities" than the preceeding one. der the plane curve C of equation y2 _ X 3 _ X 2.
We consi
~u
==L
As before, each point since ~[C]/Mp ~ and maximal ideal at P).
P ~ C(~) gives rise .to just one point ~pESpecR(l~fc]), R
Since the polynomial defining of
R[C];
furthermore,
has only one order
C
is irreducible,
(~
denotes the
{0}
is a prime ideal
it is easily checked to be a real ideal (exercise).
The orderings of ~ [ C ] are computed as follows. The ring ~[C] is the quadratic extension of ~EX] in which the equation y2 = X 3 + X 2 holds; in particular,
X 3 + X 2 e0,
i.e.
Xel.
Hence, each ordering of
~[X] satisfying this restriction gives rise to two orderings by choosing the sign of
Y.
preco es are thus generated:
of
~[C]
In geometric terms, the following prime
110
(A) If
First fix a point
P = (a,b) ~ C ( ~ ) ,
P # (i,0), P # (0,0).
and there is e <0
such that
b > 0, we have
~p+ = {F/I(C)
I Fc~[X,Y]
F(x,~x3 + x2)e0 on
~p = {F/I(C)
I F c~[X,Y]
and there is E > 0
(a,a+e)}
such that F(x,Jx 3 + x 2) e 0 on
(as,a) }.
In order to check that these sets are in fact prime precones of
~[C],
only the implication from left to right in Definition 6.1 (iv) requires verification, to see that if
as the other conditions are trivial.
For this it suffices
F 4 I(C), then the function F(x, S x 3 + x 2) has finitely
many zeros. Since the elements of ~[C] are polynomials of the form YP(X) + Q(X), our contention is clear as the equation (X3+X 2) P(X)2Q(X) 2 =0
has to be satisfied.
(Alternatively, Bezout's theorem (cf. Walker
[15 ;p. 59 ff. ) could have been used to check this point). The prime precones c o r r e s p o n d i ~ to the case replacing ~x 3 + x 2 for ~ x 3 + x 2 above. (B)
If
b <0
are obtained by
P = (i,0), we have
~p+ = {F/I(C) I There is ~ > 0 and similarly for
such that
F(x,/x3 + x 2) e 0
on (I,i+~)
 ~ x 3 + x 2.
Thus, at each nonsingular point to that of Example 6.4:
PcC(~)
we have a situation similar
P
p
p+
The points P, P+ are interpreted as the "generic" points of the halfbranches determined by
P
on the curve
C(~);
see Figure 7.
(C) Next we have two "points at infinity" corresponding to the upper and lower infinite halfbranches: ~,u
= {F/I(C)
IThere is
a>0
such that
F ( x , J x 3 + x 2) ~ 0
on ( a , ~ ) } ,
,
111
and similarly
(D)
~ ,£
_~x3
corresponding to
+x 2
P = (0,0) we have:
Finally, at the singular point
ep+ = {F/I(C) r
I There is
~ >0
such that
F(x, ~ x 3 + x 2) e0 on (0,e)},
+ = {F/I(C) P£
I There is
s >0
such that
F(x, ~ x 3 + x 2) >0 on (s,0)},
and two more points,
~p_, r
~P£
given by
 ~/x3 + x 2
instead of
~x 3 + x 2 . As an exercise the reader may check that there is no inclusion relation between any two of these sets. Obviously these points specialize on ~p, and they correspond to the four halfbranches the origin (see Figure 7).
of the curve
C(~)
through
In order to complete the analysis of this example we would have to show that (i) There are no real prime ideals in maximal ideals
Mp
at each point
~[C] other than
{0}
and the
P ~ C(~);
(ii) for each of these prime ideals there are no prime precones other than those explicitly constructed above. A simple proof of point (i) goes as follows. prime ideal of
~[C] and let
G I .... ,G£
Assume
J # {0} is a real
be its generators, where
Gi(X,Y) = YPi(X) + Qi(X). By Corollary 4.12, V ~ ( J ) # ~. As in (A) above, the points (x,y) c V ~ ( J ) satisfy the equations (x 3 + x 2) Pi(x) 2  Qi(x) 2 = 0 (i = I ..... £), and hence V ~ ( J ) is finite, say V R ( J ) = {PI .... 'Pk }" By the real nullstellensatz 4.8 J consists of all polynomials vanishing at PI .... 'Pk; but this ideal is never prime if k e2. Hence V ~ ( J ) consists of one point, P, and clearly J = Mp. above.
Finally, point (ii) is clear by the argument preceeding
(A) D
REMARKS. (a) The correspondence between points of the real spectrum and "oriented halfbranches" illustrated by the preceeding examples is a general fact, true of any variety (of any dimension) over any real closed field. The proof of this requires a sophisticated algebraic machinery based on the analysis of valuations; cf. Coste/CosteRoy [62; §7, §8]. (b)
In both the preceeding examples the length of speciali
zation chains of the real spectrum is at most 2. This is a manifestation of the fact that the (local) dimension at each point of the curves
112
under consideration B.
se in
§8
below.
D
ELEMENTARY PROPERTIES.
PROPOSITION (i)
is i, as we will
6.6.
Let
The basic open sets
quasicompact,
A
be a conmTutative ring with unit.
H(a I ..... an),
i.e. compact
in the usual
a I ..... aneA, of SpecR(A)
are
sense but not n e c e s s a r i l y
Hausdorff. In particular: (2)
SpecR(A)
is quasicompact.
(3)
The irreducible
closed subsets of SpecR(A)
are the closure of a
unique point. In particular: (4)
SpecR(A)
(5)
Let
is a
Tospace.
~, B,y ~ SpecR(A).
REMARK.
B, Y c {~}, then
A closed set is called irreducible
two closed proper subsets. PROOF.
If
(I)
order theory
if it is not the union of
set of sentences with parameters
in
A
theorem
of the firs
(in the language of unitary rings plus an additional unary
P) whose axioms are:
The axioms
for commutative

The axioms
for
P
rings w i t h unit.
defining a prime precone
We leave the proof as an exercise
(cf. Definition
6.1).
for the reader.
First check that the set
= is a prime precone. show that
{acA
that SpecR(A)
or cb
B ~ {~}
iff
~ c ~
to
is in
B
and e, and
~ , it is easiest to check
implies
~ % {B}
or
Y j B , get
beBy
and
~ c B , y,
B % {e} ceyB.
Since one of bc
we obtain either b=(bc)+c ey or
a contradiction.
REMARK.
correspondence
of such point
is To:
B i y
c = (cb)+b~B, 6.7.
= 0}
Then, use the equivalence
# If
I H(a)nF
F = {~}.
In order to show the uniqueness
(5)
y!B.
The closure of a point is clearly irreducible.

(2)
or
This is easily proved by applying the compactness
to an appropriate predicate
B ! Y
(Functorial properties
which assigns
D of the real spectrum).
The
to each commutative unitary ring its real
113
spectrum is a contravariant functor from the category of such rings with homomorphisms into the category of topological spaces (moreover, of spectral spaces, see Definition phisms.
6.9) with continuous functions as mor
This simply means that to each momomorphism of unitary rings
f:A ÷ B it is
canonically associated a continuous map SpecRf : Spec R (B) ÷ SpecR(A)
defined by (SpecR f)(B) = fl[~] It is clear that for
for B c SpecR(B), we have:
a I ..... an E A
(SpecR f)i [H(a I ..... an)]
= H(f(al) ..... f(an)),
which shows that Spec R f is continuous and, moreover,
that the inverse
image of a compact open subset of SpecR(A) is compact open.
D
In this connection note the following: FACT 6.8.
With notation as above, let
B E SpecR(B) and a = (SpeaRO(B). ~:k(a) ÷ k(B) making the
Then there is a (unique) ring monomorphism following diagram commute: f A
, B
k(a)
, k( B )
The map ~ is elementary. The easy proof is left as an exercise. It is useful to recast the content of Proposition 6.6 in the following language: DEFINITION 6,9.
A topological space
X
is called an spectral space
iff (i)
X is quasicompact.
(ii) X has a base of open quasicompact sets closed under intersection. (iii) Every irreducible closed subset of unique point.
X
is the closure of a
The Stone duality between Boolean algebras and Boolean (= compact, Hausdorff,
totally disconnected)
spaces can be extended to a duality
between the category of distributive lattices with homomorphisms and
H
114
the category of spectral spaces with continuous maps such that the inverse image of a compact open set is compact,
To each spectral space it
is associated the lattice of its compact open subsets.
Conversely, to
each distributive lattice it is associated the space of its prime filters with the spectral topology (defined exactly as for the spectrum of a ring). The fundamental result about this class of space is: THEOREM 6.10.
(Hochster).
A spectral space is homeomorphic to the
(prime) spectrum of a ring.
D
For a proof see Hochster [63]
or
Laffon [7].
In particular, the real spectrum of a ring A is homeomorphic to the prime spectrum of another ring. In the case where A is the coordinate ring of a variety over a real closed field, the ring
B
can be computed
explicitly, as we will see later (Corollary 9.10). C.
CONSTRUCTIBLE SETS.
DEFINITION 6.11, A subset of SpecR(A) is called constructible if it is a Boolean combination of basic open sets.
D
Quantifier elimination shows at once that the constructible sets coincide with the definable sets in the following sense: PROPOSITION 6.12. is an
A set
C ! SpecR(A) is constructible
Lsentence with parameters in A,
iff
there
~C = ~C(al .... ,an) , such that
C = { ~ S p e c R ( A ) I k(~) > ~C[~ (al) ..... ~ (an) I}.
D
The (easy) proof is left as an exercise for the reader. It is clear that the constructible sets form a basis for a topology on SpecR(A), called the c onstructible topology.
This topology is obviously
finer than the spectral topology and is compact Hansdorff. The main property of construetible sets is: 6.13.
THE REAL CHEVALLEY THEOREM.
a finitely presented
Let
C
where
Then
be a constructible subset of
C = {BcSpecR(B)
A
be a ring, B=A[X I ..... Xn] ~
Aalgebra (i.e. the ideal
and f:A ÷ B the canonical morphism. tible sets into constructible sets. PROOF.
Let
I
is finitely generated),
SpecRf transforms construc
SpecR(B) given by
I k(~) ~ ~C[z~(Ql/l) ..... ~ ( Q m / l ) ] } ,
Q1 ..... Qm c A [ X 1 ..... Xn].
Let I be generated by
P1 ..... P£E
115
A[X I
. . . . .
Xn].
Then we have the equality:
(SpecRf) [C] = {~cSpecR(A)
^ where for
I k(a) > ~Yl ..... ~Yn
£ ~I(~Pi)(Yl
..... Yn )=0 ^
~ C ( ~ Q I ( Y l ..... yn ) ..... ~ Q m ( Y l ..... yn))]},
F ~ A[X 1 .... ,Xn], N
F denotes the polynomial whose coefficients
are the images of the coefficients of F. The inclusion if
!
follows easily from Fact
6.8.
For the other inclusion,
Yl .... 'Yn ~ k(~) satisfy the given formula, then the correspondence
a
I
~
Xi/ll
>
~ (a)
for
Yi
i = i ..... n,
extends to
a ring homomorphism
precone of
B, B = gI [k(~)+]
such that
~ = (SpecRf)(B)
a ~A,
=
g:B + k(~).
This morphism gives a prime
{F/I I k(~) ~ (~ F)(y I ..... yn ) ~0},
and
BcC.
D
The study of topological properties of the map SpecRf is of central interest in real algebraic geometry. This study is frequently based on an elegant combination of logical and geometricotopological techniques. For example, if one needs to show that a certain constructible
set
C ! SpecR(A), given by an [formula ~C(al .... ,an) , is open, logic helps by reducing the problem to showing that the set (,)
{~n
I ~ > ~C [~]}
is open in A n (of course, ~ can be replaced by any other real closed field). The analytic techniques available in the reals often are of help in proofs of this kind. Indeed, if the set (*) is open, the open quantifier elimination theorem 5.9 implies that the formula ~c(vl ..... v n) (without parameters) equivalent in the theory RCF to one of the form ni V A Pij(Vl ..... v n) > 0 i j=l with
Pij C E [X I ..... Xn].
It follows that
is
116
C = ~ IH ( P i. l ( a l and, hence,
that
C
As an illustration result
where
PROOF.
Obviously
SpecR(K) ,
F
CASE i.
K
F = K(a),
f ~ K[X]
Let
extension
K,F
be orderable
of K, and let i:K ÷ F
is an open map.
to show that
(SpecRi)[0]
is a basic open set of
for
is open in
SpecR(F).
Note that
~SpecR(F). where
and a is algebraic
to prove
x I .... ,xn are a transcendence
over
K(x I .... ,Xn).
the theorem in the cases a algebraic
be the minimal
of SpecR(F)
generated
F = K(Xl,...,Xn,a) ,
over
it suffices
0
theorem).
Then SpecRi
it suffices
= ~nK
We may assume base of
map.
whenever
(SpecRi)(~)
Let
is a finitely
the inclusion
the following
[48].
(The open mapping
F
(al ..... an))
is open~
due to ElmanLamWadsworth
fields,
l
of the use of this technique we prove
THEOREM 6.14. denote
..... a n ) ..... Pin
F = K(a)
Obviously
and
F = K(X).
over K.
polynomial
of
a.
A nonempty
basic
subset
is of the form: £
H(PI(a) ..... P£(a)) where
Pi c K[X]
and
= {~SpecR(F)
Pi(a)
# 0.
I
Then
A Pi(a) > 0} i=l
f f Pi and we may assume
that
deg (Pi) < deg(f). Let X = (SpecRi) are equivalent:
[H(PI(a) ..... P£(a))].
(i)
BeX
(ii)
B extends
Since
K(a)
conditions
F
such that
the signchanging
criterion
are equivalent
for any order
B extends
to an order of F;
f changes
sign in
By definition have (iii)
the following
f changes
A Pi(a) >0. i=l 4.18 tells us that B of
K:
This condition
B~SpecR(K)
£ to an order ~ of
~ K[X]~f),
the following
For
Hence
and if ~ is an order of
(ii) is equivalent
sign in
is firstorder.
k(B) Let:
and
k(B)
F
extending
to : £ ~ A Pi(a) >0. i=l
B we
117
~(x,y,a O ..... an_ I ):
x
A f(x) f(y) <0,
£ ~(a,b o ..... bin_I): where
a ° ..... an_ I
of PI .... 'P£"
are the coefficients of
By continuity,
~(x,y,v o ..... Vn_ I)
~(v o
CASE 2.
and
b o ..... bm_ I those
formulas
define open subsets of
IRn+2 and
Hence, the formula
..... Vn_ l)
defines an open subset of the constructible set {BeSpecR(K)
f,
the parameterfree
P(z,w O ..... Wm_ I)
I~n+l, respectively.
is open.
A Pi(a) >0, i=l
~xy ~(x,y,v o ..... Vn_ I)
:
An.
I k(8) ~
By the remark preceeding the theorem,
~(a o ..... an_ 1 ) ^ P(a,b o ..... bin_l)}
By the equivalence shown above this set equals
X.
F = K(X).
By trivial manipulations be written in the form: H(P I ..... P£)
(cf. 4.17) a basic open subset of SpecR(F) can
=
{~SpecR(F)
I
l A Pi >0} i=l
for nonzero polynomials Pi eK[X]" Putting X = (SpecRi)[H(P I ..... P£)] we have, as before, the equivalence between : (i') (ii')
BoX, B extends to an order ~ of
where to :
BeSpecR(K ) .
(iii')
F
such that
£ A Pi >0' i=l
Below we prove that these conditions are equivalent
£ A Pi(x) > 0. i=l
With this equivalence established,
the proof is completed as above for
(iii') defines an open condition, hence an open constructible
subset of
SpecR(K). Let
L =
Conversely,
Clearly (ii') implies
(iii') implies that all the
(iii'), as
L ~
!
Pi s are positive on an interval
(a,a+c) of L. Hence the set ~a+ defined in Example 6.4 (with L replacing ~) defines an order of L(X) extending B and making the Pi's positive. This order induces an order on F with the properties required in (ii'). D
118
REMARK. The map SpecRi is also closed, since the real spectrum of an orderable field is Hausdorff (exercise). As a matter of fact, something much more general is proved by Coste/CosteRoy techniques used above:
[62;Thm. 6.2] with the
THEOREM 6.15. (The closed mapping theorem). Let A, B be rings and f:A ÷ B a homomorphism such that B is integral over f[A]. Then SpecRf: SpecR(B) ÷ SpecR(A)
is a closed map.
0
For still another application of the same technique, §7.
see Roy [32;
§2].
AFFINE VARIETIES OVER REAL CLOSED FIELDS. Now we shall study the interplay between the geometry of affine
varieties over real closed fields in particular, over ~and the topology of the real spectra of their coordinate rings. Throughout this section varieties are equipped with the euclidean topology derived from the order topology in the base field, and spectra are equipped with their spectral topology (cf. Definition 6.2). Observe that for any ordered base
field
V
over
K, there is an obvious embedding : V(K)
~ SpecR(K[V])
given by ' " ~x = {Q/I I QeK[X 1 ..... Xn] (we write 7.1
I FACT.
instead of
and
Q(x) e 0}
I(V(K))).
The map ~ is injective and continuous.
PROOF. Injectivity follows easily by considering linear polynomials. Continuity follows from the equality ~I[H(QI/I ..... QM/I)]
m n IC'~IQ~. I [ . = (0,~) ],
= V(K)
which is checked without problem.
0
m Since the family of sets of the form
V(K) n ~i Q$1[(0,~)]
for m c ~ and QI ..... Qm ~K[X I ..... x n] clearly is a basis for the topology of V(K), 7.1 says, furthermore, that the image of V(K) is a subspace of SpecR(K[V]); therefore, we may (and will) identify V(K) with its image by ~. Henceforth we also assume that K is real closed. THEOREM 7.2. With the convention above, restriction to S~~SnV(K), defines a bijeetive map between: (i)
Constructible
subsets of SpecR(K[V])
V(K),
and s.a. subsets of V(K).
119
(ii)
Open constructible sets of V(K).
PROOF.
(i)
Assume
C
subsets of
SpecR(K[V])
is eonstructible,defined
~c(QI/I ..... Qm/l), with
and open s.a. sub
by the formula
QI ..... Qm ~ K [ X I ..... x n] (see Proposition 6.12).
Then the equality (*)
C nV(K) = {x ~V(K)
shows that of
C
C n V(K)
is s.a.
I K ~ ~c[QI(~) ..... Qm(X)]},
By induction on the (Boolean) structure
one gets reduced to showing
C = H(QI/I ..... Qm/l).
(*) when
C
is basic open,
In this case one may take
and then (*) is just the equality appearing in the proof of It is clear that the map as values.
tructible subsets of SpecR(K[V]) ~c,(FI/ .... ,Fr/I)
>0,
7.1.
C~~C n V(K) takes on all s.a. subsets of V(K)
In order to see that it is injective,
and
m
~c(vl ..... Vm): i~iv _
assume that
C
defined by formulas
respectively,
and that
and
C' are cons
~c(QI/I ..... Qm/I)
C nV(K) = C' nV(K).
By
(*) this equality translates as: l K ~ V x[ ~IPj(~) = 0
(**)
j=
+ (~c(QI(~) ..... Qm(X)) <> ~c,(FI(~) .....Fr~))],
where PI .... 'P£ are polynomials generating the ideal I of V. This is a formula with parameters in K (the coefficients of the polynomials). On the other hand, for every ~SpecR(K[V]), k(~) is a real closed field containing K, and hence (**) holds in k(=). Specializing (**) to i = i ..... n, we have Pj(~ (XI/I) ..... ~ (Xn/I) = x i = ~ (Xi/l), = ~ (PJ/I) = 0, and hence: k(~) ~ ~C(~ (QI/I) ..... ~ (Qm/l)) In view of Proposition ~cC i.e.,
<
~ ~C,(~ (FI/I) ..... ~ (Fr/l)).
6.12, this means: iff
~C'
for
~ESpecR(KEV]),
C = C'.
(ii) Since the identification map ~ is continuous, it is clear that C n V(K) is open, whenever C is. The fact that every open s.a. subset of V(K) is of the form C n V(K) for some open constructible set C is an immediate consequence of open quantifier elimination (Theorem 5.9). COROLLARY 7.3.
D V(K)
is dense in
SpecR(K[V]).
120
Notation.
Given a s.a. subset
S
of
constructible subset of SpecR(K[V]) COROLLARY 7.4.
The map
SI
, S
SpecR(~[V])
S
the unique
S = S nV(K).
induces a oneone correspondence
between the connected components of In particular,
V(K), we denote by
such that
V(~)
and those of S p e c R ( ~ [ V ] ) .
has a finite number of connected components
and is locally connected. PROOF. 5.8).
Let
U I ..... U r
be the connected components of
We want to show that
SpecR(~[V]).
By Theorem
tion of SpecR(~[V]). Assume
Since the
7.2
C,C'
open (in
Ui ), nonempty and disjoint.
C,C' are clopen (in SpecR( ~[V])); hence
C,C' are finite unions of basic open sets, and
hence constructible.
By Theorem 7.2 again,
U i = (C n V ( ~ ) ) is a partition of
(Theorem
these sets are clopen and form a parti
Uj's are open, then Therefore
V(~)
are the connected components of
We only need to show that they are connected.
Ui ~ C U C', with
compact.
UI ..... Ur
Ui
u (C' n V ( ~ ) )
in nonempty open subsets, a contradiction.
The proof of local connectedness is left as an exercise. Exercise.
(a)
Prove the statement of Corollary 7.4 with a s.a. set N
S ~ V(~),
and (b)
S
replacing
The operation
V(~), SI
and SpecR(~[V]),
,S
respectively.
commutes
(i)
with the finite Boolean operations;
(ii) (iii)
with closure and interior; with images and inverse images by morphisms of algebraic varieties over
K.
(Cf. Hartshorne
[3; Ch.l]
for the notion of morphism
of algebraic variety). (c)
If
U = V(~)
open subset W of S p e c R ( ~ [ V ] )
is open
s.a., then
U
W nV(~)
= U.
such that
is the largest
Corollary 7.4 depends essentially on the fact that
V(~)
has finite
lymanyc~mected components, a property that only the real numbers enjoy amongst real closed fields. In fact: Exercise.
Prove that if
K
is a real closed field ~
connected component of one point of
Kn
~, then the
is the singleton of that
point.
D
However, using transfer on an appropriate for a fixed constructible set ted" ,
Coste/CosteRoy
C
Lformula  namely one that
expresses the property
[62; Thm. 5.5]
show:
"C
is connec
121
PROPOSITION over
K.
7.5.
Let
K
be a real closed field and
V
a variety
Then SpecR(K[V] ) has a finite number of connected components
which are constructible
sets.
The same is true of any constructible
set of SpecR(K[V]).
D
What kind of partition on V(K)?.
sub
do the connected components
of
SpecR(K[V])
induce
It turns out that the members of this partition are precisely
the components
for the following notion,
S is called s.a.connected into two disjoint,
defined for s.a. sets
S ! V(K):
if it cannot be split
nonempty,
s.a. open sets.
This notion, which clearly coincides with the standard notion of connectedness in the case
K =
~, has a deep geometrical meaning.
see this, let us consider Example
2.5(b)
the restriction
to the field
Q
this is still a twocomponent ly, the topology of
~
variety,
this property
Clear
(see Exercise above),
does.
that the notion of pathc0nnectedness
coincides
The first of these notions
ed to any real closed base field (this was done by ably,
cubic of
Manifestly,
although it has many "holes".
while the notion of s.a.connectedness It is wellknown
of the twocomponent
of real algebraic numbers.
cannot reflect
with that of connectedness.
In order to
in
]Rn
can be generaliz
Dells [70]). Remark
this generalized notion turns out to be equivalent
to that of s.a.
connectedness. Delfs and Knebusch
[71]
have introduced a theory of "restricted
topolo
gical spaces" intended to provide a frame in which the "semialgebraic" versions of some topological
notions
(e.g. that of s.a.connectedness)
may be cast in much the same terms in which the corresponding topological notions are formulated topology. §8.
See also Brocker
in the frame of general,
standard
pointset
[59; § I].
DIMENSION. We give in this section a brief summary, without proofs,
the theory of dimension
for affine varieties
K, and for s.a. subsets of Roy
[62;
§8].
K n.
of
over a real closed base field
This theory was developed by Coste/Coste
The algebraic notions of dimension used in classical
geometry were briefly reviewed in
§2.
For most of the present section we will assume that the polynomials P1 ..... P£ E K[X 1 .... ,Xn] I(V)
over
K;
determining
our variety
we will say that the variety
trical point of view this is no restriction place
P1 .... 'P£
by a (finite)
which,
by Corollary 4.11,
V
at all:
set of generators
is real;
V
generate a real ideal
is real.
From a geome
it suffices
of the ideal
to reI (V(K))
the latter obviously generate
the
122
same set of
Kpoints.
Moreover,
the rings
K[V] and
The notion of prime precone gives,
a priori,
K[X I ..... Xn]/I(V )
have the same real spectrum. A.
GLOBAL DIMENSION. a new way of measu
ring dimensions: DEFINITION 8.1. Let
A
be a commutative
affine variety over a real closed field (a)
The real dimension of
ring with unit,
and
V
an
K.
A, dimRA , is the supremum of the integers
n
such that there is a strict chain Co
of prime precones length). (b)
of
~ ~i
~
The real dimension of
~n ~ =
" ~
A(dimRA = ~
Bo B
if there are such chains of unbounded
K[V].
and ~ ! B
D imply
strict chain of prime precones of prime ideals of
A.
Hence,
these two quantities PROPOSITION
A
~ = B
(e,BESpecR(A)),
every
induces a strict chain of real
dimRA ~ dim A.
However,
for real varieties
are equal:
8.2.
Let
a real closed field of
n
V, dimR(V) , is defined to be the real dimen
sion of the coordinate ring Since
"
K.
V
be a real irreducible
Them
dim R K[V] equals
affine variety over
the transcendence
degree
K(V) over K.
D
(Cf. Theorem 2.7). In addition it follows COROLLARY quantities
8.3.
that:
For
K
and
V
as in Proposition
8.2, the following
are equal to dimR(V):
(a) The combinatorial
dimension of
V(K).
(b) The supremum of the length of strict chains of real prime ideals in K[V].
D
The combinatorial
dimension of
V(K) is the supremum of the lengths of
strict chains of closed irreducible topology
(cf. Hartshorne
chains of irreducible
[3;Ch.
subvarieties
subsets of
V(K) with the Zariski
i]); or, in other words, of
V(K).
Proposition
proving the equality of this quantity with dimR(V). quantities
(a) and (b) follows
of strict 8.2 is used in
The equality between
from the Real Nullstellensatz
4.8: the
map I~~VK(I) is a bijective correspondence between real prime ideals of K[V] and irreducible subvarieties of V(K).
123
Note that the preceeding results are false for nonreal varieties: if is given by the polynomial dimR(V) = 0 is
X 2 + y2, so that
V(~)
since the only prime precone of
~ = { F/I I F(0,0) ~0}.
However
A =
V
= {<0,0>}, then ~[X,Y]/I , I = (X 2 + y2)
A contains the chain (o) ~ (X/I,Y/I)
of real prime ideals. The results above show that, as far as measuring $loba.l dimensions is concerned, the use of prime precones yields the same results as the tools of classical commratative algebra.
However, prime precones provide the
means of constructing a theory of local dimension capable of explaining the phenomena of
"fall of dimension" observed in the examples of
§2;
this cannot be done with the classical tools. B.
LOCAL DIMENSION. Definition 8.4.
field
K, and
dimR(V,x), chain
Let
x ~ V(K).
V
be an affine variety over a real closed
The (local) real dimension of
is the supremum of the integers
n
=
a o ~al~
of prime precones of ing to
x;
cf.
an
 . .
K[V] ending in
V
at
x,
such that there is a strict
a
X
a~(= the prime precone correspond
§7).
Comparing this definition with Definition 2.6(c) one may wonder whether dimR(V,x) coincides with the real dimension of the ring
K[V]M_.
This
x
is not true in general, but we have: PROPOSITION 8.5.
dimR(V,~ )
=
dimR(K[V]M_) h, X
where
Ah
denotes the Henselization of a local ring
A.
[]
The proof of this result requires some nontrivial arguments developed by Coste/Coste Roy [62].
For the construction of the Henselization of
a local ring, see Lafon [6]
or Nagata [Ii].
Next we state the central geometric theorem on local dimension: THEOREM 8.6.
Let
V
be an affine real irreducible variety over a
real closed field K, and x ~ V(K). Them dimR(V) = dimR(V,x) iff belongs to the closure (in the euclidean topology) of the set of nonsingular points of
V(K).
[]
The proof is done in Coste/CosteRoy analogue of Theorem 2.8 Examples
2.9
[62;Thm. 8.9].
for real varieties.
This result is the
Looking back at the
we can see now that our notion assigns the correct dimen
sion to the origin in both cases
: 0 in the first example, 2 in the
124
second. C.
THE DIMENSION OF SEMIALGEBRAIC
One of the remarkable
SETS.
features of the local theory of real
dimension is that it assigns a dimension only to varieties but, more generally, provides a notion of dimension DEFINITION and
x e S.
8.7.
Let
K
to s.a. sets.
be a real closed field, S
at
PROPOSITION
8.8.
sup{dimR(X,x)
sion of the closure of
S
in
Kn
I xES}
is the supremum
in S
ending in
~.D
is equal to the (real) dimen
with the Zariski topology,
It follows that the natural notion of (global)
real dimension
set is that of the real dimension of its Zariski closure; defined,
it
of a variety.
S c K n a s.a. set,
x, dimR(S,x),
of strict chains of prime precones
not
In particular,
for the connected components
The real dimension of
of the length,
(both locally and globally)
D for a s.a.
this is well
for such a closure is a variety by definition.
PROPOSITION
8.9.
Let E
S c Kn
~
be a s.a. set.
dimR(S,~), For a given integer
is upper semicontinuous.
defined on S,
Then the function
k ~ i, the
set {xe S
I
dimR(S,x) < k}
is s.a., open in So
D
Looking at Example result;
the set
2.9(b) , we see that Proposition {xeV(~)
I dimR(V,x)
should be, since it coincides with COROLLARY neighborhood
8.10.
If
U
x
of
S c Kn in
S
CONTINUOUS
{<x,y,Z>
such that
SEMIALGEBRAIC
The study of continuous
U
V(~),
as it
xeS, then there is a s.a. equals
the (real)
K n.
FUNCTIONS.
s.a. functions has only begun recently.
The subject is still largely unexplored, the past concentrated
gives the "right"
I x=Y =0 ^ z ~ 0}.
dimR(S,x) in
8.9
is open in
is sla. and
dimension of the Zariski closure of §9.
= I}
as algebraic
geometers have in
on the study of analytic s.a.(=Nash)
functions,
which lie closer to the geometrical phenomena and have better algebraic properties.
Nevertheless,
line the increasingly Furthermore,
the investigations
carried out so far under
important role of continuous
the study of these functions
s.a. functions.
leads naturally
to that of
Nash functions. We begin by introducing
a class of functions which,
in most interesting
125
cases,
turns out to coincide w i t h that of (continuous)
DEFINITION We say that
9.1.
f
Let
S c
is globally
£ ~ i and polynomials such that the equation
A n be a s.a. set and
algebraic
is verified
S c
for all
PROPOSITION ~n Then:
(I) If
f
Let
(I)
finition
f:S
then
f
(2) If f is continuous f is s.a. PROOF.
Pi nonzero,
f(~)£I + ..... + Po(X~
+ ~
be a function
is globally
and globally
By trivial manipulations
V(Pi(~,y) i
= 0
Pi,Qik c ~ [ X I ..... Xn,Y].
junct defines It suffices nomial
iff there is
some
= 0
defined on a s.a.
set
algebraic.
algebraic,
and
the graph of
S
is open,
f, Gr(f),
then
has a de
of the form:
(*) with
with
a function.
~ ~ S.
9.2.
is s.a.,
functions.
f:S + ~
(over polynomials)
Po,...,P£ ~R[XI,...,Xn],
Pz(x~ f(~)£ + P£_I(~)
s.a.
a nonempty
Furthermore,
occur.
i.e.
we can assume
that each dis
set.
to show that each disjunct
equation,
actually
^ A Qik(~,y) > 0) k
of
that a polynomial
For then,
setting
(*) contains Pi
P(X,Y)
of degree
a nontrivial e i in
= ~Pi(X,Y),
Y
poly
does
we have
1
P(x,f(x)) Assume
= 0
(2)
5.7
at
holds
contains
at
no nontrivial
<Xo,Yo >,
S.
= PZ(X)~
+
... + Po(X)
a partition =
Ao,A 1 .... ,Am of
... = P£(x)
and for each
= 0
i = l,...,m,
(jj) There are continuous
decomposition
a nonzero
Under the present hypothesis,
(j) Po(X)
that
f
equation.
If
is not single
Xo' a contradiction.
may only occur when the sign of
arrange
polynomial
then (*) shows
We shall use now the cylindrical
P(X,Y) on
xES.
that one disjunct
this disjunct valued
for all
theorem
polynomial
5.7
annihilated
the first alternative is zero.
Therefore,
S
into s.a.
sets,
all
by
f
of Theorem
P
for
with
we can
so that:
XeAo,
we have: s.a.
functions
~,
giving exactly
the real roots of the polynomial
In particular,
P(x,Y)
has constant
sign ~ 0
.... ~ i : A i ÷ P(x,Y),
~' £i ~ i,
for all
x e A i.
in each of the intervals
126
(_~. ~(~)), (~(~),ij+l(X)) . . . ,
.(x)' + ~ ) . By Theorem 5.8 1 in assuming that A 1 ..... A m are connected.
there is no loss of generality Since
f(x) is a real root of
with one of the
~(x),
and the continuity cise). The foregoing
of
say f
argument
..... (~
P(x,Y),
by (jj) it coincides,
j = s i.
Now,
imply that
shows
si
for
the connectedness
of
is the same for all
that the graph of
xcA i, Ai
xEAi(exer
f FA 1 o...uA m is defined
by the formula: V ,,y i=l ~i (~) ^ where
~i
interior
(in
interior
~n),
as
P(x,Y)",
of
f fAo.
(j) shows that
Po .... 'P£"
V(~)
in S,
intersects
root of
A i
In fact,
defined by
has empty
th
about the definability
of continuity.
V is the variety
~ Ao
si
is a formula defining
Next we need to worry argument
is the
A o ! S n V(~),
In particular,
has this property.
and each neighborhood
A 1 u... uA m.
f(x)
By continuity
=
lim
This is just an
Since (in
where
A o has empty S
is open, A
o
S) of a point
we have
f(y)
e AIO. • .uA m Since
f rA I u...
f ~A O
is also s.a.
uA m is s.a.
and the definition
There is a notion of minimal polynomial This is a consequence PROPOSITION of R,
and
beR
I b = {P~A[Y] then
Ib
9.3.
of the following Let
R
an element
J P(b)
= 0}
for globally algebraic
be a ring, algebraic
of limit is firstorder,
A a unique
over
is principal.
If
algebraic
functions.
result: factorization
A.
Then the ideal
R
is an integral
subring
domain,
is also prime.
The proof, which
D
is just a variant
Palais
[30;
applies
to the case under
§3]
and in Dickmann
of standard [72;
consideration
Prop. V.3.1].
DEFINITION
appears
in
This result
by setting:
R = the ring of realvalued , continuous, S ! A n (henceforth denoted C(U)); A = the ring
arguments,
s.a.
functions
on a s.a. set
~ [ X I ..... Xn]. 9.4.
The minimal polynomial
of a continuous
s.a.
(or,
127
more generally, a globally algebraic) function
f
on
S
is defined to
be a generator of the ideal If = {P c~[X,Y] such that the
g.c.d,
I P(~,f(~)) = 0
for all
~S}
of its coefficients is i.
The minimal polynomial is, of course, a polynomial of lowest degree in If, and is unique, but not necessarily irredu.c.ible; for example, the minimal polynomial of the absolute value function on
~
is
(YX)(Y+X).
However,
it has the following properties: PROPOSITION 9.5. and let
Let
P = P1 ..... P£
P e~[X,Y]
be the minimal polynomial of
be a decomposition of
P
f ~ C(S),
into irreducible factors.
Then: (a)
The Pi are distinct.
(b)
If
F i = P/
and
Pi
U i = {x~S I Fi(x,f(x)) ~0}, then the
pairwise disjoint, open s.a. subsets of on (c)
S, and
Pi
Ui
are
vanishes identically
Ui . S  6 Ui i=l
open, then
has empty interior £ S ! i__~Jl.=l~i.
For a proof, see
Brumfiel
[65;
(in
~n);
in particular,
if
S
is D
Prop. 8.13.15].
The proofs of some of the basic properties of continuous s.a. functions which we will consider below, use a technique depending on the fact that the
real roots of a polynomial equation
have a convergent
P(X,Y) = 0
in one variable X
Puiseux series expansion (cf. Example
3.2).
Precise
ly: PROPOSITION 9.6. and
p
Let
a real valued
[Xo,Xo+e),
for some
P c~[X,Y]
be a polynomial in two variables
function defined on an interval e>0 ,
and such that
P(x,p(x)) = 0 Then
o
for all
x~ [Xo,Xo+e).
has an absolutely convergent Puiseux series expansion p(x) =
for all
x
for all
k ~ N, and
kE N
ak(Xxo)k/P
in some interval (Xo,Xo+~) , a N ~ 0.
function p defined on expansion.
0 <~ ~~
Here
p ~ I, N ~
,ak~
In particular, any globally algebraic [Xo,Xo+S )
has such a Puiseux series 0
128
The existence of a formal solution is a consequence Walker
[15; Ch. IV,
§ 3].
It is a remarkable
series which is a solution of an equation convergent. Chenciner
of T h e o r e m 3.3;
P(X,Y)=0
is a u t o m a t i c a l l y
A neat discussion of the question of convergence [2; Ch. VIII,
§8.6].
see
fact that a formal Puiseux
See also Hormander
appears
in
[18; Appendix].
As an application of this technique we prove: THEOREM 9.7 (The Zojasiewicz s.a.
set an f, g
(i) For all
continuous
s >0,
{xEC
(ii) Z(f) ! Z(g)
functions
I Jg(x) l ~
(where
Let
defined on
c,r > 0
such that
~ m2
[ @ xeC
(u = Ig(x) l ^ v =
is given by a disjunction of conjunctions
H
is a s.a. set contained in the positive quadrant of
certain polynomials, e >0
say
Pl(U,V),...,Pt(u,v).
Let us consider Igl ~ s
Puiseux series expansions
the set
on C.
Hn((0,e)
× ~).
In the last case,
Assume,
Lgl
then,
If it is empty,
that
Hn((0,s)
× ~) ~ 9.
that for
x e C
also that
v >0
I Ig(x) l = u}
it follows that
v
g = 0
the m i n i m u m of
Ill
(i)) and the
This set is bounded below on PI,...,Pt,
implies
on
if(x) i~v(]g(x)l),
and
inf (Hn({Ig(x) i} × m ) . u~(o,E),
then
is a compact set (by(i)) on w h i c h
f ~0
Ifl
say v(u).
we have:
v(ig(x) i) =
v(u) ; hence
either
in C.
0 < Ig(x) i < e
Observe
have abso
(by assumption
(0,e) by one of the roots of one of the polynomials This means
on
9.6 there
(0,e).
C is compact
and the result follows easily by considering m a x i m u m of
in
~.
of sign conditions
By P r o p o s i t i o n
such that all the real roots of these polynomials
lutely convergent
that
on C.
]f(x) I)}.
Then
If
such that:
Ift eclgl r
H
{x~C
C
be a closed
E} is compact.
Clearly
or
~n
Let H = {
is
C c
Z(f) = fl[0]).
Then there are constants PROOF.
s.a.
inequality).
(0,s);
for if
has a m i n i m u m ~ 0
(by(ii));
on this set, which equals,
v(u) > 0.
is bounded b e l o w by Ifi e clgl
Let us assume that
for some v
~ >0
on (0,e),
then an easy argument
shows
c >0.
is not bounded away from 0, i.e.
Using the Puiseux series expansion of
v
we get:
lim+ u+0
v(u) = 0.
129
(*)
v(u) =
[
akuk/p
=
a N uN/p (i +
[
k=N
for
~(0,~).
Let
T(u) denote the series expnasion in the last term;
since its exponents are positive, then 0 <6 ~ ~ , so that from (*) that Putting
=
c'
I + ~ e 1/2
aN > 0 . aN/2
and
Igl r
{x~C J Jg(x) J ~ 6}.
Put
REMARKS.
(a)
N/p
implies that N > 0 .
_ jr IfJ > c'Jg
on
for some
c">0
on the compact set
c = min {c',c"}.
D
The idea of the proof above goes back to H~rmander
[18;
Len~na 2.1 ]
(b) When the set
for
v(u) >0, it follows
v(U)u+~0
we have
C
is compact, the statement of Theorem
fact firstorder, as one may take 3.4].
Since
Choose 6 ,
The argument used at the beginning of the proof
JfJ ~c"
Appendix;
r =
lim+ y(u) = 0. u*0
on (0,6).
The fact that
{xeC J 0 s Jg(x) t< ~} shows that
ak/aN ukN/p)
k=N+l
r
9.7
rational, cf. Dickman
is in
[72; Prop.V
By transfer, the inequality is valid in each real closed field, C
closed and bounded. Dells [70;
Lemma
3.23
gives an elementary
proof valid for arbitrary real closed fields.
D
In the remainder of this section we sum up other results obtained by application of the same technique, and study their effect on the structure of the rings C(S). We shall denote set S c K n, topology.
by
cK(s) the ring of
Kvalued s.a. functions on a s.a.
K a real closed field, which are continuous in the euclidean
The following result is proved by CarralCoste then, by transfer, for any real closed field PROPOSITION 9.8. and
gEcK(sz(f)).
prolonged by
0
COROLLARY Z(g) ! Z(f).
on 9.9.
S.
first for
~
Let
S
such that the s.a. function fmg,
Let
S
be as in 9.8 me i
V
0 and
f,gecK(s)
such that g
be as in 9.8. if
and
be a locally closed s.a. set, fEcK(s) m ~i
Z(f), is continuous.
In particular,
Spec (cK(v(K))) PROOF.
S c Kn
Then there is
COROLLARY 9.10. phic to
Let
Then there is
[603, K.
be such that
divides fm in
Then Spec(cK(s))
cK(s).D is homeomor
is an affine variety over
K, then
is homeomorphic to SpecR(K[V]).
This is an application of the duality between spectral spaces
and distributive lattices mentioned in
§6.B.
It suffices to prove that
the lattices of compact open subsets of Spec (cK(s)) and of
S
are
130
isomorphic. By Theorem 7.2 subsets of
the latter is isomorphic to the lattice of open s.a.
S.
The former is simply
{D(f)
] fccK(s)}, since Corollary 9.9 implies that
D(f) uD(g) = D(f 2 + g2). The map the required isomorphism
D(f) I • {xcS If(X) ~0}
establishes
: it is injective by Corollary 9.9, and it is
surjective since for a given open s.a. set
U ! S, the function ds_ U
(= distance to SU) is continuous. COROLLARY 9.1].. Let
S c Kn
be a locally closed s.a. set.
Then
dim (cK(s)) = dimR(S).
D
This corollary shows that the rings cK(s), c(S), are radically different from the rings of arbitrary continuous functions : the Krull dimension of the latter is one or infinite, whatever the underlying space; inclusion chains of prime ideals in this case are of length 1 or at least 2~I (cf. Gillman  Jerison
[85; Thm. 14.19]).
Thus, we see that rings of conti
nuous s.a. functions are wellbehaved objects which reflect geometric properties of the underlying spaces. The results above have an effect on the structure of the ideals of COROLLARY 9.12. an ideal of (i)
Let
cK(s).
S
be as in Proposition 9.8, and let
I
cK(s). be
The following are equivalent:
I is real.
(it) I is radical. (iii)l is a zideal (i.e. Z(f) = Z(g) and The residue rings
cK(S)/p,
where
P
g~l
imply
fcl).
is a prime ideal, have the following
properties, similar to those holding in rings of arbitrary continuous: PROPOSITION 9.13. of (i)
Let
S c K n be a
s.a. set, and
P a prime ideal
C~(S). The relation f/p e 0
iff
there is
defines a total ordering on (it) The ring (a)
gccK(s)
such that
g>_0 on S and
f/p = g/p,
cK(s)/p.
cK(S)/p has the following properties:
It is a local ring (i.e.
P
is contained in exactly one
maximal ideal). (b)
Every nonnegative element has a square root.
(c)
Every monic polynomial of odd degree has a zero.
131
In particular: (iii)
If
M
The proof,
is a maximal
ideal,
given in Dickmann
then
[84],
cK(S)/M is a real closed field.
is a "definable" version of an ~
argument known in the case of rings of (arbitrary) see GillmanJerison
C
is an algebraic
proved in Dickmann closed ring, COMMENT.
[84];
introduced
curve over
the residue rings
~
and
it establishes
P
is a prime ideal, is
[82].
One may consider classes of continuous
type are familiar in analysis.
conditions;
s.a. functions obtaned
many conditions
condition
ments;
for example,
for
finite may not be a natural
r
differential However,
differentiable
9.14.
Let
f
be a
C ~ (i.e. infinitely
function defined on an open, connected
s.a. subset of
~n D
[65;
Prop.
8.13.16].
NASH FUNCTIONS.
f:U ~ ~
I0.I.
Let
U
be an open s.a. subset of
~n
A function
is called a Nash function if it is s.a. and analytic on U.
denote by
N(U)
the ring of Nash functions
defined on U.
These functions,
first considered by Nash [29],
prime importance
in real algebraic geometry.
Efroymson observe,
[25; p. 214],
algebraic properties
[26]
[25] is a comprehensive subrings.
constitute
but better geometric properties.
also be found in Roy
survey of the algebraic
a wealth of material. [32]
BochnakEfroym
to the subject, while BochnakEfroymson theory of
Many of the basic results were first collected
which contains
a tool of
The point is, as Bochnak
dealing with Nash functions.
is an introduction
We []
that Nash functions have the good
of polynomials,
There is a vast literature
[28],
differentiable)
is analytic.
DEFINITION
son
s.a. functions
class from the point of view of
The proof is implicit in Brumfiel §i0.
in connection with other require
rfold continuously
the following is important:
realvalued f
Semialgebricity
geometry.
PROPOSITION Then
of this
As far as we know, nothing has been done
in this direction beyond the study of Nash functions. not always be a natural
C(C(IR))/p,
a link with the notion of real
in CherlinDickmann
by imposing further "regularity"
my
functions;
[85; Thm. 13.4].
A result of geometric nature concerning where
continuous
and Palais
these two papers are very different
[30];
Valuable
N(U) and its in ~ojasiewicz
information
can
the points of view of
from the one adopted here.
In this survey we shall only consider Nash functions
defined on connect
132
ed open domains
in
An.
This will be quite sufficient
although much of the theory below applies definition A.
(see BochnakEfroymson
for our purposes,
to more general domains of
[25]).
BASIC ALGEBRAIC PROPERTIES. Many of the good algebraic properties
from:
10.2. Fundamental
fact.
If
U
of Nash functions
is open and connected,
follow
then N(U)
is an integral domain. PROOF.
Let
f, g eN(U)
of these sets, say f = 0
on
U
be such that
fg = 0, i.e. Z(f) uZ(g) = U.
Z(f), has nonempty
by the principle
interior.
Since
of analytic continuation
U
One
is connected
(Dieudonn~
[17;
9.4.2])
D
An immediate COROLLARY Nash
iff
consequence
is :
10.3.
f
Let
be an analytic
it is locally algebraic;
neighborhood V of such that P(x,f(x)) By Proposition
function
on
i.e. for every
~oo and a polynomial = 0 for all x ~ V .
9.3 the minimal polynomial
U.
x~U
Then
f
is
there is a
P ~ [ X I ..... Xn,Y] , P = 0, D of a Nash function is irreducible.
This has a number of simple but important algebraic
consequences;
we
mention the following: COROLLARY g:U ÷ ~
10.4.
Let
an analytic
f:U x ~ ÷ ~
f(x,g(x)) Then
g
Let
Since
f ~ 0,
follows nomial.
for all
P e ~ [ X I .... ,Xn,Xn+I,Y]
x ~ U.
then
for all
and hence
P
is not divisible by
=
P(~,g(~),0)
=
of
Y;
it
P(~,g(~),
f(~,g(~)))
= 0 D
10.5.
function on
COROLLARY i = i. . . If
. .
f
~f satisfies ~x°
f.
0) is not the zero poly
x c U.
COROLLARY analytic
P ~Y
be the minimal polynomial
that Q(X I ..... Xn+ I) = P(X I ..... Xn+!, We also have: Q(~,g(~))
PROOF.
= 0
is Nash.
PROOF.
and
be a nonzero Nash function and
function satisfying
Let U
F c N(U)[Y] satisfying
10.6. The ring N(U) :N (u). n, then ~f ~x i satisfies
be a nonzero polynomial F(g) = 0.
is differentially
the polynomial
the equation
Then
equation
g
and g an
is Nash. stable:
P(~,f(~))
D
if F E N (U)
= 0, then
133
De
~f  = 0 ~~P (~,f(~)). ~i(x)
(~,f(xD) +
~X i
which has Nash coefficients.
Hence it is Nash by 10.5.
Another consequence of
is:
PROPOSITION 10.7. open
s.a.
and
10.4
(Implicit function theorem).
~e~n
b ~R
a Nash function such that is a s.a. neighborhood such that
g(a) = b
V
and
f(a,b) = 0 of
Let
be such that e U.
a
in
and ~n
f(x,g(x)) = 0
Q
U c ~n+l Let
~f (a,b) ~ 0. ~Xn+ I
be
f:U ÷ ~
be
Then there
and a Nash function g:V + for all
x E V.
PROOF. By the implicit function theorem for analytic functions (see Dieudonn4 [17; 10.2.43) there is V, which we may take s.a., and an analytic solution g as above; Palais
[30; §i]
g
is Nash by 10.4.
D
shows that this statement is equivalent to more general
versions of the implicit function theorem, for a variety of situations including, of course, Nash functions. The following algebraic property is a consequence of Corollary 10.5: PROPOSITION 10.8. a quotient
closed 10.5,
N(U)
is integrally closed;
that is, if
f/g of Nash functions satisfies a monic polynomial equation
with coefficients in PROOF.
The ring
N(U), then g
divides
The ring of analytic functions on (cf. Dickmann
[72; Ch. V]);
f
U
in
N(U).
is known to be integrally
hence f/g is analytic on U.
f/g is Nash.
By 0
Now we mention, without proof, an algebraic property of crucial importance. THEOREM 10.9.
The ring
N(U) is noetherian.
The original proof, due to Risler [31],
D
is basically of algebraic nature.
A proof using complexification techniques is sketched in BochnakEfroymson
[25; Thm. 3.1].
COROLLARY i0.I0.
We shall use later the following consequence:
Let A be a subring of
nomials, and k I an ideal of that Z(1) = £'h Z(fi). i=l PROOF.
The ideal
I.N(U)
A.
N(U) containing the poly
Then there are fl ..... fk EI
generated by
I
in
such
N(U) is finitely generat
ed.
Each of these generators is a linear combination of members of
say
fl,.,.,fk.
The conclusion follows at once.
The algebraic properties considered above are valid for all (open) domains U. On the contrary, unique factorization in N(U) is an
I,
134
algebraic property which depends essentially on the geometry of the domain
U;
namely, on the triviality of its first cohomology group
(cf. BochnakEfroymson and Risler B.
[31]
[25;
§4],
where further references are given,
for simple examples).
NASH FUNCTIONS AND REAL ALGEBRAIC GEOMETRY. In order to understand the relevance of Nash functions for real
algebraic geometry, we underline the basic fact that in the classical theory of algebraic curves, the notion of a branch of analytic parametrizations;
see Walker
For example, the branches of the curve Example
is defined in terms
[15; Ch. IV, §2].
y2 _ (X 3 + X 2)
considered in
6.5, are given by the functions
b i : (i, + ~ )
÷~,
i = 1,2,
(i) i
~
defined as follows:
+ X2
for
I<X~0
~X 3 + X2
for
0 ~X
bi(x) (l) i+l
(where ~Fdenotes the positive value of the square root). tions are analytic branches
(exercise), and they are obviously s.a.
I I
The s.a. function
These func
f:[l, +~) ÷If
\bz
defined by
f(x) = min {y c]R I y2 = x 3 + x 2} is continuous but not analytic, thus not a branch.
f
135
As a second example (a).
Its only branch b(y)
defined on
consider
the curve
y2 _ X 3 + X 2
of Example
2.5
is given by the map × such that
y
2
= x
3
2  x , and
=
the unique
x > 1/2
A
(we have to give it in this form in order to have a func
tion). It is a remarkable by the preceeding
fact that the description examples
in any number of variab!es braic variety. Proposition
and
(i)
f
(ii)
There
nomials
is
q
q ~ n+l,
Let
U
a "branch"
of continuous
of an alge
s.a.
given by
be an open,
connected,
s.a. sub
are equivalent:
variables,
is
an irreducible a continuous
P ~A[X1,...,X gtale
over
q]
V
(b)
(pr FV(A))
o
jection
the first
onto
affine variety
s.a.
such
map
V
given by poly
s = < Sl,...
,
Sq > :U
÷
VOR)
that:
A. s = idu,
where
n
pr
: A q ÷A n
denotes
the
pro
coordinates.
f = Pos.
The proof is far outside and Roy
[32]
Condition
(a)
see Raynaud sequences
The functions
(jj)
The map
is classical Exercise.
exercise
f
[17; 9.11.1].]
s I ..... Sq
of Theorem
we con
I0.i!, we have:
are Nash.
is locally
injective.
D
gives a clue as to why Nash functions
do not appear
geometry.
f:C n ÷ C
is globally then
algebra;
(a)  (c) above:
pr r V ( A )
algebraic
Let
but the mere presenta
from commutative
but point out for later use the following
With the notations
(j)
The following
[24]
Since this condition will not be used explicitly,
explanations,
FACT.
significance,
the use of heavy machinery
of conditions
10.12
see ArtinMazur
and further uses of this characterization.
has a deep geometric
[12].
omit further
the scope of this survey;
for details
tion of it requires
f
sense,
illustrated
Every Nash function,
is as follows:
The following
(a)
9.1),
form.
is Nash.
in
(c)
result
(ArtinMazur).
f:U ÷ A .
and a polynomial
If
is, in a suitable
The precise
I0.ii.
An
of Nash functions
their general
(Compare w i t h the description
9.2).
THEOREM set of
gives
be a holomorphic
algebraic
(=complex
over polynomials
is a polynomial.
[Hint:
analytic)
function.
(in the sense of Definition
use Liouville's
theorem,
Dieudonn~
136
C.
THE SEPARATION THEOREMS. Now we will consider
the problem of finding a simple class of
s.a. functions having the following separation property: joint closed s.a. sets CI, C2, that
f ~C I >0
and
given two disin the class such
are not sufficient
to separate closed s.a.
the sets of Figure ii provide a counterexample.
that Nash functions

f
f ~C 2 < 0.
It is known that polynomials sets;
there is a function
 even Nash functions of a particularly
have the required DEFINITION
It turns out simple form
separation property.
10.13.
Let
U !
An
be open s.a.
We define
the smallest
subring of
C( IRn) containing
(i)
If
f e ~ R(U) 2
is such that
f >0
on
IRn, then I/f E R(U).
(ii)
If
f ~ ~ R(U) 2
is such that
f >0
on
U, then
The ring
R(U) can be constructed
B ¢_ C ( A n ) , ~,
let
with =
B (n+l) B (~) Let
SB
such that
as follows:
and
f >0
on U.
B
and such that:
~E
R(U).
for a ring by all functions
Let
B =
=
inductively
B (I) denote the ring generated over
f ~ ~ B2
B (°)
the polynomials
R(U) to be
(B(n)) (i) U B (n) n~
denote the multiplicative f >0
IR[X I ..... Xn](~)
on
I~n.
Then
at the set
subset of
B
of all functions
R(U) is the localization of S
~[x I .....
Xn ](~)
f ~ ~ B2
t37
Observe that the functions condition #f
in
R(U) are continuous
10.13 (ii) is introduced
THEOREM 10.14. Let
CI,
C2
f.
be disjoint
f c R ( ~ n)  even
A n and Nash on U;
in order to make this true, since
is not analytic at any zero of
Then there is
on
closed s.a. subsets o f ~ n.
f E (~[X]s
)(I) _ such that f ~C 1 > 0 REX]
and
f rc 2 < o.
D
PROPOSITION
10.15.
such that
f r u>0
A relative
separation
THEOREM 10.16. subsets of U.
Let
and
U !
~ n be open s.a.
f r(~nu)
theorem is obtained Let
U _cA n
Then there is f c R(U)
= 0. from 10.14 and 10.15:
be open s.a. and
Then there is f ~ R(U) such that
CI,C 2 be closed s.a.
f FC I > 0
and
f FC 2 <0. D
Various proofs of these results, Efroymson
[27; §i]
due to Mostowski,
and Bochnak  Efroymson
have the following result of Risler COROLLARY is an open,
10.17.
Let
connected
P
As a corollary we
[31]:
be a prime ideal of
s.a. set.
can be found in
[25; §5].
Then
N(U), where
U i
An
Z(P) is connected.
PROOF.
By Corollary i0.I0 there are gl ..... gr eP such that r Z(P) = ~ Z(gi). Hence Z(P) is s.a. and has finitely many connected i=l con~onents
(Theorem 5.8).
s.a. subsets Let
CI,C 2 of
If
Z(P), closed in
f E R(U) be such that
Then
hlh 2
f rC 1 > 0
rE gi2 cP, i=l
=
but
Similarly h 2 ~ P, contradicting REMARK.
f ~C 2 <0.
Let us now define
for
P
CIUC 2 = Z(P).
i = 1,2.
h I l'C1 = 0
and
h I~C 2 >0.
is prime.
D
The proof shows that the result holds for any subring A of
which contains §Ii.
and
h 1 % P because that
there are disjoint
A n, such that
/~ gi2 + f2 + (l)i f i=l
=
hi
Z(P) is not connected,
~,
whenever
THE SUBSTITUTION
f e A and
THEOREM;
f >0
N(U)
on U.
"STELLENSATZE".
In this paragraph we prove another central result in the theory of Nash functions: a number of A.
the substitution
"stellens~tze"
THE SUBSTITUTION If
K
theorem.
for the rings
We use it later to derive
N(U) and many of its subrings.
THEOREM.
is a real closed field containing
~,
S c ~n
is a s.a.
138
set and
f:S ÷ ~ a s.a. function,
parameters define,
in
~) defining
respectively,
ed function on
S
it is clear that the formulas
and
Gr(f) are interpretable
a s.a. subset of
S K, denoted by
fK.
K n, denoted by
The substitution
any ring homomorphism ~: N(U) + K some point of U K, Precisely:
an
K
Then
Kvalu
U c R n be an open, ~, and
4: N(U) + K
:
<~(~i ) .... , ~(~n)> E U K,
maps,
Let
a real closed field containing
R = a l g e b r a homomorphism.
(i)
S K, and a
theorem says that
is the evaluation homomorphism at
THEOREM II.I (The substitution theorem). connected s.a. set,
(with
in K, and
where
~i ..... ~n denote the projection
~i(Xl ..... x n) = x i.
(ii)
For every f E N(U),
PROOF.
Observe that
~(f) = fK(~(~ I) ..... ~(~n)).
(i) gives a sense to (ii).
The proof proceed in the
following steps: (I)
We prove (ii) for
~n,
f E R(U).
Since any such
this makes sense even in the absence of
(II)
We prove
(III)
We prove
STEP I.
Condition
truction of
(ii) is clear for polynomials.
R(U) it suffices
to prove that for
holds for
h
and
h rU>0,
(b) if
(ii)
holds for
h
and
h >0,
PROOF of (a). Since
e0
(*)
By the inductive consh~ c(~n),
=
then (ii) holds for f = ~ ;
then (ii) holds for f = i/h.
hK(~(~))
~(T) = <~(~i ) ..... ~(~n)>.
= fK(~(~))2,
In order to conclude that ~(f)=fK(~(~))
to show that both sides are nonnegative,
for we have assumed that ¢(f) >0
h e0:
f2 = h, we have
~(f)2 = ~(h)
it suffices
(i).
(ii) for arbitrary f e N(U).
(ii)
is
is defined on
(i).
(a) if
where
f
f ~ 0.
for any f E N(U) such that
Indeed, as we have I / ~ e N ( U ) ,
then
The righthand side
But we also have:
~(f).#(I/~)2
f > 0 on = !;
U.
this implies that
~(f) >0. The proof of STEP II. f ~(~n (**) holds in
(b) is similar.
By Proposition 10.15 let _ U) = 0.
f e R(U) be such that
f ~U > 0
Then the firstorder statement Yx (x E U<>f(x) > 0)
~, and hence in K.
By (*) we have
%(f) > 0, and by (I),
and
139
fK(~(~)) >0, STEP III.
It follows
Now we use the ArtinMazur
(Theorem i0,ii). polynomials
s.a. function
P
to prove
X = V(~)
s[U] obviously of
s[U] in
continuity
s(pr(y))
and
for the functions
is connected, component
0 c~q
Using 10.11(b) our contention.
Vx~U V ~ in
~.
prK(~)
Put
theorem
if
y
belongs
to the
for some sequence u n ~ U.
Hence ucU,
closed in X. In
and using and
pr
10.12
(jj)
is injective
on
0 n X ! s[U], which
10.16 to get It follows
that
s[U]is
h c R(W)
such that
from this situation
that
q [z ~ V n W
^ h(~) > 0 it holds
and
^ pr(z)
in K.
= ~ ÷ z = s(x)]
Specializing
~ = <~(~i ) ..... ~(nn)>
of (***)
to
= ~(~)
(which is in U K K ~(s i) = si(~(~)),
is the equality It suffices,
then,
to check that
of (***) hold in K.
= p r o s = idu(10.11(b)).
hK(z) > 0.
By (I),
~(hos)
(*) shows
that

 and
statement
By transfer
Irmnediate from 
(a)(c)
s I ..... Sq.
we conclude
at once that
h T (X  s[U]) <0.
the consequent
the premises
Indeed,
s(u) c 0 n X
i = 1 ..... q, which we want to prove.

X).
~ = lim s ( u n~
it is checked
= <~(s I) ..... ~(Sq)> by (i)),
continuous
and we prove next that it is clopen
of
so that
the separation
and
the firstorder
holds
(ii)
X, then
0 nX.
Now we invoke
I0.II
a
nW.
p r o s = idu(10.11(b))
proves
h F s[U] > 0
P1 .... ,P£~ ~ [ X 1 .... ,Xq] be
so that conditions
s[U] is open in X, let
choose an open set
of Nash functions
s = <s I .... ,Sq >:U + V ( ~ )
= lim s(u n) = y, i.e. 7 ~ s [U].
order to see that
(***)
characterization
(jj)  hold.
X (hence a connected
closure
~(~)E U K.
q ~n+l, V,
a polynomial,
(j),
and
let
a variety
it suffices
The set
By
and
10.12
W = prl[u]
in
Therefore,
defining
hence also Clearly,
from (**) that
z
Since
=(hos) K (~(~))
= hK(z).
Since
h >0
on
s[U],
then
~(hos) >0.
e ( V n W) K.
x = prK(z) E U K,
the variety
V
and ~
then
zcW K.
As the polynomials
P1 .... P£
s[U] i V ( ~ ) , we have £ VycU A Pj(s(y)) = O, j=l
that is, the Nash functions
P.os ]
vanish on
U;
therefore
define
140
V
0 = ~(Pj o s) = P~(~(S))
=
The substitution theorem is due to is due to Coste [67]; §7].
B.
for
j = i ..... £.
Efroymson [27].
D
The present proof
for a different proof see BochnakEfroymson
As shown in this paper,
of subrings of
P~(z)
[25;
the result applies as well to a wide class
N(U).
APPLICATIONS:
"STELLENSATZE"
;
COMMUTATIVE ALGEBRA OF NASH
FUNCTIONS. Now we show how the substitution theorem can be used to derive, by a uniform method, all the "stellens~tze" holding for rings of Nash functions.
These derivations
are purely algebraic
 in fact, they are
a sophisticated elaboration of Robinson's method to prove Hilbert's nullstellensatz

and work for any ring of s.a. functions
for which a
substitution theorem is available. A "stellensatz"
is a result establishing an equivalence between a con
dition of the form ~
V ~ c U [A gi(~) ~ 0 i
^ A hj(~) > 0 ^ A fk(~) = 0 j k
+ f(~)? 0]
where gl ..... gr' hl'''''hp' fl .... 'fq' f c N(U) and ? is a sign condition (> , ~ or = ) and, on the other hand, an equation P(gl .... 'gr' where of
P
hi .... ,hp,
is a polynomial with coefficients
r,p, q
f) = 0
fl .... 'fq' in
N(U).
Here, one or more
may be zero.
Thus we have: THEOREM 11.2. gl,...,gr, (A)
(General stellensatz
h I .... ,hp,
for Nash functions).
Let
fl .... 'fq' f e N(U).
The following are equivalent: (1) (2)
R > V~U[A There are
i
gi(~) _>0 ^
A hj(~) > 0 ^ j
t,S,Uk~N(U )
and
sj ~{0,i}
A fk (~) = 0 k
÷f(~) >0].
(l_<j _
where s,t are of the form F(g I .... 'gr' hl ..... hp), with F a polinomial whose coefficients are squares in N(U), such that ft = s + H he.j + j J (B)
Similarly,
f(~) >_ 0
is equivalent
~fk~k"
a condition of type (A.I) with to a polynomial equation
f(x) >0
replaced by
141
E.
ft = (Hh. j) f2£ + s + ~ fkUk , j J k for some
£>0
and
s,t, u k, sj
A condition of type
(O
as is
(A.I) with
(A). f(x) = 0 instead of f(x) > 0
is
equivalent to ( ~ h~ j ) f2£ with
£ el
and
+ s + kl fkuk = 0
S,Uk, ej as in (A).
Each of the equivalences "formal stellens~tz".
(A), (B), (C) is derived from a corresponding
For example, the result needed for
PROPOSITION 11.3. (Formal positivstellensatz). tive ring with unit, and: (a)
S a subsemiring of
A(= a subset of
Let
A
(A) A
is:
be a commuta
closed under sum and
(b)
product, but not necessarily under difference) containing the squares. M a multiplicative subset of A containing i.
(c)
I
(d)
f c A.
an ideal of
A.
Then, the following are equivalent: (i') For every ring homomorphism field, we have: ~[S] ~0, (2')
~[M] >0
and
~:A ÷ K, with
~[I] = 0
imply
K
a real closed
~(f) >0.
There are
s,t e S, m £ M and u e I such that f.t=s+m+u PROOF of : Proposition ! 1 . 3 implies Theorem II.2(A). The implication (A.2) implies (A.I) is checked without difficulty. (A.I) implies
(A.2).
As it is obvious, we apply
11.3 with
A = N(U) and:
S = the subsemiring generated by the gi's, the h.' ,~ s s and the squares. M = the multiplicative subset generated by the hj . I = the ideal generated by the
fk's.
It suffices to derive condition (I') of 11.3 from condition (A.I) of Theorem 11.2. This is done using the substitution theorem, as follows: Let ~ be a homomorphism verifying the assumptions of (i'); thus we have: ~(gi ) ~0,
~(hj) >0,
~(fk ) = 0
for all i,j,k.
142
By the substitution theorem we get: giK(*(Y)) e0, The assumption
h~(~(~)) >0,
(A.I) then yields
again, we conclude
f~(~(~)) = 0
fK(~(T)) >0, and by substitution
~(f) >0.
D
The "formal stellens~tze" are, in turn, a consequence of: 11.4 The pivotal lemma. T, N be subsets of
A
Let
A
be a commutative ring with unit and I,
such that :
(a)
T is closed under multiplication.
(b)
I c T is an ideal of A.
(c)
i)
ioN.
ii) iii)
x,ycN implies (xy)eN. (N). Z TA 2 c ETA 2 .
Then, the following are equivalent: (i")
N n ETA 2
#
0.
(2")
For every prime ideal the fraction
field of
N/p n E
P
of
A containing
I, if
L denotes
A/p, we have:
T/p " L 2 # 9.
D
We omit the purely algebraic proof of this result, which can be found in Dickmann
[72; Ch. V].
This is a generalization of ColliotTh~lene
[44; Lemma l.bis], who proves it in the case
I = (0) and
N = {i}.
However, we give the PROOF
OF
:
Lemma 11.4 implies Proposition 11.3.
The implication
(2')
implies (I') in 11.3 is evident. (i') implies (2').
Assume not (2'); this just says:
(*)
M n ((Sf S) + I) = 9.
We apply Lemma
11.4
with
N = M
and
T = (S=f.S) + I, leaving as
an easy but tedious exercise for the reader the task of checking that the assumptions
(a)(c) of 11.4 are fulfilled.
The equality (*) amounts to the negation of condition (i") of 11.4. Then, by (2"), there is a prime ideal (**)
containing I such that
N/pn Z T/p .L 2 = 9.
Since ~,
P
fl cM, then
which makes
i
~ Z ~
T/p e0.
L2 . ~Let K
This means that
L
admits an order,
be the real closure of
~:A ÷ K the canonical homomorphism,
~(a) = a/p.
Then we have %IT] e0,
143
which implies: (***) as
~[S] e0,
S ! T,
~(f) ~0
and
f • T by definition
By (**) we get
~(x) # 0 for
(****)
and
x•M,
M . E T A 2 ! ETA 2 by (c.iii). i.e.
~[M] > 0.
Since
I c p
we have
(*****)
$[I]
Conditions
(***),
Proposition
(****)
and (*****)
PROPOSITION
11.5 I
f vanishes
(ii)
f • ~I.
In particular,
the numerous
N(U)
on
and
(ii) implies
i0.i0 there are
condition
(i) is equivalent
of
PROOF. • Z(M);
that
Let
N(U)
by
Z(I) = N Z(fk); hence k
f(x) = 0]. to
£ _> i,
R~coincides
with
Rad(I),
U c 1%n be an open connected
is real,
the radical
s.a. set.
Every
and the map:
: M~a = { f ~ N ( U )
a oneone
correspondence
I f(a) = O} between
points
of
U
and maximal
N (U) .
A maximal
also real.
Conversely,
f • I.
we have:
11.6.
a;
ideals
÷
= 0 implies
f c R¢~.
ideal of
establishes
= 0
for some
The proof shows
COROLLARY
such that
(with r,p = 0) this is equivalent
In particular
maximal
are equivalent:
f rZ(I)
to
f2£c I implies
the following
(i) is trivial.
fl .... 'ft ~ I
R ~ V ~ • U [Ifk(~) ii.2(c)
11.2.
for Nash functions).
f ~ N(U),
I is real iff for every f • N(U),
The implication
By Theorem
of Theorem
Z(I).
Corollary
REMARK.
(I'), which proves
corollaries
(Real nullstellensatz
of
(i)
of I.
contradict
D
a few amongst
Given an ideal
PROOF.
= 0
11.3.
We mention
which
~[M] ~0,
Since then
PROPOSITION
ideal is always radical; M
M _c M~ 11.7.
is proper,
by the remark above,
11.5 also shows
and, by maximality, (Solution
of Hilbert's
that
Z(M) # #.
it is Let
M = M~. 17th problem
for Nash
144
functions). Let
U !~ n
on
U, then
PROOF.
be an open, connected s.a. domain and f
f e N(U).
If
is a sum of squares in the fraction field of
By Theorem
II.2(B)
where I is an integer and
(with r = p = q = 0), we get
s,t
sums of squares in
f ~0
N(U).
f.t = f2l + s,
N(U).
D
Further results in the same vein are reported in BochnakEfroymson [25]. As remarked earlier, Theorem 11.2 holds for any ring of realvalued s.a. functions for which the substitution theorem holds. the case for the polynomial ring
IR[XI,...,Xn].
Trivially,
this is
Thus we get back, with
a uniform proof, generalized versions of earlier semialgebraic
stellensatze:
the real nullstellensatz 4.8 (as in Proposition 11.5) when there are no gi ' s or
hj'
s
but we allow the
fk s;
the equivalence (c) of Theorem
11.2, when there are no fk ' s but we allow the
hj' s, is the
gi ' s and the
semialgebraic real nullstellens~tz of Stengle [58]; fk's or hj's but we allow the gi's, the equivalence
when there are no II.2(B) is Stengle's
nichtnegativstellens~tz.
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(1981) pp.25.
ON
THE
(%)K
Carlos A.Di Prisco
Wiktor Marek
I.V.I.C.
Dept.Computer
Departamento Apartado Caracas
§i.
Lexington,
1827
on
order type of P = ~} appears naturally
and is specifically
As it is shown in [S.R.K.]
target ~ (c.f. also [B.D.T.]) ultrafilter
(%)K
KY 405060027
U.S.A.
(%)< = {P i ~ I
in the study of large cardinals
Science
University of Kentucky
de MatemAticas
1010A, Venezuela
The space
huge cardinals.
for
SPACE
if there is a
containing
related to the so called
, a cardinal
K is huge with
Kcomplete normal nontrivial
the sets of the form
p = {Pc(%)
p ~ PK(~).
Since any ultrafilter
on the set
ties mentioned above
(Ncomplete,
[%]K = {p ! %1
IPI = K} with the proper
normal and fine) concentrates
the existence of such ultrafilter
on [l]r is also equivalent
on ( % ) K
to hugness
of K with target ~. In
[DP.M2] we constructed
trivial
filter on [%]K
Large cardinal property
Related combinatorial structure.
This raises
F~,%, the least Kcomplete,
, which permitted us to complete
Normal Measure on (K measurable)
Normal Measure on
Closed unbounded subsets of
Closed unbounded subsets of
P~(~)
Normal measure on [%]K
(K÷(~))
could the table be also completed putt
structure on (%)K instead of FK,%?.
"absolute" notion on (%)K
FK,~
PK(X)
this note is to show that there is no such possibility: natorial
the following table
(K %supercompact)
the following question,
ing a combinatorial
fine, normal non
approximating
The purpose of there is no combi
normal measures on (%)K
In this way we present results which stress the differences and
[%]K studied by Mignone in [Mig.].
pointed by Mignone pertains minateness choice.
is accepted,
between
(~)<
One should note that the difference
to the situation in which the axiom of deter
whereas
in this paper we work under the axiom of
We would like to thank M. Magidor for providing us with the exact
statement of Theorem 4.
152
Let us recall some results and definitions from be regular uncountable cardinals. subset of PK(%), let of elements of X}. of PK(%)}
If
[DP.M2]:
Let
K<%
X ! PK (%) is a closed unbounded
AX = {Pe[k]KIP is the union of an increasing Kchain The sets of the form
{AxIX
closed unbounded subset
generate a Kcomplete, normal, fine,nontrivial filter on
[%]K
called FK,%" It is interesting to note that if in the definition of sets of elements of ([DP.M2]).
X
instead of
AX
we use directed
Kchains we obtain the same filter
The following propositions are proved in [DP.M2].
PROPOSITION i.
If
V = L
then
(%)K is not ~,~stationary.
PROPOSITION 2.
FK, k is the least
Kcomplete,
0
fine, normal, non
trivial filter on [~]~.
0
From these propositions we obtain the following METATHEOREM:
If
ZFC
is consistent then
plete, normal, fine, nontrivial filter on PROOF.
ZFC
"There is a
Kcom
(~)K,,
Assume the contrary, so ZFC + V=L proves as well that there is a
nontrivial, G on
Kcomplete,
fine, normal filter F on ( % ) K
Define a filter
[%]K as follows XcG
Then G
is a
Kcomplete,
~=~
Xn(%) K e
F. [ ]K and
fine, normal, nontrivial filter on
therefore by proposition 2,FK,~! G.
But (%)KEG and so (~)K
is
F
stationary contradicting proposition I.
O
The next statement clarifies the situation. PROPOSITION 3.
The following statements are equivalent:
a)
There exists a
b)
FK,% t(k) K
c)
(%)~
KCOmplete, normal,
fine, nontrivial filter on
(~)~. is
is such a filter.
FK,xstationary.
PROOF. a) ~
b):
The argument of the metatheorem shows that (%)K
stationary and therefore
FK, ~ induces a filter on (~)K
implications follow easily. The results in is
[DP.M2]
FK,%stationary or
is FK, %
The other O
suggest that perhaps we have that either (%)K F~,%
is the CLUB filter on PK+(%) restricted to
[ ]K , or equivalently that either (%)< is F K ,~stationary or ~ belongs to FK, ~ (where ~ = {Pc[%]KI~!P}). (In fact, the first part of the dis
153
junction holds if K is huge with target % and the second occurs if V = L or, as it will be shown in the next section if ~ = K+). This alternative does not hold in general but we have something close to it: THEOREM 4. there is PROOF.
Given regular cardinals
K<%'~X
such that (%')< v ~ ~ F~,%
Suppose
is
~FK, %
or
FK,X, stationary.
then, its complement in [X]~, i.e. the set
S = {P ! [X]
K<% then either
f(P) = first element of
P
Considering the function f:S ÷ X defined
after the first gap, and using the normali
ty of FK, % we see that there is an FK, % stationary S I ! S such that for all PcSl,f(p ) is a fixed value ~. Since F is fine (i e. contains v K,X " all cones p = {PeE%]KIpIP} for pcPm(%)) this ~ cannot belong to K. th Thus ~eK. So, if an element P of S' has order type >K, its element must be above K.
Therefore,
then a stationary subset
S2
greater than
g:S 2 ÷ % by
K.
Define
is a stationary subset g(P) = ~,
S 3 i $2
From this we conclude that given X' =
SI
is not
FK, % stationary
contains only elements of order type g(P) = K th element of P.
and a
(x,)K is
There
X'<% such that for every p~S 3,
FK, ~, stationary.
is closed and unbounded in
S3nA X, is thus
PES3nA X, Pn~'
This is so because
(X)K
is not
PnX'~Ax,n(X) K.
T<X)actually occurs.
<
A = {P~[x]KIy+[Pny]=Py} "just like"
Py).
(i.e.
0
FK, % stationary and ~ is not in FK, ~
(but (y)K is FK, Y stationary for some example due to J. Henle. Consider
PK(X) SO AX,CFK, X.
FK,xstationary.
has order type K and so
The possibility that
is
(%)<
X ! P~(X') closed and unbounded the set X' ! P<(X) defined by {pePK(x) Ipn%'~X}
The set For
of
if
A is the set of
This set is in
X = {p~pK(%)Iy+(pny)=py} is club and stationary. We also have that ~v F < , x
We give an
The set
P~[X] ~ such that
Pny
F<,% because the set A X = A. Thus (x)K is not FK, %since if ~ e F< X then there is
a closed and unbounded set X ! PK (%) such that ~ ~ A X. But since K<X~y there is a bijection f from X to y which is the identity on ~. The set
Y = {f"PlPeX}
is a closed and unbounded subset of
PK(y) such that
Ay is in F~,T and is contained in ~. But this contradicts that in this case (y)< is F<,yStationary. §2.
We consider in this section the case in which X = K+.
PROPOSITION i. Then
K+(y) since
A X n Ord
Let
X
be a closed unbounded subset of
is an unbounded subset of
m+
P (~+).
154
PROOF. stages than
Given
y
a directed
let
pEX
subset of
be such that
X
y~p.
We will
whose union is an ordinal
in
construct in + K greater
y. STAGE i: For each ordinal
containing
it.
Put
~ in
upp
let
be an element
p~
X.
X
Dl={p~l~upp}u{p}. D2
STAGE 2: Close D 1 to obtain D 2 ~ D 1 such that set of
of
(Note we can obtain
such a directed
set
is a directed
D2
sub
such that
}D21=ID!I+ N0).
STAGE 2n+l: For e v e r y ~ E sup(uD2n)UD2n l e t containing it.
Put
STAGE 2n+2: containing
D2n+l=D2nU{P ] ~
Close
It follows
D =
closed and unbounded,
subset of
2.6
AXnK +
of
~ < K +,
PK(K +)
Therefore
if
and
+
a filter on
Kcompleteness,
the filter,
~, p
XsP
(K+), X
K +, thus we have
subset of
fine filter on
and so
(K+) K
CLUBK+ _c FK,K+ rE +
K + (it is fine because is a closed unbounded
is not
if
X
FK,K+ stationary.
is closed unbounded
i shows
which provides
normality,
or fineness.
that if we want
an "absolute"
The easiest
if we can maintain
sub
On
then the
to produce
counterpart
to
to give up seems
unboundedness
is that,
with respect
if
K÷(%)
such a filter modifying
X ! PK(%)
if for all
that
cones
to be in
Another
condi
of the filter
this fact.
a construction
is endunbounded
P Se q (where p Se q means
to it).
the elements
to any normal measure witnessing
to
of the elements
(That is to say: even if we do not require
they must be stationary
We will define We say that
that for each
On the other hand,
of section
(%)K
tion we wish to maintain must belong
in
we must drop one of the three usual properties:
specially
of our filter.
is an ordinal
(AXo { $}) nK + .
The m e t a t h e o r e m
be fineness,
[DP.M2]
K c FK,K+ rE + because
(%)K,
uD
A^ ford c K+~). p
set K^ = ~ K contains
on
such that
pePK(K +) contains
FK,K+ ~ = C L U B
the other hand,
X
is a closed
since this is a ~+complete normal
such that
X, D2n+2,
0
FK,~+ rK + __c CLUBK+.
given any
(in ZFC)
subset of
u D nn~
from the lemma
shown that
a measure
X
sup(uD2n)UD2n}.
D2n+l to obtain a directed
Clearly D is a directed + K above y.
§3.
be a set in
D2n+l.
Finally we put
set of
pe
q
from [DP.M2].
p~PK(%)
there is q~X
is an end extension
of p).
155
A
Clearly every endunbounded set is unbounded. not endunbounded (with the exception of filter generated on
PK(%)
The cones
$=P<(%)).
p={qJp!q} are
Call SUPERCLUB the
by the endunbounded and closed subsets of
P<(~). PROPOSITION. The filter SI~ERCLUB is trivial.
<complete, normal and non
It is not fine but all its elements are unbounded subsets of
P<(%). PROOF.
Let us first recall a definition from
filter on
P<(%) generated by sets which are
[DP.MI].
with respect to unions of end extension chains of length is
Kcomplete and normal but not fine
ECLUB is the
end unbounded and closed
([DP.MI]).
<<.
This filter
It is easy to verify
that SUPERCLUB=CLUB n ECLUB and thus we have only to prove that this intersection is non empty. For this, note that the sets = {p~PK(~)lorder type of Since
p>~} are in SUPERCLUB. + SUPERCLUB is not < complete.
n{~:~e<}=~,
D SUPERCLUB is not the
least <complete, normal filter in P<(%) containing {~I~
peP<(%) there is
~<~
such that
pu(~up)eX). Using the filter
ideas
on
o f [DP.M2]
(~)<.
Given
we d e f i n e
X ! PK(~),
now a
Kcomplete
normal nontrivial
let
A"x={PE(%)K I There is an end extension chain {p~}~
If
X
is endunbounded then
GK, % the the filter on
THEOREM 3.
G <,% is a
up~ =P}
(X) K
A" X #~.
generated by
D
{A"xIX superclub}.
Kcomplete, normal, nontrivial filter in
(~)K PROOF.
The
Kcompleteness and normality are easily verified.
To see that
G
is nontrivial, in the sense that it does not contain small subsets K,% of (%)<,we prove the following lemma. LEMMA.
If
B ! (X)K
is such that
[%uB[=%
then B is not G <,%
stationary. PROOF.
It is enough to show that there is a superclub set ~
A"~nB=~.
We put
~ = { p E P K ( X ) j for all
PeB(pP#~)}.
such that
156
Clearly
~
is closed, and since
kuB
is cofinal in ~, it is super
unbounded. Therefore, of (~)K
0 if
K
then
of cardinality
GK, ~
contains the complements of all subsets
<~.
0
We will now show that the elements of respect to normal measures on (k) < LEMMA. PROOF.
If
[
Axn(k) K.
elements of X.
are of measure one with
is closed and unbounded then
The inclusion
belongs to
and
X
GK,k
is obvious.
Then
P =
v
A"xAxn(%) < .
For the other one, assume that
p~
where
Since we can assume that (P~)$
is a continuous chain
P = K it is not difficult to construct a subchain of
is an end extension chain. COROLLARY 4. sure, then
If
P
0
A e GK, ~, K ÷ (~) and ~ is a witnessing normal mea
A e ~.
PROOF.
There is an
(k)K~
SO
X c SUPERCLUB such that
A" X ! A.
But then
AXC~ and
A" X ~ ~.
0
Finally let us mention that if
X~
is closed and unbounded for each ~<~,
A A" X~ = ~<~
A" ~<~ A
X~
Note that both sides could be empty unless the sets
X~
are superclub.
REFERENCES [B.D.T.]
[DP.MI]
Barbanel, J., DiPrisco,C.A.& Tan,I.B. DiPrisco,C.A.& Marek,W.
Manytimes huge and superhuge cardinals. J. Symbolic Logic 49(1984),pp 112122. Some properties of stationary sets. Dissertationes Mathematieae CCXVIII(1982), pp.l37.
[DP .M2]
Di Prisco,C.A.& A filter on ~rek,W. 591598.
[Mig]
Mignone, R.J.
[k]K Proc.Amer.Math. Soc. 90(1984), DD.
On the ultrafilter characterization of huge cardinals. Proc.Amer.Math.Soc.
90(1984), pp.
585590. [S.R.K.]
Solovay,R., Reinhardt,W. Kanamori,A.
Strong axioms of infinity and elementary embeddings.
Annals of Mathematical Logic 13(1978),
pp. 73116. This work was partially supported by CONICIT grants SII129 & Capt 1584
THE MODEL EXTENSION THEOREMS FOR ~ 3 THEORIES
Itala M.L. D'Ottaviano Universidade Estadual de Campinas Instituto de Matematica,
Estatistica e Ciencia da Comp.uta$~o
Caixa Postal 6155 13100 Campinas, SP,
Brasil
Two generalized versions of Keisler's classical Model Extension Theorem are obtained for
~ 3 theories.
These theories are threevalued with
more than one distinguished truthvalue, reflect certain aspects of modal logics and can be paraconsistent.
~3theories
were introduced in
the author's doctoral dissertation. §i. INTRODUCTION.
A theory
T
is said to be inconsistent if it has
as theorems a formula and its negation;
and it is said to be trivial
if every formula of its language is a theorem. A logic is paraconsistent if it can be used as the underlying logic for inconsistent but nontrivial theories
(see [I] and
[2]).
In a previous paper (see [4]) we introduced a threevalued propositional system,
~ 3 ' with two distinguished truthvalues, which is para
consistent and reflects some aspects of certain types of modal logics. In another paper (see [5] ) we axiomatized
~3
and established relations
between this calculus and several known logical systems, as, for example, intuitionism.
We especially emphasized the close analogy between
and Lukasiewicz'
~3
threevalued propositional calculus £3"
We also introduced the corresponding threevalued firstorder
~3 
Theories in whose languages may appear other equalities in addition to ordinary identity, and which extend the firstorder predicate calculus , ~ 3 =" We proved the Completeness Theorem and the Compactness Theorem for ~ 3  t h e o r i e s . The model theory for ~ 3 theories reflects much of the classical model theory.
We were able to obtain in our doctoral dissertation generalized
versions of the following classical results: Keisler Model Extension Theorem, Lo~Tarski Theorem, ChangEo~Suszko Theorem, Tarski Cardinality Theorem, LSwenheim  Skolem Theorem, JointConsistency Theorem,
158
Craig Interpolation Lemma, BethPadoa Definability Theorem, the Quantifier Elimination Theorem and many of the usual theorems on complete theories and categoricity. In some cases, as for example the Model Extension Theorem and the Definability Theorem, we proved more than one generalization of the classical results, all of them compatible both with the manyvalued aspects and the modal aspects of ~ 3  t h e ° r i e s " Our aim here is to present two generalized versions of Keisler's classical Model Extension Theorem, asserting both the conditions to extend a structure ~ for a L31anguage In
L
of
T
to a model of the ~ 3  t h e o r y
§2, after giving some basic notions and the axiomatization for
T.
~3
theories, we emphasize several results and theorems which are necessary for the development of the last part of the paper. In
§3, after adapting the concepts of isomorphism and elementary equi
valence, we define
Fsubstructures and discuss the case of certain
special sets of formulas ro After this, we present the two Model Extension Theorems for ~ 3 theories. Both theorems generalize the classical Model Extension Theorem of Keisler and their proofs involve specific characteristics of manyvalued and modal logics, reflecting the existence of more than one distinguished truthvalue in the matrices defining ~ 3  t h e ° r i e s " Finally, we give a threevalued version of Lo~Tarski Theorem, which characterizes universalopen ~ 3 theories; Tarski Lemma, on the union of elementary chains; and ChangLo~Suszko Theorem, characterizing universalexistential ~ 3  t h e o r i e s . In this paper, definitions, theorems and proofs, when similar to the corresponding classical case, are omitted. For definitions and theorems we use the nomenclature of [13]. We have also extended some of the above results about ~ 3  t h e ° r i e s ~ntheories,
to
3 ~ n ~ ~o"
The results about Joint nonTrivialization,
Definability Theorems,
Complete Theories, Quantifier Elimination Theorems, Categoricity and the mentioned results about ~ 3  t h e o r i e s will appear elsewhere. The content of this work is part of the results of our doctoral dissertation.
159
§2.
FIRST ORDER
~3THEORIES. The symbols of a firstorder
guage are the individual symbols, V,
the primitive
variables,
the function
connectives
~ , v
and
symbols,
~31an
the predicate
V, the quantifiers
@
and
and the parentheses.
The identity
= must be among the predicate
In particular
cases,
other predicate
symbols
symbols
of every
L31anguage.
can be considered
as equali
ties. We use f c
x,y,z
and
w
as syntactical
and g, for function for constants.
The definitions
p
of term and atomic
order languages; The definition a formula,
symbols;
variables and
formula
q,
then
VA
variables;
symbols,
variables
for terms.
is also the usual plus the condition:
is a formula;
A,B,C,
and
are the usual ones for first
a,b ..... etc, are syntactical
of formula
for individual for predicate
etc, are syntactical
if
A
is
variables
for formulas. By a L 3  1 a n g u a g e of [13]) whose
we understand
logical
The truthfunctions;
symbols Hv, H V
a firstorder
language
are the ones mentioned and
H~
(in the sense above.
are defined by the following
tables:
AvB •
0.,
,
A
VA
0
0
0
0
I
½
1
½
½
1
i
i
I
0
{0,½,1}
and the set of distinguished
{½,1} are denoted by "V" and "Vd" respectively.
A
&
B
abbreviations
=def ~ ( T A v
will be used:
~B)
AA =def IV ~ B ~*A =def ,VA A >+ B =def VdAy B A = B =def A ~+ B =def
~A
½
The set of truthvalues The following
A
~VA v B (A >+ B) & (~B >+ ,A)
A = B =def(A = B) & (B = A)
truthvalues
160
is called weak negation or simple negation, implication, We present
the table of some of the non primitive A
AA
~*A *A
A
~ * strong negation, = basic
~ basic equivalence.
A
AA
0
0
connectives: ~~
g
1
0
o
½ I
i
I
0
1
I
I
½I
I ½o
½
A=B
A=B
_a °
I
o
L
o
0
0
t
1 0
0
0
0
Free occurrence
of a variable,
open formula,
I
closed formula,
variable
free term and closure of a formula are defined as in [13]o We let
b x l , . o o , X n [al,...,a n]
replacing
all occurrences
respectively; ed from
A
and we let
of
be the term obtained x I .... ,xn
by the terms
from
b
by
a I ..... a n
AxI ..... Xn [a I ..... an] be the formula obtain
by replacing all free occurrences
of
x I ..... x n
by
a I ..... a n respectively. Whenever
either of these is used,
x I ..... x n
are distinct variables
Axl ..... Xn [a I ..... a n ] , a i D E F I N I T I O N 2.1. sists of : i) a nonempty
it will be implicitly and that,
is substitutible
iii)
I ~71 n
to
x i,
i = I .... ,n. ~31anguage
con
f
of
L, an nary function f ~
I~I;
for each nary predicate
nary function
for
I ~ I called the universe of ~ ;
ii) for each nary function symbol from
in the case of
A structure ~ for the firstorder set
assumed that
p~
from
l~I n
symbol to V.
p
of
L, other than
=, an
161
As in [13] ~f(a) ~ V
, we construct
the language
for each variable
L(~)
free term
a
;
of
and define a value
L(~f),
and
ginstance
of a formula A. We use
£
and
j
as syntactical variables
for the names of individuals
of ~f. DEFINITION in
L(~f) i)
if
A
contrary, ii)
= p~
2.2.
The truthvalue
~f(A) for each closed formula
A
is given by: is
~(A)
if
A
is
a = b,
then
~(A)
= I
if
~f(a) =
@£(b);
on the
= 0; p(a I ..... an) , where
( ~ ( a I) ..... ~ ( a n ) )
p
is not
=, then
~f(A) =
;
iii)
if
A
is
~ B, then
~£(A)
is
H (~(B)) ;
iv)
if
A
is
VB,
~(A)
is
HV(~f(B));
v)
if
A
is
B v C,
vi)
if
A
is
~xB,
then
~(A)
= max { ~ f ( B x [ Z ] / Z E L ( ~ f ) } ;
vii)
if
A
is
VxB,
then
$(A)
= min { @f(Bx[Z] Z c L ( ~ ) } .
7
DEFINITION for every
2.3.
then
A formula
~instance
DEFINITION
then
2.4.
A' of
~(A)
A
of
is
Hv(~f(B), ~f(L));
L
is valid in ~ i f ,
A, ~(A') belongs
A firstorder
~3the°ry
to
and only if
V d.
is a formal system
T
such
that: i) ii)
the language of the axioms of
T, L(T), T
further axioms, iii)
is a
•31anguage;
are the logical axioms of the nonlogical
the logical axioms and rules of
T
AXIOM
i :
A(A ~+ (B ~+ A))
AXIOM
2 :
A((A ~+B) ~+
((B ~+ C) ~+ (A ~+ C)))
AXIOM
3 :
A((~A~+,B)
~+ (B ~+ A))
AXIOM
4
A(((A ~+ ~ A) ~+ A) ~+ A)
AXIOM
5: :
AXIOM
6
7 •
x
AXIOM
8
Ax[a]
AXIOM
9 :
AXIOM i0
:
A(A(A ~+ B) ~÷ A(AA ~+ AB))
AXIOM
: :
Vx(x = x) =
y ~ (A[x] ~ A[y]) ~ @xA
VxA = A x[a] ~xA   ~ V x ~ A
and certain
are the following, w i t h the
usual restrictions :
:
L(T)
axioms ;
162
A X I O M ii :
VxA : m ~ x ~ A
A X I O M 12
: [email protected] ~ V x ~ A
A X I O M 13
: ~VxA
 ~x,A
A X I O M 14 :
V~xA = ~xVA
A X I O M 15
VVxA E VxVA
RULE
:
A,A(A ~+ B)
RI :
B
RULE
R~
RULE
R3
VA
:
L
T A~
(@introduction) :
C
@xA~ RULE
R4
(VIntroduction):
C
C = A C ~ VxA
A
is a Theorem of
T, in symbols
:
~ T A,
is defined in the standard
way. Observe that it is possible connectives
7
and
to define
~÷, instead of
L 3  1 a n g u a g e s , using the Eukasiewicz' v, V
and
m
So, there is a close analogy b e t w e e n Lukasiewicz's (see [3],
[7], [8] and
THEOREM 2.1. connectives
7
~3
is a n o n  c o n s e r v a t i v e
and
THEOP~M 2.2. positional
[9]) and the underlying
~3
is a conservative
calculus with connectives
[6]),
the matrices
~3
In the case of classical is not the case for ~T~A
~
DEFINITION
£3
with
extension of the classical *,
, &, =
and
~ has to be gradually
pro
~. fortified
(see
have more than one d i s t i n g u i s h e d logic,
the equivalence
to the other logical ~3the°ries'
~ behaves
symbols°
truthvalue.
as a congruence
Unfortunately
since it is possible
to have
this
~T A ~ B
~B.
Hence, we introduce ce in
extension of
in order to obtain a relation compatible w i t h the fact that
defining
relation with respect
and
logic £3
~+.
The initial notion of equivalence [5] and
threevalued
logic for ~ 3  t h e o r i e s .
theories 2.5.
a stronger equivalence, and A ~
~*, called strong equivalen
which is a ~ ]  c o n g r u e n c e
relation.
B =def(A ~ B) & (~A z m B ) .
For this strong equivalence we obtain an Equivalence
T h e o r e m for
163
~3the°ries
(see [5] and
[6]).
The following closure theorem and the theorem on constants are identical
to the classical ones.
THEOREM 2.3. if, If
for ~ 3 theories
If
A' is the closure of
A,
then
~TA
if,
and only
~TA' . T'
is a ~ 3  t h e o r y
for every formula
A
obtained from
of
T
T
by adding new constants,
and every sequence
then
el,o°.e n of new constants,
~T A if, and only if, ~T'Ax I, ... ,xnEel ..... en]" We observe
that if F is a set of formulas
in the theory
T, then
is the theory obtained from F by adding all of the formulas nonlogical
T[F]
in F as new
axioms.
THEOREM 2.4.
(Reduction Theorem for nonTrivialization):
non empty set of formulas
in a
~3theory.
trivial if, and only if, there is a theorem of tion of negations
Let F be a
Then the extension T
T[F] is
which is a disjunc
of closures of formulas of type
VA,
with
To finish this section, we discuss how to convert a formula
A A
in
F.
into a
formula in prenex form. DEFINITION operations
either
2.6.
on
If
A
A
is a formula in a
the classical prenex operations,
b)
replacement or
L, the prenex
are:
a) ~x
L31anguage
of a part
VQxB
with the usual restrictions;
of
A
by
QxVB, where
Qx
is
Vx.
By the above definition we can obtain the prenex form of every formula A, based on the following result. THEOREM 2.5.
If
prenex operation,
A'
then
is obtained from DTA
A
with
A
in
L(T), by a
~* A'.
~3theories
are threevalued
truthvalue,
they can be paraconsistent
theories with more than onedistinguished and reflect certain aspects of
modal type logics. The details of a further study, this paragraph [5] and §3.
the proofs of the theorems mentioned
and other results about
~3theories
in
can be found in
[6]. THE THEORY OF MODELS° The theory of models
of classical model theory.
for
~3theories
reflects a great deal
164
In some cases, theorems,
there is more than one generalization
all of them compactible
of the classical
with the characteristics
of ~I 3 theo
ries. In the following, we give the two versions Model Extension Theorem for dl3 theories. The proofs
of
manyvalued
both theorems
and modal
logics,
reflect
appropriate
and the conventions
3.1.
Let ~f and
such that for
a I .... ,an in l~f I:
ii) p
If
from
u
[13],
w i t h the
~B is a bijective
for the
mapping
•31anguage
~ from
I • I to
L. I~ I
P~f (al ..... an) = P~B (~(al) ..... ~(an)) , for every predicate of L.
of I~fl
JB .
of
the existence
~(f~f (al ..... an)) = fdB (#(al) ..... ~(an)) , for every function of L;
If ~ is a mapping a
of
£B be structures
of ~f and
symbol
characteristics
involve
adaptations.
DEFINITION
i) f
Keisler
truthvalue.
An isomorphism
symbol
the specific
and they specially
of more than one distinguished We shall use the notations
of the classical
u
from
l~fl
, we use i ~
to
THEOREM 3.1.
and i is the name of an individual
to designate
is an expression
by replacing
IJB I
of
the name of the individual
L(~f),
each name i by
u~
= JB (a ~) for every variablefree
term
for every closed formula
A
L(~f).
The following
are similar
definitions
~(a) of obtained
i#.
Let ~ be an isomorphism of
is the expression
a
of ~f and of
~ .
L(~f),
Then
and
to the classical
~(~f(a))
~f(A) = ~ (A ~)
ones°
DEFINITION 3.2. Two estructures ~ and ~ for L are elementarily equivalent, in symbols ~ z ~, if the same closed formulas of L are valid in
@£ and
~o
As in the classical
case,
then they are models A
is a closed
~f(A)
if ~
of the same
formula of
and
~
are elementarily
313 theories.
But if
L, it is not possible
equivalent, ~f
to conclude
z
JB and that
= ~B ( A ) .
DEFINITION from
3.3.
l~f I
to
An embedding I~B I
tion 3.1 hold for all nonlogical al,...,a n
in
l~fl .
of
~f in
~
such that conditions symbols
f
is an injective
mapping
(i) and (ii) of Definiand
p
of
L
and all
165
D E F I N I T I O N 3,4. an e x t e n s i o n of m a p p i n g from
We say that ~f is a s u b s t r u c t u r e of ~
l~fl
when
to
I @fl is a subset of
I~ I
I~ I
is an e m b e d d i n g from
~ , or
~f in f;
ii)
p~f (a I ..... a n ) = p ~ (a I ..... a n ), for every
p.
In the conditions of the d e f i n i t i o n
T H E O R E M 3.2.
is
~ , that is:
f ~ (a I ..... a n ) = f ~ (a I ..... a n ), for every
are models of some
~
and the identity
i)
D E F I N I T I O N 3.5. ~B
,
3.4,
if ~f and
dl3 theory T, we say that ~f is a submodel of ~B .
Let ~f and
~B
I~I.
be structures
for L, and let ~ be a
mapping from
l~fl to
Then ~ is an e m b e d d i n g of ~,f in
~
if,
and only if,
~f(A) = ~ (A~), for every v a r i a b l e free formula
A
of
L(~). Let F be a set of formulas in the ture for set of
L.
~finstances of formulas in
D E F I N I T I O N 3.6. is a subset of ~f is a if
Let
l~B I,
~
and
implies
T H E O R E M 3.3. formula
A
L
~
~
in
be structures for L, such that
and that
~
is a
F,
~ VA
is in
F.
If ~
in
A
L
is a
L.
I~I
We say that
r  e x t e n s i o n of
JB (A) e Vd, for every formula
if, and only if,
~f be a struc
designates the
r.
Let F be a set of formulas
~(A) e V d
and let r(~f)
and let r be a set of formulas in
P  s u b s t r u c t u r e of
~[(A) ~ V d
then
]L31anguage
As in the classical model Theory,
in
~f
r(~) .
such that for every
r  s u b s t r u c t u r e of
~B (A) E V d for every formula
A
.~, in
r(~). PROOF.
If
~(A)
= 0, then
If r is a set of formulas b e l o n g s to
A
in
F,~A
F, it does not seem p o s s i b l e to obtain the result above.
or
if
~f (A) = 0
~B (A) = i
In
we can conclude only that
~ (A) = 0.
On the other hand, we can prove that and
0
such that for every formula
fact, u n d e r such conditions, (A) = ½
~ (~VA) = i.
implies
~(A)
= ½
implies
~ (A) = ½,
~f(A) = Io
It seems impossible also to prove the above result w h e n r is a set of formulas such that for e v e r y formula But,
if for every formula
A
in
A
F,~ A
in
F,
and
VA is in VA
r.
b e l o n g to
F,
then
we can prove a stronger result. T H E O R E M 3.4. formula of ~ ,
A then
Let P be a set of formulas in
in
F, I A
~(A)
and
= ~ (A)
VA
are in
F.
for every formula
L If A
such that for every ~f is a in
Psubstructure
r(~f).
166
PROOF.
By T h e o r e m
3.3,
as
sufficient to prove that A
in
implies
~ (A) = ~ , it is
~ (A) = I for every formula D
(I)
If r is the set of all open formulas
are structures for is a substructure of
and ~
is a
and ~ and
If
in
Furthermore,
and
~
L,
~f and
~ , then
~f
~f. L
and
~
is
are e l e m e n t a r i l y equivalent.
If F is the set of all formulas in an L 3  1 a n g u a g e
• , or that
T H E O R E M 3.6.
in
P  s u b s t r u c t u r e of
is an e x t e n s i o n of
~ , then ~
~
• , we say that
~
~f is an e l e m e n t a r y
is an e l e m e n t a r y e x t e n s i o n of
is a elementary substructure of
e l e m e n t a r i l y e q u i v a l e n t to A
•
F  s u b s t r u c t u r e of
substructure of
formula
is a
If F is the set of all formulas in
F  s u b s t r u c t u r e of D E F I N I T I O N 3.7.
L ~
(2)
L
= ~ implies
r(~ ) .
T H E O R E M 3.5.
a
~(A)
~f(A) = I
~
and
~(A)
= ~ (A)
~
~ , then
°
~f is
for every closed
L.
in the case of
L 3  1 a n g u a g e s , it is convenient to observe
that if ~f is a substructure of
~
and e l e m e n t a r i l y equivalent to
there can be closed formulas
in
L(~)
A
such that
~f(A) and
~ ,
• (A)
are d i s t i n g u i s h e d values, but different ones. T H E O R E M 3.7 ~f is a If ~
If
x = e
belongs
P  s u b s t r u c t u r e of
is a s t r u c t u r e for
to the set F of formulas in
~ , then
L
and
~f(e) = ~ (e).
L, we define the
F  d i a g r a m theory of
~f as
in the classical case° D E F I N I T I O N 3.8. The structure such that,
Let
~f
~,f and
for the
~
be structures for
~31anguage
L(~)
if £ is the name for an individual
L
with
~fc ~
.
is an e x p a n s i o n of a
of
I~I
then
(~f) (£) = a. D E F I N I T I O N 3.9.
The
F  d i a g r a m theory of
~ 3 theory w h o s e language is the formulas
A
LEMMA.
Then
a model of
~
:
~[ is a
DF(~f) , is the
and w h o s e n o n  l o g i c a l axioms are ~f(A)
belongs
to
V d.
Let F be a set of formulas in L
such that
F  s u b s t r u c t u r e of
~
l~fl
L, and
is a subset of
if, and only if,
~f
is
DF(~).
formula of the form A
L(~f)
such that
be structures for
D E F I N I T I O N 3.10.
formula
F(~)
(Diagram Lemma)
let ~f and I ~ I.
in
~f,
in
F,
A set F of formulas of x = y
or
x #
y
L
is in
every formula of the form
is a l m o s t  r e g u l a r F,
if every
and if for every
A [ x I ..... x n]
is in F.
167
THEOREM 3.8. L,
T
a
(Model Extension T h e o r e m I):
~3theory
formulas
in
L.
w i t h language
Then ~f has a
PROOF.
Suppose
that such a
is not valid in A 1 ..... An,
~f,
fore
~ (~*A~) = 0
what
is impossible,
On the other hand,
T
A 1 .... ,A~ i,
Vd,
for
from
There
= 0,
T.
T'
as new axioms.
of
1 ~ i ~ n.
~ (~*Aiv...v~*A~)
Hence
An
Let
T'
be the
L(~f)
as new
by adding all the non
We shall show that
T"
is non
then by the Reduction Theorem for n o n  T r i v i a l i z a t i o n
*Aiv...v~*A~,
and
~~*AlV..oV~*A ~ T'
~f(A~)
Hence by the Theorem on Constants, by replacing
~*AlV...v~*An
!
by adding all the names of obtained
such that
of
if,
and ~ * A l V . . . v ~
that the condition holds.
T"
T
If
F,
~finstances
is a model of
~
in
to
since
~
regularity
exists.
i, 1 ~ i ~ n.
there is a formula of type
from A~
~
formulas
for every
DF(~f)
If not,
Fextension
~f(A~) belongs
from
set of
~f.
then there are
and let be
logical axioms of
and F an a l m o s t  r e g u l a r
for
which is a disjunction of strong nega
A 1 ..... A n
suppose
obtained
constants; trivial.
with
such that
~3theory
T
in F is valid in
~T ~*AlV" " " v ~*An,
@,f be a structure
Fextension which is a model of
and only if, every theorem of tions of formulas
L
Let
with
belongs
AI, ' .. "'A'n to
Vd
LT~*AlV...v~*An,
the names by new variables
F, we have that is valid in
A 1 ..... A n
~f.
Therefore
~(A~)
for
r(~f),
i = 1 ..... n.
where A i results
Then
belong
in
by the almost
to F and
= 0
so
for some
i, w h i c h
is a contradiction. By the Completeness proof becomes
Theorem
identical
the r e s t r i c t i o n
~
T"
has a model
to the classical
of
~ '
to
L
~ '
one.
is a
and from here on the
That is, we prove that
Fextension of ~f, w h i c h is
a model of T. COROLLARY. A
D
Let F be an almost regular set of formulas
be a set of formulas
is a disjunction structure
for
Fextension PROOF. If
Let
A 1 ..... A n
Len~na where DA(~F),
~f~ B
L
C of F'
containing every formula
of strong negations and ~
~
is a
of formulas
be the set of formulas are in
F'
is a model of
and
~
F.
~ * A l V . . . v ~*An,
Dr,(~)
Dr(~f). So [email protected]
then by the D i a g r a m Lemma
~f(B)
If
in
B
~
let
where
A
is a
~f. L(~).
then by the Diagram
(B) c V d and As
and
then there is a
extension of
AE£ 1 ..... £n ]
is the closure of ~ * A l V . . . v ~ * A n .
L
VXl...VXnA , in
Aextension of ~ ,
which is an elementary
in
~f~S)
=
~'(B),
is an axiom of
is distinguished.
Hence,
168
by Closure Theorem Therefore DF,(~)
~*AlV...v~*A
there is a
is valid in
n
r'extension
6'
of
~
,
~,
which
and the proof follows as in the classical
is a model of
case.
If in the statement of T h e o r e m 3.8, we had "disjunction of negations of formulas
in
F", then we could not prove it by the same method.
Given an almost regular L(T),
set r of formulas
in
if we suppose that there is a model
~
tension of ~f
~
~AlV.
formulas i,
~, •
then we can not prove that .v~A n.
In fact,
A 1 ..... A n in
1 ~ i ~ n,
of the characteristics On the other hand, .~ ~ A l V . . . v ~ A n Theorem 3,8, if
such that
of
~ (A~) belongs
of the basic negation
~
of
By the observations
and if we construct
the theories
and
~ ~
~AlV...V~An,
we cannot conclude
with that
above,
almost regular DEFINITION
~
of
sets w i t h special
3.11.
DEFINITION
T'
and
A
~f(A') i
L
in F and
= 0 for some
theorem
we have to consider
A set F of formulas
is
in
of
and if for every formula
L A
L
if F is al
VA belongs
is in
of
Vregular
F,
Aregular F, AA
to
F.
if F is
belongs
to
r.
is regular if F is
Aregular.
THEOREM 3.9. L
In fact,
A I ..... A n
~3theories,
A set r of formulas of
3.13.
and
T", as in
characteristics°
3.12. A set F of formulas
DEFINITION
PROOF.
implies
in order to obtain a model extension
most regular and if for every formula
almost regular
for every on account
classical Keisler Model Extension T h e o r e m and which
uses the primitive n e g a t i o n
valid in ~ ,
Vd
= 0,
~3the°ries"
~T~AlV...V~An
F(~f),
w h i c h generalizes
language
to
~(~Aiv.o.wA ~)
that
rex
implies
Ai ..... A 'n of
~finstances
i,l~i~n,
theorem of
w h i c h is a
then we can not prove that T" is nontrivial•
in
Vregular
T
if we suppose that
~T ~ A i v ' ' ' V ~ A n
A i ..... An
F,
and a structure $ for
~T~AlV...V~An
if there are
we cannot conclude
L(T)
Let
~ be a structure
and let r be a T
Vregular
which is a disjunction then ~ has a
We construct
the
for
L, T
a
~3the°ry
set of formulas of L. of negations
with
If every
of formulas
in r is
Pextension which is a model of T. ~3theories
T' and
T"
like in the proof
of Theorem 3.8. If
T"
is trivial,
there are formulas
by the Reduction Theorem for nonTrivialization, A',.. ,A' I " n
in
r(~)
such that
~T~*AI v "
'
"
v~*A' n
169
and
~(A~) c V d,
3.8,
i = i .... ,n.
By the Theorem on Constants,
we can obtain formulas A I ..... A n such that
By hypothesis, some
~f~
i i = i,
~VAlv...v ~VA n . n, and
That is a contradiction,
A~
in
so
as in T h e o r e m
~ T ~ *AlV.v ~ *An"
Therefore,
we have
~(A~)
= 0 for
F(~)
T"
has a model
~ '
The proof then follows as in Theorem 3.8. THEOREM 3.10. language has a
L
Let
~f be a structure
and let F be a
rextension w h i c h
If
~(AA~) If
is a
(A~) and
~ ~A~)
~T~AlV...V~An
L, T
a
in L.
and only if, of formulas COROLLARY
belong
~
~
is a
COROLLARY guage L
2o
i, I ~ i s n .
for every
has a
containing
So,
i, 1 ~ i ~ n.
Hence
D
Theorem II). L
Let
T
which
for
set of
is a model of
T
if,
is a disjunction of negations
~.
every formula of
~ be a structure
and let F be a regular
Fextension w h i c h
of formulas
Fextension
Let ~
and
Therefore,
Let F be a regular set of formulas
is an elementary
P,
A i ..... A'n
~AlV...v~A n .
every theorem of
set of formulas and
Vd,
w i t h language
junction of negations which
for every
in
which is a model of T, we have that
to
in r is valid in
I.
~f
~f.
Winstances
which is a contradiction.
Then
If
then every theorem of T which
A I .... ,An
= 0,
with
L.
in F is valid in
then there are
(Model Extension
~3the°ry
formulas
of
with the formulas
of ~
implies ~
THEOREM 3.11.
~3the°ry
i, 1 ~ i ~ n .
Fextension
~ A i ~ . . . v ~ A ~) = 0,
a
~f,
~f~A~)
= i, for every
~
of formulas
~T~AlV'''v~An'
A 1 ..... A n such that
L, T
set of formulas
is a model of T,
~ A l V o. .v~A n is not valid in of
for
Aregular
is a disjunction of negations PROOF.
D
~,
in
ro
be a structure
L
VXl.ooVxnA , If
and let A be a where
A
~ is a structure
then there is a
extension of
in
Fextension
is a disfor
L
~ of
~f. for L.
and let F be an almost regular
T a ~ 3theory with lan
set of formulas
in L, such that
~A and VA belong to F for every formula A in F. Then, ~f has a Fextension which is a model of T if, and only if, every theorem of T which
is a disjunction of negations
of formulas
The Model Extension Theorem I (Theorem 3.8) ~3theories,
seems to be natural
in ~f . for
on account of the role played by ~* and the Reduction
Theorem for non T r i v i a l i z a t i o n The Extension
in F is valid
in ~ 3  t h e ° r i e s "
T h e o r e m II (Theorem 3.11)
seems to be more compatible
170
with the main characteristics for
of
~3theories,
besides more meaningful
~ 3 theories and general manyvalued theories and model logics.
But it seems not to be possible to obtain each one of the two versions from the other.
That is, it seems that Theorem 3.8, does not imply
Theorem 3.11, and viceversa. It is convenient to observe that if r is an almost regular set of formulas in F,
L
such that
~
A
and
AA belong to F for every formula A
in
then we can prove a result similar to Corollary 2 to Theorem 3.11,
though F can be not regular. We yet observe that, in the proofs of some of the following theorems, as the case of Lo~Tarski Theorem, in the Joint nonTrivialization
ChangLo~Suszko
Theorem for
Theorem and also
~ 3 theories, when we apply
one of the model extension theorems, we need only a sufficient condition to
Fextend a structure
~
for
L(T) to a model ~
of the
~3theory
T. In these cases, considering the special characteristics of the sets F , we can indistinctly apply Model Extension Theorem I, or Model Extension Theorem II. Now we study threevalued versions of Lo~Tarski Theorem and ChangLo~Suszko Theorem. THEOREM 3.12. to an open T
is a model of
PROOF.
(Lo~Tarski Theorem): A ~ 3  t h e ° r y
~3theory
If
T
T
is equivalent
if, and only if, every substructure of a model of
T.
is equivalent to an open
show that every substructure
~3the°ry
~X of a model
~
By Theorem 3.2, every open formula valid in
of ~
T', it suffices to T' is a model of T'
is valid in
~f , so
is a model of T'. On the other hand, we first prove that if F is the set of open formulas in T, if
F'
is the set of formulas in r which are theorems of
if every structure for
L(T) in which all the formulas of
is a model of
T
T, then
logical axioms are in theorem of
T
F.
is equivalent to a Then,
is valid in ~ ,
~3theory
F'
T, and are valid
whose non
it suffices to show that if every open then
@X is a model of T.
a regular set, by the Model Extension Theorem II, ~f has a
So, as F is Fextension
which is a model of T. DEFINITION 3.13.
D
A sequence
chain if for each n, ~ n + l
@~I' ~ 2 .... of structures
is an extension of
~n"
for
L
is a
171
DEFINITION
3.14.
Given a chain of structures
the chain is the structure verses n
of the
~n;
if
such that all of
~ whose universe
are individuals
V d, for every DEFINITION
3.16. n,
THEOREM 3.13.
A'
~
Lemma):
formula of A.
in
L(~fn),
If
A
is atomic,
If
A
is of type
hypothesis If
A
Let
A
%(VB)
~fn(A)
the result ~B
or
~fl' ~2'''"
~(A).
~xB.
L(~n) ;
is an elementary
extension
BVC,
if
the result
tables
of
= 0
~(~EB) = ½ ;
follows
H
that
and
A
chain, ~fn"
is a closed on the length
from the induction
H
~n(VB)
= 0
{f(B) = i
if, and only if,
or
~(B)
if, and only if,
= ½ , then
such that
~k(Bx[i])
of each
is obvious.
~f(%xB) k
of ~fn"
We use induction
= i if, and only if,
we choose
and we have that
=
VB, we observe
be of type
L(~),
If
~fl' ~f2 ..... such
extension
case, we must show that if
and the definition
for each i in in
then
is of type
~[(B) = 0;
chain is a chain
of the chain is an elementary
As in the classical
.... and belongs to
A, n = 1,2...
is an elementary
PROOF.
is an
..... ak).
of
An elementary ~n+l
(Tarski's
then the union
then there
~fn and we set
If ~ is the union of the chain ~ i , ~ 2 L, then A is valid in ~ if L ( A ' )
~ninstance
that for each
of
(a I ..... a k)
n
P ~ (a I ..... a n ) = p ~ n ( a l DEFINITION 3.15. is a formula of
L, the union of
a I ..... a k are in this union,
a I ..... a k
f ~ ( a I ..... a k) = f~f
A
for
is the union of the uni
k>n
if
= ~.
@£n(Bx[i])
~[(Bx[i])
= ½ for some i
and i is a name in
~f(@xB)
= I
= 0
we proceed
L(~k) , similar
ly.
D
THEOREM 3.14. equivalent
to a
(ChangLo~Suszko ~3theory
whose nonlogical
and only if, the union of every PROOF. T'
As in the classical
whose non logical
that the union T'.
Let
L ( ~ k ).
~f of a chain
~Xl...~xnA
For large enough Hence
axioms
k,
Theorem):
A
axioms
chain of models
case,
if
T
~3theory of
be an ~finstance
~(qXl...~XnA)
cVd
and
to a
of T'
of a non logical
~fk(A[il ..... £n ]) e V d for some
if,
is a model of T. ~3The°ry
then if suffices
9,fl, ~2 .... of models
is
are existential T
is equivalent
are existential,
T
axiom of T'
i I ..... i n
~f is a model of
to prove
is a model of
T'.
in
172
Now suppose that every union of a chain of models of T.
It also suffices to show that if ~
T
is a model of
is a structure in which every
existential theorem of T is valid, then ~f is a model of T. as in the classical case by constructing a chain ~[I = ~' ~ n of
~f2n+l"
If
~
valent,
is a model of
T
and
~f2n+3
is the union of the chain, as then ~[ is a model of
~ and
~fl' ~f2 ....
We proceed such that
is an elementary extension
•
are elementarily equi
T.
The second part of the proof is possible because the set of all universal formulas of
L(T)
is regular, in the sense of Definition 3.13. REFERENCES
[ I] ARRUDA, A.I.
A survey of paraconsistent logic, Mathematical Logic in Latin America, NorthHolland, Amsterdam, (1980), pp. 141.
[ 2] ARRUDA, A.I.
Aspects of the historical development of paraconsistent logic (To appear)
[ 3] BORKOWSKI,L.
(ed.) Selected Works of J. Lukasiewicz, NorthHolland, Amsterdam (1970).
[ 4 ] D' OTTAVIANO, I.M.L. & da Costa,N.C.A.
Sur un probl~me de Ja~kowski, C.R. Acad.Sc.Paris,
[ 5] D ' O T T A V I ~ O , I.M.L.
The completeness and compactness of a threevalued
270A (1970), pp. 13491343.
firstorder logic.
To appear in Mathematical Logic,
V Latinamerican Symposium on Mathematical Logic, MarcelDekker. [ 6]
Sobre uma teoria de modelos trivalente (Thesis) Universidade Estadual de Campinas, Campinas
[ 7] ~.UKASIEWICZ,J.
On the principle of contradiction in Aristotle, Review of Metaphysics
[
8]
(1982).
XXIV (1971), pp. 485509.
Philosophische Bermerkungen zu mehrwertigen systemen des Aussagenkalk~lls,
C.R.Soc.Sci.Lett.
Varsovie 23, (1930), pp. 5157 (Translation to English in [3] , pp. 153178).
173
[ 9] ~UKASIEWICZ,J. & Tarski, A.
Untersuchungen Nber den Aussagenkalk~ll, C.RoSoc. Sci.Lett.Varsovie 23 (1930) pp. 3950 (Translation to English in [3] , pp 131152).
[I0] RASIOWA,H.
An algebraic approach to nonclassical logics, North Holland, Amsterdam (1974).
[II] RESCHER, N.
Manyvalued logics,
McGrawHill, N.York (1969).
[12] ROSSER, J.B. & Turquette,A.
Manyvalued logics, (1952).
NorthHolland, Amsterdam
[13] SHOENFIELD,J.R. Mathematical Logic. (1967).
Addison Wesley, Reading
COMPLETENESS THEOREMS FOR THE GENERAL THEORY OF STOCHASTIC PROCESSES Sergio Faj ardo ~* University of Wisconsin Department of Mathematics Madison, Wisconsin 53706 U.S.A.
§0.
INTRODUCTION. Adapted Probability Logic, denoted Lad, is a logic adequate for
the study of stochastic processes.
In this paper we show how to axio
matize in Lad the basic notions of the general theory of stochastic processes
(i.e. stopping time, martingale,
adapted, progressively mea
surable, optional and predictable stochastic processes) and prove a "completeness" theorem for each such notion. The study of stochastic processes has been for the last forty years one of the main topics of research in probability theory. The development of the theory of martingales and Markov processes has brought many new ideas into this field. Concepts such as filtration of oalgebras and stopping times are now of fundamental importance after the work of Levy, Doob, Ito, Chung, Hunt and others. more advanced tools has appeared.
Along with these ideas the need for The "General theory of Processes" is
the branch of probability that has as its subject of study all these new objects.
Among the many concepts studied by "general theorists"
(besides the basic notions mentioned above) are: stopping times of different types, local martingales,
semimartingales,
generalized stochas
tic integrals, stochastic differential equations, etc.
Good sources of
information about these topics are [D], [DM!] , [DM2] , IMP] and the Lecture Notes in Mathematics published regularly by the members of the Strasbourg school of probability
~*
Current Address
:
Department of Mathematics University of Colorado Campus Box 426 Boulder, Colorado 80309
(U.S.A.)
175
After the introduction of the Loeb measure in [Lo] and Anderson's nonstandard construction of Brownian motion [An],
nonstandard analysts
have developed a new approach for the study of stochastic integration (see [KI], [Pa], [Li], [HP], [Pe]) ([KI] , [HP], [C])
and stochastic integral equations
that is more intuitive and seems to be very well
suited for direct applications. What does logic have to do here?
Hoover and Keisler realized that many
of the concepts in [K I] could be naturally expressed in a language with integral quantifiers and conditional expectation operators, and that some of the properties of the hyperfinite adapted spaces introduced in that paper were general model theoretic theorems in the logic constructed with this language.
Thus, adapted probability logic was born.
logic first appeared in [K2].
This
Later Rodenhausen in his thesis [R] proved,
among other things, a completeness theorem for Lad. Hoover and Keisler in [HK] introduced different notions of elementary equivalence for Lad and using a model theoretic approach gave direct applications to probability theory. Keisler in [K 3] tied
Lad with the two previously known probability
logics L A P and LAf (see [K3] , [K4] , [HI] , [H2] , [H3] , [H4]) by introducing a new logic, the so called Probability Logic with Conditional Expectation of LAE
LAE.
Lad was then reintroduced as a two sorted form
and in this way Keisler in [K3] , [K 5] gave a new (simpler) proof
of Rodenhausen's completeness theorem for Lad. Keisler [K 3] is an uptodate account of the development of the different probability logics including
Lad.
For more on
LAE
see Fajardo [F].
A description of the contents of this paper is as follows
: Section 1
contains an introduction to adapted probability logic that "almost" does not require a previous knowledge of the other probability logics, but Keisler's survey paper [K 3] is strongly recommended.
In section 2
we show how to axiomatize the concept of stopping time and prove our first completeness theorem.
In section 3 we give a set of axioms and
prove a corresponding completeness theorem that simultaneously covers adapted, progressively measurable and optional stochastic processes. We also give a different set of axioms for predictable processes and prove its completeness.
Finally we present a completeness theorem for martin
gales. The reader is assumed to be familiar with the basic concepts of probability theory such as random variable and conditional expectation which can be found in any elementary textbook in probability
(for example
176
[As]). We do not expect readers to know the general theory of processes, so all the basic definitions are presented. Elliot [El is a good introductory book covering the concepts studied here.
We do not use nonstan
dard analysis and only require some familiarity with basic model theory. §i.
DEFINITIONS AND BACKGROUND: We introduce adapted probability Logic
original presentation in [K2].
Lad
following Keisler's
An alternative way of defining
Lad as
a twosorted form of logic with conditional expectation LAE was given recently in Keisler [K3]. We follow the former approach since it allows us to go directly into the basics of
Lad
without assuming a previous
knowledge about the other probability logics.
Basic notions from adapt
ed probability logic such as adapted model, interpretation and adapted elementary equivalence axioms for
Lad
(weak and strong) together with a complete set of
are given.
We also present all the basic definitons
from the general theory of processes that are relevant to our exposition. Those readers familiar with Keisler [K 3] will realize that we do not worry about admissible fragments LAa d of Lad. Instead, we just work with A = HC (the hereditarily countable sets), but all results generalize in the obvious way to fragments with A c HC and ~ s A. Throughout this paper we assume that socalled "usual conditions" of a filtration of oalgebras (see definition I.I below). These are, as the name suggests, the kind of assumptions that probabilists working with the type of structures that we are going to be dealing with usually make.
A
theory of processes that does not assume these conditions has been partially developed but it is at a very early stage and it does not yet have important applications.
For more on this subject see [DMI].
DEFINITION i.I (a)
An Adapted Probability Space or Stochastic Base is a structure = (~,(Ft)t~[O,l],P) where: (i) (ii)
P
is a complete probability measure on ~.
(Ft)t~[0,1]is an increasing filtration of
oalgebras of
Pmeasurable sets satisfying the Usual Conditions, (I)
F0 is Complete: in
(2)
F 0 and
(Ft)t~[0,1]is Right Continuous Ft
i.e.:
all the Pnull sets are contained
=
n F s>t S"
: For all t c [0,I]
177
(b)
Let
M
be a Polish
on ~ if
is measurable
X(t)
X
Let
instead of X
and
Y
X(w,t)
(i)
(ii)
processes from
= Y(w,t)
(i)
X(w,t) of
definition
Y
Y
if for almost all
if for all
on
ws~
for
t s [0,i]:
of adapted probability
Symbols
of
logic.
Lad are:
Countably
(2)
One Probabilistic
many time variables:
First Order Function each
(iv)
Intesral
i ~)
listic variable
and from
quantifier
Conditional
The NonLogical
connectives:
connectives
ne~
functions (iii)
variable:
tl,t2,...,Sl,S2,...
w
Connectives: (2)
~n
:
the function
into
symbol:
Expectation
Sy~ols
~ ,A •
of
for
~.
f
operator
Lad
w i t h two arguments,
symbol:
E[I].
are stochastic
the first argument
and the second for time variables.
seen to w o r k
symbol F
F ~ C ( ~ n) = the set of continuous
carried out using just one stochastic for countably
process
process
symbols
for the probabi
Most proofs
symbol
many stochastic
are
X, but everything processes.
1.3.
The set of (i)
on ~ taking values
a.s.
(i)
(I)
(a)
we write
Variables:
(ii)
DEFINITION
Sometimes
1.2.
The Logical
is easily
to the product measure
Then we say:
t s [0,I]:
= Y(t)
space).
Process
X = (X(t))ts[0,1 ].
be stochastic
X is a Modification
DEFINITION
(Xi(...):
and
metric
Stochastic
on [0,I].
all
Now we give the formal
(b)
with respect
X is Indistinguishable
X(t)
separable
is an Mvalued
~ is the Borel measure
the same space M.
(a)
(i.e. complete
X:~ × [0,I] ÷ M
P x B where
(c)
space
A function
Ladterms
For each
is defined
r ~ 4 + , [X(w,t)Fr]
inductively
as follows:
is an atomic
term
(see remark
1.9). (ii) (iii) (iv)
(v)
Each time variable Each real
ra~
If T is a term, If
T 1 ..... T n
t
is a term.
is a term. so are are terms
fTdw, and
fTdt
and
F E C ( ~ n)
E[TIs] then
(w,t).
F(T 1 ..... ~n )
178
is a term. (b)
Free and Bound variables the integral
/~dx
in a term are defined
binding
x, in
of s are bound and w and t are free, first occurrence we shorten
of s is bound
this expression
to
as usual, with
E[~Is](w,t) and in
E[~Is](w,s)
and the second E[TIs]
or
all occurrences is free.
E[Tls](s).
the Sometimes
A closed
term is a term with no free variables. (c)
The set of
LadFormulas
(i)
For each Ladterm
(ii)
If ~ is a formula
(iii)
is the least set such that ~,[Te0] so is
If ~ is a countable variables
then
is an Atomic
formula
~ ~.
set of formulas
A ~ is a formula.
with finitely many A Sentence
free
is a formu
la with no free variables. (d)
There
is one natural way of extending
la given in (c). r c ~n[0,1] guageo
We can add probability
and allow formulas
does not increase
of [K 3] for a rigorous things
tifiers
smoother.
in situations
quantifiers
of the form ( P ~ r )
As for the Logic with Integral
addition makes
the definition
the power of
(PEer) with
~ in our lan
Quantifier Lad
proof of this fact) their presence
L w f, this i (see Section 3.6
but sometimes
We will not hesitate where
of Ladformu
it
to use these quan
helps
the reader
to
have a better understanding. Remark
3°9
suggests
other possible
extension
of the definition
of
Ladformula. DEFINITION
1.4.
An Adapted Probability A = (~,(Ft) t~[O,I],X,P) process
Structure
for Lad is a structure
= (~,X) where X is a real valued
on the adapted probability
Before we can give the definition Lad in a model,
(Model)
stochastic
space ~. of the notion of interpretation
we need to introduce
the main concepts
of
of the general
theory of processes° DEFINITION
1o5.
Let ~ = (~,(Ft)ts[0,1],P) X a real valued (a)
stochastic
be a fixed adapted probability process
X is Right Continuous all
we~,
space and
on ~.
w i t h Left Limits
the path X(w,):[0,1]
÷~
(r.c.l.l.)
if for almost
is right continuous
with
left limits. (b)
X is Adapted with respect
to (Ft)t~[0,1 ] if for all t~[0,1]
the
179
random variable X(t) is Ftmeasurable. (c)
X is a Martingale
(d)
if the following two conditions
(I)
For each t, X t is integrable
(2)
For all
s ~ t, E(XtlF s) = X s
X is Progressively
the restriction of
measurable,
where
[0,t].
A !
function
B([0,t])
~ x[0,1]
tic process.
(e)
X
to ~ x[0,t]
is the
is F t xB([0,t])
oalgebra of Borel sets in
is said to be a Progressive
IA(W,t)
is a progressively
The Pr0gressive
(Ft)ts[0,1 ] is the progressive
a.s°
Measurable with respect to (Ft)tg[0,1 ] if for
all ts[0,1]
dicator
hold:
~algebra
oalgebra on
set if the in
measurable
M associated
stochas
to
~ x [0,i] generated by the
sets.
A random variable
S:~ ÷ [0,I] is a S t o ~ w i t h
respect
to (Ft)ts[0,1 ] if for all t s [0,i] {ws~:S(w)< t} ~ F t . ing times are sometimes called Optional Times) o (f)
The
oalgebra
sets to
B
0 on
~ x [0,i] generated by the
such that for each
F t and
Bt =
n
Bs
respect (g)
The
Predictable
P on
A x Is,t] with
Galgebra with respecto
As u Fr is called the r<s to (Ft)te[0,1 ].
if it is measurable with respect
In the probability
literature
the Optional
and Predictable
ing sections
in [D],
oalgebra
to P.
there are other ways of defining
oalgebras.
[DM I] and IMP].
See for example the correspond
The definition of Optional
that we have given is taken from
[K 3] and we think it is easier
to understand
for the reader not familiar with stochastic
The following
theorem gives an equivalent way of introducing
nal and Predictable them.
with
...................
~ x [0,i] generated by the sets of the form
AsF 0 and
X is Predictable
NOTE:
~
belongs
oalgebra
if it is measurable with respect to 0.
oalgebra
A x {0} with
(i)
t g [0,1]B t = {w:(w,t)sB}
to (Ft)ts[0,1 ]
X is Optional
(h)
p x ~measurable
is called the Optional
s>t
(Stopp
oalgebras
A proof can be found in
and establishes
processes. the Optio
the relationship
between
[DMI].
THEOREM 1.6 Let (a)
~ = (~,(Ft)te[0,1],P) The Predictable
be an adapted probability
(Optional)
oalgebra on
by the real valued continuous stochastic processes
on
~.
space,
then:
~ x [0,i] is generated
(right continuous)
(Ft)adapted
180
(b)
P ~ 0 ! M. (These inclusions are usually strict).
DEFINITION
1.7.
(Keisler [K3]):
An Interpretation
of Lad i n a n L a d model A is a function
that assigns to each Ladterm T(w,t) a P × Bn measurable (~n is the Borel measure on [0,I] n) T A such that:
[r (i)
If T is [X(w,t)~r]
then
if if
TA(w,t)=~r
~ X(w,t) (ii) (iii)
If T is
(v)
/edt
then
LEMMA Lad.
otherwise
For each term
T A = /eAdB.
(~,w,s) the following conditions hold:
(E[T(~,w,s) Is](w,a)) A
is
0 × Bnmeasurable,
Borel oalgebra on [0,I] n, n the Optional oalgebra. (b)
X(w,t) e r X(w,t) <  r
If T is a time variable then TA(a) = a(as[0,1]). If T is /@dw then T A = /TAdp.
(iv)
(a)
function
For each ntuple
b
where B n is the
is the length of
~
and 0 is
we have:
i
(E[T (~,w, s)] s] (w,a)) A = E [ r A (~b , ' , ' ) 1 0 ] ( w , a )
2
For all as[0,1], (E[~(~,w,s) Is](w,a)) A = E[TA(b,,,a) IFa](W)(P a.s.)
1.8.
(Keisler
For each term
tions agree at
[K3]): T(w,~)
T(w,~)
~
in
for Palmost all w.
B a.s°
[0,I], any two interpretaIn particular, in
A
if
agree at
w
is T(a)
We can see from the definition of interpretation of
in a model that the interpretation process.
×
Every model A has an interpretation of and all
not free in T(~) then any two interpretations for all ~ in [0,i]. REMA~( 1.9.
P
Lad
of each term is a bounded stochastic
The process X may be unbounded but the atomic terms [X(w,t)Fr]
truncate X at r and consequently all the terms, built from the atomic terms, are bounded. DEFINITION i.i0. (i)
(Keisler [K3]):
The Logic Lad has the following axiom schemes: (0)
Axiom schemes for twosorted
(I)
Axioms for the conditional a
LAf(see [K3])
e~pectation operator:
E[T(u,~)Iu](w,s ) = ~[T(v,t) Iv](w,s ) v do not occur in t.
b  ~E[~ I t](w,t)
where
u
and
 Tdw = /E[~It](w,t ) .E[TIt](w,t)dw
181
(2)
Axioms
for the filtration
a  fF(t)dt = r. flF(x)dx 0~
For any
and stochastic F:[0,1] ÷ ~
processes:
continuous
with
= r
b 
s
÷E[TIs]
= E[E[TIs]It].
C 
A V //flT(W,S) m n
 ~(w,t) l"
max(0,11stl'n)dsdtdw
1
<
m°n
Lad has the following
(ii)
a 
Modus ponens:
b 
Conjunction:
c 
Generalization:
l~÷[T(w,s,t)
Rules
of Inference:
~,~ ~ ~ ~ {~ ÷ ~I~ E F} ~ ~ ÷ A
e 0]
I~÷f~(w,s,t)dwe
0
where w is not free in ~. 2~÷[T(w,s,t) ~ 0] l~÷fT(w,s,t)ds e 0 where THEOREM
Ioli.
!Rpd~enhausen's of
s is not free in ~.
c o m p l e t e n e s s a n d soundness, for
LadSentences
PROOF.
See
To conclude
[K 3]
has a model
ed in [K3].
two notions
They were introduced context
Here we just present
and their
of elementary
by Hoover
importance
the definitions
equivalen
and Keisler for
in
Lad is explain
and some properties
proofs.
DEFINITION
1.12.
(Hoover
and Keisler
[HK])
Let A and B be models. W
(a)
A is said to be weakly LadEquivalent if for every LadSentence
to
B, denoted by
A ~ B,
0:A ~ @ if and only if B ~ @. S
(b)
A is strongly each free:
THEOREM
1.13
~
in
LadEquivalent [0,i]
A ~[~] (Hoover
W
(i)
A ~ B
to
and formula
B, denoted by ~(~)
A ~ B, if for
of Lad in which w is not
if and only if B ~ [ ~ ] . and Keisler
Let A and B be models.
set
B
this section we present
in a probabilistic
without
A countable
and [K 5]
ce for adapted models. [HK]
Lad ) .
if and only if ~ is consistent with Lad.
[HK])
The following
are equivalent:
182
(ii)
~±~ere is a set ~
each
(iii)
(b)
If
~
B
A
=>
X
and
s
A ~

If
A ~[~]
if and only if
[HK~:
Y
of
B if
B
are stochastic processes X
then
(~,X)
and only if
over time variables (d)
~(~):
W
fication (c)
and formula
(Hoover and Keisler
S
A
T ~ [0,i] of measure one such that for
T
For each L ~term T(~) with no integrals over time variables • A 'ta) ~ = T B ta) '~ for almost all ~ in [0,I]
THEOREM I. 14. ~a)
in
for
Y
is a modi
5 (~,Y).
each
and all
Ladterm T(~) with no integrals in [0,I], TA(%) = TB(~).
a
S
and
T c [0,I] has measure one 
and
on
s
A ~ B
with time parameters
W
restricted
to
T
§ 2. A COMPLETENESS
then
A ~ B. "Just as the seemingly
THEOREM FOR STOPPING TIMES.
trivial definition of the derivative
contains
in germ all of the calculus,
and it its discovery may have involved as much genius as the whole development
that followed it, the seemingly
(due to Doob)
is the cornerstone
~ith this statement,
Dellacherie
te the study of stopping times it seems appropriate
trivia± notion of stopping time
of the "General Theory of Processes" and Meyer in [DM I]
(sometimes
called optional
some few lines to this basic concept. and it immediately
suggests
stochastic processes
a natural
on
~ × [0,i].
This is precisely
times take into account
bability
space.
set of axioms.
is very simple
We first have to
defined on ~ and not
We do not work with
L~I,
X(w,t)
the because
the time structure of the adapted pro
Instead of artificially
using a stochastic process
and giving an axiom that characterizes
processes
that are time independent
primitive
symbol the random variable
Ladterms
(definition
in place of
the content of
logic designed for the study of random variables,
stopping symbol
Thus
logic, we dedicate
The definition of stopping time (def lo5.d)
observe that stopping times are random variables probability
times).
that in this paper where we intend to study the
general theory of processes using adapted probability this section.
"
(page 115) initia
those stochastic
(i.e. random variables), symbol
S(w).
we take as
The definition of
1.3) is modified in the obvious way with [S(w)~r]
[X(w,t)~r].
All the results
valid with this modification.
stated in section
Let us first introduce
that we can follow the usual probabilistic
practice.
i remain
some notation so
183
NOTATION. (a)
From now on we are going to shorten the expression [S(w)~r]
(b)
Given
to
Xr(t)
Ladterms
formula
and
~ and ~ we write
719  ~Idw
7(4  min(~,~))dw
~ 0
[X(w,t)rr]
and
S r respectively.
and
~ = ~ a.s. instead of the Lad
~ ~ ~ a.s.
instead of
~ 0.
DEFINITION 2.1. A model A = (~,S) = (~,S,(Ft)te[0,1],P) is said to be a Stopping Time model if S is a stopping time with respect to (Ft)te[0,1]" DEFINITION 2.2.
Let
ST
be the following set of axioms in
(ST1)
Axioms for
(ST2)
A + (0 ~ S r ~ i a.s.) rs~
(ST3)
A + (Pt e l)(E[min(Sr,t)It](t) re~ (Intuitively, is
Lad:
Lad (see definiton 1.9)
for almost all
= min(Sr,t)a.s.)
t
the random variable
Sr A t
Ftmeasurable).
THEOREM 2.3
(Soundness)
Let
A = (~,S) be a stopping time model then
A ~ ST. PROOF.
It follows trivially from the definition of stopping time.
THEOREM 2.4. Lad. PROOF.
(ComRleteness)
Let ~ be a countable set of sentences
in
If ~ uST is LadConsistent , then ~ has a stopping time model. If we go back to Keisler's proof of the completeness
theorem
for Lad(See [K 3] and [K5]), it is easy to see that by adjoining axioms (ST 2) and (ST 3) to (ST I) we can get a model A = (~,S) of ~ having the properties written in parentheses.
This model is
"almost" the
one that we want but we have to make sure that these properties hold everywhere and not only almost surely. (ST 2) (I)
we can find For all
weU
and
If we recall that for all function S': ~ ÷ 0,i
We proceed as follows:
From
U i ~ of measure one such that all
r ~ +, Sr(W ) e [0,i].
wen
S(w)= lim Sr(W) r÷~ defined by
S'(w) = I S(w) 1
if
w ~U
if
w ~U
then using
(I) the
184
has the following
properties:
(i)
S' is measurable
(the probability
space is complete)
(2) (ii) CLAIM.
S' = S a.s.
S' is a stopping
order to prove random variable exists (3)
S'A t
t sT,SA
We can strengthen For all
This
the function
is a model of
Ftmeasurab!e.
~:
ts[0,!]
In the
By (ST 3) we know that there
one such that:
t is Ftmeasurable.
t c [0,I],
SA 0
te [O,I]\T.
A' = (~,S')
(3) to
is easy to see:
find
is
T c [0,I] of measure For all
(4)
time and
this we just have to check that for all
SAt
is Ftmeasurable.
First observe is trivially
Since
that we can assume
Fomeasurable.
[O,I]\T has measure
zero,
0 c T, since
Now let's
for each
consider
n ~
we can
m n e T such that (i)
(5)
for each
(ii)
For each
ws~
lim n+~
nc ~ m n < m n + I < t
and
mn = t
we have
S ( w ) A t = lim S ( w ) A m n.
By (3)
we know that
n+~
for each n the function
S A m n is
that for each n,
is
Ftmeasurable Finally, usual of S. A'
SAm n
as we wanted
§3.
Ftmeasurable ;
we can conclude
This proves
that
COMPLETENESS
predicctable
process
this question;
T on
processes.
predictable
by optional theorems
therefore
SAt
equivalent
holds with
time and by
to and so
(2.ii.)
Similar
we prove questions
or progressively
can be proved.
the model
A' b ~.
PROCESSES.
In this section
the
S' in place D Given a theory
have an adapted model A = (~,X) with ~?
is also
(Ft)te[0,1 ] satisfies
(4)
S' is a stopping
in other words,
predictable completeness
that
T H E O R E M FOR STOCHASTIC
Lad, when does
and by (5.i) we have
to show.
given the fact that the filtration
conditions,
is weaklyelementarily
T in
Fmnmeasurable
(theorem
3.12)we
a completeness
X
a
answer
theorem for
can be asked by replacing measurable,
and corresponding
The main difficulty
in finding
185
correct axiomatization notions
of these notions arises from the fact that the
of predictable,
optional
in terms of measurability
and progressive
conditions with respect
product space ~ x [0,i] but the conditional only
"talks"
about
oalgebras
no "obvious" ways of axiomatizing from Probability
processes to
oalgebras
expectation
defined on
~.
these notions
operator
Therefore in
are given
Lad.
gales. the
for Lad
there are Using tools
theory and the model theory of adapted probability
logic we can find the correct axioms and prove the completeness To conclude
on the
this section we present a completeness
In handling
"obvious"
this notion it is, in principle,
axioms but we immediately
tifier over time variables about this problem);
is needed.
nevertheless,
realize
theorems.
theorem for martineasy to write down
that a universal
quan
(See remark 3.9 for some comments
we are able to avoid this obstacle
by making use of a simple probabilistic
construction.
Here are the
details° DEFINITION
3°1o
structure
and
(M,B(M))
where
Generated
in
completion
X
(Skorokhod
Let
a stochastic process B(M)
is the Borel
(M,B(M))
under
[Ski):
P
if
~
be an adapted probability
taking values in a Polish space
oalgebra on M.
o({X(t):t E [0,i]})
of a countably
Now we state the main results
generated
from probability
S is Countably
is contained
in the
oalgebra. theory that we use in
this section. THEOREM
3.2.
generated (a)
(Skorokhod
Mvalued
X has a predictable predictable
(b)
For all
THEOREM 3.3.
[Ski)
stochastic
~
X = (X(t)) t [0,i] be a countably
modification
The following are equivalent: (i.e. there exists
Y
on
such that for all t, Y(t) = X(t)a.s.)
tE[0,1],
X(t) is Ftl measurable,
(Dellacherie
and Mexer
adapted stochastic process on Y on
Let
process.
such that for all
~
[DM2]):
where
Ft_ = o( u Fs). s
If X is a measurable
then there exists an optional process
t~[0,1],
X(t) = Y(t) a.s.
(i.e. X has an
optional modification). REMARK
3.4.
generated
The obvious question is:
stochastic processes?
gave an argument nerated.
are there enough countably
In an earlier version of this paper we
showing that the optional processes
are countably
The referee has pointed out that it is easy to prove that
any measurable
process
is countably generated.
The importance of Skorkhod's
theorem
3.2
for us is that it gives a
ge
186
a characterization of
"predictable modification"
rability condition with respect to the
in terms of a measu
oalgebras
(Ft~)ts[O,l ].
The following theorem is going to show us a natural way of expressing this condition in the language of adapted probability logic. THEOREM
3.5.
Let
structure and (i.e. for all
~
= (~, (Ft)t~[O,l]P)
X = (X(t))ts[0,1 ] an integrable stochastic process on te[O,l], / X(w,t)dP < ~). The following are equivalent:
(a)
X(t)
(b)
~ ~ plnlE(X(t) IFt_I/p)
PROOF.
be an adapted probability
is
Ftmeasurable X(t) I < I/m
a.s.
Condition (b) is just a "suggestive" way of writing:
(I)
E[X(t) IFt_I/n] n~X(t)
For each
a.s.
n e ~ let: I
FtI/n
if
L
F0 otherwise
t e i/nan d X(n) = E[X(t) IF(n)]
F(n)=
Clearly, the sequence (X(n))ns ~ is a discrete martingale with respect to the increasing family of oalgebras (F(n)) n c~" Since X(t) is integrable, we can apply a well known martingale convergence theorem due to Levy (see for example [DM 2] Thm. 31, page 28). With our notation the theorem says: (3)
X(n ) = EEX(t)]F(n )3 n÷ E[X(t) l O ( n ~ F ( n )]
a.s.
Using this fact our theorem can be easily proved as we now indicate. Suppose
X(t)
is
E[X(t) IFt_] = X(t)
Ft_measurable. a.s.
with
This by definition, means:
Ft_ = O(s~tF s) = ~( u n ~
F(n)),
therefore by (3) and (a) we have X(n ) = E[X(t) IF(n )] ~ X(t) Suppose (b) holds, then E[X(t) IFt] = X(t) a.s.
a s
and this is
X(n ) ÷ X(t ) a.s.
(b) (i e. (I))
and by (3) we can conclude D
NOTE° Observe the importance of the integrability condition on the stochastic process X = (X(t))te[0,1 ] in the previous theorem, in particular, if X is bounded then it is integrable.
187
DEFINITION 3.6.
A model
A = (~,X) is said to be a Predictable Model
if the stochastic process X is predictable with respect to (Ft)tg[0,1 ]. Similarly we can define adapted, progressively measurable and optional models.
By Theorem 1.6, every predictable model is optional, every
optional model is progressively measurable and every progressively measurable model is adapted. LEMMA 3.7. ture.
(a)
If
Let
~ = (~,(Ft)t~[0,1],P)
be an adapted probability struc
A = (~,X) is an adapted model then there exists an optional
stochastic process
Y
on
~
such that the optional model B =
(~,Y)
is strongly Ladequivalent to A. (b)
If X(t) is
Ft_measurable then
X(t) is
Ftmeasurable.
PROOF. (a)
Recall that in our definition of stochastic process only deal with measurable stochastic processes°
(def l.b) we
Then if
adapted, Theorem 3.3 provides an optional modification s B = (~,Y) ~ A = (~,X).
X Y
is of
X
and Thm 1.14ob shows that (b)
Trivial.
By definition,
F t_ ! F t
We can now present the first completeness theorem.
It basically says:
"If X is adapted this is enough in order to get an optional model". At first sight it may seem strange that from an adapted model we can go to an optional model (compare the definitions), but the force behind this is Theorem 3.3 THEOREM 3.8.
which is a nontrivial probabilistic result. (Soundness and Completeness for Adapted, Progressively Measurable and Optional Processes):
Let
T
set of sentences in Lad. (a)
If T has an adapted model then T U{rAE~+(Pt ~ l) (E[Xr(t) It] (t ) = Xr(t ) a.s.) is consistent in Lad°
(b)
If T u{ A r~+
(et > l)(E[Xr(t)It](t ) = Xr(t ) a.s ) 
is consistent in
Lad
then T has an Optional model.
PROOF. (a)
Immediate from the definition of adapted model.
be a countable
188
(b)
If we add the axiom T u { A + (Pt ~ l) (E[Xr(t) It] (t) = Xr(t) re~
to the list of axioms completeness
for
Lad
theorem for
(definition
For each From
(2) Let
r s2 + ,
r
2+
A =
for almost
(i) we can find
For each
(~,X) all
CLAIM.
X'(w,t)
for each
t
U, Xr(t)
X'
A' = (~,X')
is adapted
rable and this property By the definition for each
lim Xr(W,t)
Then by
t c [0,i]
=
lim
by proposition
lemma 3.7.a
is
Ftmeasurable. If
tc[0,1]
is
(Xr(t))tg[0,1 ] we know that
and almost all
1.13.d.
=
w s ~:
lim Xr(w,t ) = X(w,t)
r÷~ w A' ~ A and so modification
A' ~ T.
Finally by
X ~ of
In the above proof we did not immediately
model of T due to the fact that with the integral make sure that a given property
holds
necessarily
A research
[K 3] is to develop universal
the argument perties
t ~ [0,i].
X'
of the previous
of stochastic
naturally
expressed
sec. 4.4 in [K3]).
logic with
theorem
processes
we can only
te[0,1]
but not
in Keisler
the addition
of the
If this were done,
could be simplified
that hold for "all times"
in this extension
is an
get an adapted
problem proposed
ranging over time variables.
and the A"
quantifier
for almost all
adapted probability
quantifier
Ftmeasu
:
we can find an optional
for all
ws
to the limit.
model A" = (~,X") is such that A" ~ A' and consequently optional model of T as we want to show. REMARK 3.9°
and
and 0 other
It is easy to
r s ~+ E[Xr(t) IF t]
by passing
process
E[Xr(t) IFt](w )
r÷~ Therefore
one such that:
= X(w,t).
(2) and (3) for all t s U
X'(w,t)
Ftmeasurable.
if the limit exists
for each
of the truncated
w e ~ and each
(3)
since
is
is an adapted model of T:
is preserved
ex
such that: Xr(t)
as follows:
= lim E[Xr(t) IFt](w) r~
The model
see that
t~[0,1],
proof of the
can be naturally
of T,
U c [0,i] of measure
X' = (X'(t))ts[0,1 ] be defined
then let wise.
1.9), Keisler's
Lad (see [K 3] and [K5])
tended in order to get a model (i)
a.s.)
and many procould be
of adapted probability
logic
(see
189
DEFINITION (PI)
3.10.
Axioms
Let
for Lad
P
be the following
(see definition
set of axioms
in Lad.
1.9)
(P2) (et~l)( A rs~ +
A ms~
V nsiN
A (Pu~l)(tI/p
(intuitively, for almost all t, ~(t) THEOREM 3o11.
(Soundness
of P)
lu](u)Xr(t)
is F t_measurable).
The axioms
of P are true in every pre
dictable model. PROOF.
It follows
from the soundness
theorem for Lad and theorem
3.2
and 3.4. We are now ready for our main completeness THEOREM 3.12.
(Completeness
be a countable
set of sentences
then
T
PROOF.
has a predictable Let's
quantifier)
Theorem
for Predictable
in Lad.
If
T uP
Processes).
Let T
is consistent
in Lad
model.
first make an observation Consider
theorem.
structures
for
about L
•
L i f (Logic with integral
of the form ~I f
A = (~,(Yn)
,Y,P) where
!
the Yn s
and Y are unary random variables
n siN and add to the axioms Then Keisler's
of L
~if
completeness
the axiom
theorem for
^ v A m n p~n
i Y
L w /(see
p
 YI < I/m a . s
[K3])
can be extended
1
to obtain models
A as above where
The idea here is to introduce symbols
(Xn)n e ~
(E[X(t) IF
n
~ Y a.s. n÷~
in ~.
countably many new stochastic
that are going to represent
]) ti/n
Y
.
process
the processes
So, for each n let
ts[0,1],n s ~
X n = (X$ (t)) r t~[0,1]
be defined by E[Xr(t) IF
] (w)
if
t > I/n
if
t< i/n
tI/n (I)
X n (w,t) = EEXr(t) IF ](w) 0
As we noticed /l main
/l nslN
xn(t)~~Xr(t)
in Theorem
3.5,
the expression
/l IE(Xr(t) IF )  Xr(t) I < i/m a.s. p >n ti/p a.s..
Axiom
Just means
(P2) is the formal way of expressing
this
t90
(for almost all t) in the language of Lad. axiom (P2) holds (2)
For all
Therefore
any model in w h i c h
is such that:
rg~ + for almost all
tc[0,1]
the r a n d o m variable Xr(t)
is Ft_measurable. By the above remarks T
such that
it is clear that we can get a model
(2) is true of X.
Using a similar argument
given in the proof of theorem 8.b. X' = X'(t))t~[0,1 ] on (3)
For all
~
A = (~,X) of to the one
we can find a process
such that:
tE[0,1]the r a n d o m variable X'(t)
is Ft_measurable
and
w
(4)
A' = (~,X')
~ A.
By lemma 3.7 and theorem 3.3 we can find an optional m o d i f i c a t i o n X" = (X"(t))ts[0,1 ] of X' such that s
(5)
A"=
(~,x")
z A'
By the characterization
(Theorem 1.13.c)
property
in
(3) also holds
A".
of strong equivalence
Then by theorems
and
(P2)
3.2 and 3.4 we can
obtain a predictable m o d i f i c a t i o n X''' = (X'''(t))t~[0,1]of X" such that the model A''' = (~,X''') s~ A . . . Thus . . A _ ' is the desired predictable model of
TuP.
D
If we take a close look at the above proof we will see that the theory T plays a "cosmetic" role because proof of the completeness theorem.
theorem,
its importance
is only felt in Keisler's
for Lad and we are just quoting his
We can isolate the m a i n model
theoretic
feature w i t h the follow
ing T H E O R E M 3.13.
If A
~ "for almost all t, X t is Ft_measurable,"
there is a predictable
model B such that
Let's remark that all the other completeness can also be put in this form. reader.
The two previous
then
A ~_ B.
D
theorems
We leave the details
from this section
to the interested
theorems are stated in a form that is familiar
to logicians but we can use the same proof in order to present some results
that are along the lines of the probabilistic work of Hoover
and Keisler
in [HK].
undefined notions T H E O R E M 3.14.
Let
We refer the reader to that paper where all the
that appear in the following X
and
different adapted spaces) (a)
Y
theorem can be found.
be stochastic processes
and suppose
X
(on perhaps
is predictable, then s If X and Y have the same adapted distribution (X ~ Y in our lan
191
guage) then
Y
has a predictable modification (but
Y
itself is
not necessarily predictable). (b)
We can find a predictable stochastic process
Z
on the adapted
S
Loeb space such that
X ~ Z.
PROOF. (a)
Just read carefully the proof of Theorem 3.12.
(b)
Use (a) and the saturation property of the adapted Loeb space. D
Finally we study the notion of martingale which is of central importance in probability theory and crucial for the development of the general theory of processes.
A thorough exposition of the theory of discrete
and continuous time martingales can be found in [DM 2]
We are going to
prove a completeness theorem that gives us martingales with time parameter [0,I)
instead of
point on the right.
[0,i].
The problem arises because i is an end
After the proof of the theorem we indicate how to
take care of this problem. DEFINITION 3.15.
Let
M
be the following set of axioms in Lad:
(M I)
Axioms for Lad (see Definition 1.9).
(M2)
(et ~ I) ^ v A (f[Xq(t)Xp(t)Idw < I/n) n m qepem
(M 3)
A+ r~
(Pst>l)(s~t
÷ (E[Xr(t) Is](s) = Xr(s)a.s.) )
Intuitively, axiom (M2) says that for almost all
t
the random variable
X(t) is integrable. THEOREM 3.16.
Let
T
be a countable set of sentences in Lad°
is consistent in Lad then PROOF.
T
If
has a martingale model.
As in previous theorems the first step is easy:
use
M
to
extend Keisler's proof of the completeness theorem for Lad in order to get a model
A = <~,(Xt)ts[0,1],(Ft)tE[0,1],P>
(i)
For almost all t, X t is integrable
(2)
For almost all (t,s)s[0,1)xE0,1), X s a.s.
We now show how to find a martingale
if
s ~t
of
then
T
such that:
E(XtlF s) =
(Yt)tE[0,1) living on
W
(~,(Ft)t~[0,1],P) From (i) (3)
and
For all
(2)
such that
A ~ B = <~,(Yt)ts[0,1],(Ft)ts[0,1],p>.
we can get G ! [0,I) of measure one such that:
t s G, X t is integrable
T uM
192
(4)
For all
t EG
the following holds
in
A:
(Ps > l)(s ~ t ÷ E(XtlF s) = X s a.s(3) gives us the
It is easy to verify that
(Xt)ts G
integrability
So we just have to clneck:
For all
condition.
is a martingale.
s,tsG(s
In order to see this let rsG such that
s < t < r
Using
s, tsG with s< t0
(4) we can find an
and
(5)
E(XrIF s) = X s a.s.
(6)
E(XrIF t) = X t a.s
Thus we have: E(XtlFs)a?sE(E(Xr;Ft) IF s) by a.s.= E(XrIFs) = X s by a.s.
Now the definition s t > to
Let
Yt
of
since
(6), F s _
(5)
(Yt)ts[0,1).
If
ts[0,1) Xt
if
tgG
E(Xst IF t)
if
t ~G
=
It is a simple exercise to show that we leave it to the reader.
(Yt)te[0,1)
Finally observe
and since G has measure one theorem A w= B as we w a n t e d to show. Let's
find s t s G such that
t ~G
is a martingale
that for all
1.14.d.
0
indicate what to do if we want to have a m a r t i n g a l e with
Fortunately
in Lad we can express (Xt)ts G is uniformly
one
from
ts[0,1]. to
[0,I]
the fact that on a set G c_ [0,I) of integrable.
We do it by adding the
following axiom to the list of axioms of the definition A n
[0,i)
integrable martingale.
measure
(M 4)
t s G, X t = Yt
tells us that
The idea is that in order to ensure the transition we have to have a uniformly
and
3.15:
V (Pt e i) (fl t) Xm(t) Idw < I/n)) m (q>Am Xq( 
Once this axioms holds G an increasing
in our model
sequence
(see above proof) we can choose from
(dn) n slq converging
to 1 and define
193
YI = n_~=limXdn.
It is now easy to show that (Yt)te[0,1] is a martingale
and we leave the details to the reader. REFERENCES [An]
Anderson, R.M., A nonstandard Representation of Brownian Motion and It8 Integration, Israel J.Math 25(1976), pp. 1546.
[As]
Ash, R., Real Analysis and Probability, Academic Press (1972)
[C]
Cutland, N., On the Existence of Strong Solutions to Stochastic Differential Equations on Loeb Spaces, Z.W,V.G., 60(1982), pp. 335357.
[D]
Del!acherie, C., Capacites et Processus Stochastiques, SpringerVerlag, (1972).
[DM I] Dellacherie, C. and Meyer, P.A., Probabilities and Potential, Vol. I, NorthHolland, 1978. [DM 2]
, Probabilit~s et Potentiel, Vol. II, Hermann, 1980.
[E]
Elliot, R., Stochastic Calculus and its Applications, SpringerVerlag, (1982).
[F]
Fajardo, S., Probability Logic with Conditional Expectation in: Annals of pure and applied Logics (To appear).
[H I ]
Hoover, D., Probability Logic, Ph.D. Thesis,Univ. of Wisconsin (1978)
[H 2]
, Probability Logic, Ann.Math.Logic 14 (1978),pp. 287313 .
[H 3]
, A Normal Form Theorem for Lw, p with Applications, J.S.L. 47(1982), pp. 605624.
[H4]
, A Probabilistic Interpolation Theorem.
(To appear)
[HK]
Hoover, D. and Keisler, H.J., Adapted Probability Distributions. T.A.M.S. (To appear).
[HP]
Hoover, D. and Perkins, E., Nonstandard Construction of the Stochastic Integral and Applications to SDE's. I and II, T.A.M.S. 275(1983) ,pp. 136 and 3758.
[K I]
Keisler, H.J., An Infinitesimal Approach to Stochastic Analysis. Memoirs Amer.Math.Soc. 297, Vol. 48(1984).
194
[K 23
Keisler, H.J., Hyperfinite Probability Theory and Probability Logic.
[K 33
[K 43
Lecture Notes, Univ. of Wisconsin
(Unpublished)(1982).
, Probability Quantifiers, To appear in Abstract Model Theory and Logics of Mathematical Concepts, ed. by J. Barwise and F. Feferman, SpringerVerlago , Hyperfinite Model Theory, pp. 5110 in Logic Colloquium 76, NorthHolland
[K 5]
(1977).
, A Completeness Proof for Adapted Probability Logic. (To appear)
[Li]
Lindstr6m,
T., Hyperfinite Stochastic Integration,
I,II, and III.
Math. Scando 46(1980), pp 265333. ELo]
Loeb, Po, Conversion from Nonstandard to Standard Measure Space and Applications to Probability Theory, T.A.M.S. 211(1975), pp. 113122.
[MP]
Metiver, M. and Pellaumail, Press, 1980o
[Pal
Panetta, L., Hyperreal Probability Spaces: Some Applications of the Loeb Construction, Ph.D. Thesis, Univ. of Wisconsin (1978)
[Pe]
Perkins, E., Stochastic Processes and Nonstandard Analysis. L ~ 983.
[R]
Rodenhausen, H., The Completeness Theorem for Adapted Probability Logic, Ph.D. Thesis, Heidelberg University (1982).
[Ski
Skorokhod, A.V., On Measurability of Stochastic Processes, Theory of Probability and Its Applications, Vol. XXV,(1980), pp.139141.
J., Stochastic Integration,
Academic
I wish to thank my advisor, Professor H.J. Keisler for suggesting the subject of this paper and for his invaluable cormments. This research was partially supported by a University of Wisconsin Research Assistantship.
A BARREN EXTENSION J.M.Henle
A.R.D. Mathias
Smith College
Peterhouse
Northampton,
Cambridge,
Mass.
England
U.S.A. W o Hugh Woodin California Institute of Technology Pasadena
California U.S.A.
ABSTRACT.
It is shown that provided w ÷ (w) m,
extension adds no new sets of ordinals. the same extension preserves elucidates
a wellknown Boolean
Under an additional assumption,
all strong partition cardinals.
the role of the hypothesis
This fact
V = L[R] in the KechrisWoodin
characterization of the axiom of determinacy. §0.
INTRODUCTION. Let
B = Power(m)/Fin be the quotient of the Boolean algebra
of all subsets of m by the ideal
Fin
the set of nonzero elements of
of finite sets, and P = [m]m/Fin
B, with the induced
partial ordering.
We shall study the Boolean extension that results from using notion of forcing: with a famous theorem AC
of (ml,ml*)
gaps in
P
[5]
P
as a
about the existence under
in mind, we shall call this the Hausdorff
extension. Our underlying set theory is ZermeloFraenkel.
We shall be working in
contexts where the full axiom of choice is false; use DC, the axiom of dependent choices, states that if
Q
then there is a map
is a relation on f:m÷Power(m)
notation of Boolean extensions Mathias
[15],
3.7
of < of order type
and %,
3.8.
at times, we shall
or DCR, its weaker form, which
Power(n)
such that
such that Vp~ q Q(p,q), Vn Q(f(n),f(n+l)).
Our
and forcing follows that described in We write
[<]%
for the set of subset
and follow Supercontinuity
[8]
in our notation
of partition relations. It has long been known that under
DCR
the Hausdorff extension adds no
new sets of integers: what it does add is a Ramsey ultrafilter on ~. (Cf [15],
Theorem
4.2).
For a recent discussion under
AC
of
P,
see Dordal [3]. Since without
AC
there are difficulties
in the simultaneous
choice of
196
representatives
of equivalence classes
, it will be convenient to take
as our forcing conditions
infinite subsets
standing that if
q
p
and
p~q, they force the same statements, tion than
q
if
p\q
p,q
of
~, with the under
have finite symmetric difference, written and that
p
is a stronger condi
is finite.
In section i we prove in the theory
ZF + w ÷ (m)m
that the Hausdorff
extension is barren in the sense that every map in the extension from an ordinal into the ground model lies in the ground model: no new sets of ordinals are added.
in particular,
We characterize this latter proper
ty in terms of the GalvinPrikry notion
[63
of completely Ramsey
families. In section 2, LSU and EP.
we consider three settheoretic principles, LU and
LSU are the weak and strong uniformisation princi
ples discussed in Mathias [16], valent in
ZF + DCR
holds and
V = L[R] or if
to
where it is shown that
m ÷ (m)m + LU, ADR
and that
holds, or if
V
"all sets of reals are Lebesgue measurable". this section from
called LU,
LSU
in the theory
logical terminology of Ellentuck
LSU
is equi
LSU is true if
AD
is Solovay's model for
EP, which we derive in
ZF + DCR, is, to use the topo
[4] ,
the assertion that the inter
section of any wellordered collection of comeagre sets is comeagre. In section 3,
we prove in the theory
ZF + LU + EP that the Hausdorff
extension preserves every partition property of the form K ÷ (~)%~, where
K > ~,
Finally,
2 ~
< K, and
0 < ~ = ~
~ ~.
in section 4, we comment on the implications of the results
of section
3
for the recent characterization
by Kechris and Woodin,
[123
of
AD
in
L[R]
and discuss some problems related to our work:
reference to this discussion is made
in the text by the string
[PROB]. Assumptions are given in full in the statements of theorems, but may be omitted in Lemmata and Propositions when the flow of the narrative demans it. §i.
The end of a proof is signalled by
THE BARRENNESS OF THE HAUSDORFF EXTENSION.
THEOREM 1.0. N
ordinals;
M
be a transitive model of
P0
Then
M
moreover every sequence in
and N
N
ZF + ~ ÷ (~)~ and
have the same sets of
of elements of
It will be sufficient to prove in the theory ^
if
Let
its Hausdorff extension.
PROOF.
4.
^
l~ f:~ ÷ V, then for some
Fix then such
M
ZF +
lies in M. m ÷ (~)~that
^
q0 ! Po'
qo
P0' f' ~ and suppose that no such
T~ fcV. q0 exists.
Then for
197
each
p c [P0
there will be at least one ordinal
xcV
exists with
For
P ! ~, define
enumeration Now define ing as
of
p
I~ f(~)
P~\{k}
p~(n)
for
equal ~.
Q = P\{k},
~(P) = 0,
ty, so
~(P)
Put now
we have
~(Q) = 2;
can only equal
v = ~(P).
= p(2n),
that
x # y, q
choosing
q,
but
is compatible
Since
sr
Since
s~ ~ q,
forces
the value
As in Happy Families sup{n+lJn ~ s},
DEFINITION (p e A
and for
defined
of
q ~ [p]~ The
CR
tement
1.2.
The
r,
~(s~) > ~(s r)
there is a
always
Following
r,
q ! P
P, ~(s r) =~. and
q
and thus,
Isl ~ np,
Isl for
we write
Is,p]
for
of
[~]~ will be called p' ~ A;
(p' ~ domain(F)
Galvin and Prikry
invariant
and
if
F(p)=F(p')).
[6] we call a subset
CR, if for all or
a function
[m]m will be called
implies
or
invariant
similarly
<s,p>
there
is a
[s,q]nP = 0. of
to the assertion
[15],
1.3.
The sta
that all subsets
of
Ramsey. We shall
is a
call a subset
q~[p]~with
q c [p]~ with
sets are those in
in the topology
from
s~ ! q,
absurdity.
subset of)
is equivalent
1.3.
x,y c V
se[P] ~ by
and extends
since
sets are chosen in ~H in the notation
there CR
and
implies
Ramsey,
<s,p>
<s,p>
q
s ~ [m]<~, we write
A
always
Is,q] ! P
DEFINITION
Define
q, the fourth
and so on, so that
with
with either
m * (m)m
~;
[0,p] = [p]~.
[~]m completely
[~]~ are completely
all
for
A subset
p~p')
and
q,r e [P]m and
~ y.
from
~(q) >~, So
p ~]~
and p~p')
DEFINITION
s
an evident
Thus
on (an invariant
(p ~ domain(F)
P
i.i.
and
I~ f(~)
of
but
[15],
{x~[~]~l s ! x ! sup}.
F
r
f(~).
~(s) = 2 # 1 = ~(P) = ~(s),
if
and
with both
for
P~
^
~(s~) e ~(q); x
~(pr ), and let
~(P) = ~(Q) by homogenei
of ~ there are
then one from r,
(st) r c r "
than
I.
~ x
(Sr) ~ c q,
~(p) = 0, 1 or 2 accord
Q~ = Pr' Qr = P~\{k}; 2, 0;
the first three elements
then three from
= p(2n+l).
and so have the same value under
if
By definition
I~ f(~)
Pr(n)
~(P) = I: for putting
^
with
q for the monotonic
to, or greater
Notice
force the same statements,
thus if
to be the least such ~.
~:[p0 ]~ ÷ 3 by setting
is less than,
and
~(p)
by writing
q i m and setting
PE[p0]~ be homogeneous k = min P
define
P~ ! m, Pr ! ~
a partition
~(p~)
~ x:
~ < ~ for which no
I H in
P
of
Is,q] c P, and [s,q]nP [15],
defined by Ellentuck
[~]m CR + CR
if
for
if for all
= 0. 1.4, and are the meagre
in [4].
sets
198
1.4
In a manner
tivized versions is
CR+ on
Is,p]
there is an
r
familiar
to readers
of these concepts: if for all
in
[q]~ with
We are now in a position
of
[15],
we shall want rela
we shall for example with
s ! t
and
say that
P
(t\s)uq ! P,
It,r] ! P
to reformulate
a weaker
form of the conclusion
of 1.0 as PROPOSITION
CR
1.5
(ZF)
The following
(1.6)
The Hausdorff
(1.7)
Let ~ be an ordinal
subsets
of
[~]0J
extension
adds no new sets of ordinals.
and
Then there is a
< ~, C~ is, relative
to
[p]~,
either
CR +
(1.6) ÷ (1.7)
:
filter
in
added by the Hausdorff
B = {v < ~ IP c G(Cj,
relative
to
a sequence
p~[~]~
PROOF.
Power(u)
Fix
are equivalent:
or
CR.
and let
G
be the generic
extension.
[p]~,
of invariant
such that for each
is
In
V[G] put
CR+)} ^
By (1.6), shall
there is an
show that for all %# in
that for all%#not contrapositives Let
v < ~.
CR
there is C , ^
~
If
in
A,
of these
If
of q
A c ~, and a
such that
C%# is
is CR
CR +
q
I~ B  A.
relative
relative
to
to
We
[q]m,
and
[q]~, by proving
the
statements.
C%# is not
Is r ] c _
A,
C %#
q
CR +
[0,q]
[r] ~ nC%# is empty,
relative
with
to
Is,r] ^n C
and so r
%#
i~v~
[q]W,
then since
empty,
so by the invariance
B;
hence
q
~
Cv
~e ?B,
is
so
^
%#eA, and thus %# ~ A. C v is not
CR
relative
to
[q]~,
there
is
Is,r] ! [0,q]
with
^
Is,r] c C , q
~v~
so by invariance
B, so
q
[r]~i C%#,
~ A, and so
and so
r
i~ v c B;
hence
%#~A. A
(1.7) ÷ (1.6):
Suppose
Then each iff a
C is invariant. ^ %# l~%#eB, and C%# is
p p
satisfying REMARK
for
n~, 1.9
1.8. let
Notice
(1.7)
that
CR relative
the conclusion
of
Set C
%# to
(1.7) will
is false without
C
= {ql q
I~
~ c B}.
is
CR + relative to [p]m ]~ ^ [p iff ^p ] ~ v ~ B: so
force
B c V.
the hypothesis
of invariance:
C n = {p I n ~ inf p}.
The conclusion
squarebracket
I~ B ! ~"
partition
of Theorem relations:
1.0 may be derived write
m ÷ [~]~,
from certain where
2 ~ % ~ w,
199
to mean that for any
~:[~]m + %
some
y ~ [x] ~, ~(y) # v.
v < %
and all
from a sequence n, ~n
<~n I 2 ~ n
families,
~ ÷[~]~
not known whether n e 3 [PROB]
0 ~ i < B,
to m ÷ [m]m~
for countable
~ ÷[~]~n+2.
It is
~ ÷ [~]~n
for
{p(mB+i) Im~ ~}.
and
application
than n,
Thus if
(q)Bi = (P)0 \{~(0)}"
of
1.0.
Vj
~((p)i)}.
w + [~]~n,
a
is of cardinality
for some
i,j
Set
p c [p0 ]~ may be found for
less than n. p e [p]~,
But then since if
less than B , by the primality
~((p)0 ) = ~((p)j)
< ~((p)i )
of B and the invariance
the B values
*(p),
,(p\{p(0) }),
,(p\{p(O) ,p(1) }) .
,(p\{p(0),p(1) ..... p(Bl)}) Now put
~ = ~(p).
q I~ f(v) (s)0n q
that
for each
1.0 may be derived as follows.
let (P)i be
... = ~((p)B_l ) for any
~,
AC
from any
of theorem
~((p)l ) =
of
n < ~,
f' ~' ~ be as in the proof of Theorem
@[[p]~]
where
Let B be a prime number very much larger
~(p) = { i l i < B
which
,
a counterexample
that for some
, (q)i = (P)i+l'
By (repeated)
has observed
of [m]~
DC, or at least
m ÷ (~)~ can be derived
p ~ [~]~,
P0'
implies
m ÷ [~]mn.
q = p\{p(0)} Let
m ÷ [mien'
but the conclusion
Suppose and for
to
thus assuming
x e [m]m such that for
Kleinberg
<~> of partitions
is a counterexample
may be constructed:
there is an
is infinite,
$((s) 0) = v
while
established REMARK
two transitive
they might
differ
an extension
(s) 0 n r
q, r e [~]m, x,
x # y.
Find
is infinite,
~((s) I) z ~(q)
.
> 9,
y e V,
s c qur
(s) I i q
with
so that
Then
contradicting
the property
of
paragraph. V
It is a theorem of Vopenka and Balcar
models,
axiom of choice,
there are
~ y, and
in the previous
i. I0.
.
will be distinct.
As before,
E x, r I~ f(~)
.
of which at least one is known
have the same sets of ordinals, if neither
satisfies
of that result,
see Monro
AC [17].
[21]
that if
to satisfy
they coincide.
is due to Jech
the That
[9]: for
200
2.
THE L A R G E N E S S OF THE INTERSECTION
DEFINITION continuous
on
2.0. A function [s,p]
F(q) n(k+l)
formised on
2.1.
[s,p]
DEFINITION R
as above,
on
[0,x]) is
2.2.
a sequence of
CR +
([15];
D
n C
[6].
uniformised
I~ < e>
CR +. ZF + DCR, and is due
A proof may be given by the methods
suppose
Proposition
of Happy
I.i0).
the theorem true for all sequences
of
e , and that it fails at O for the sequence of e and the first sentence of this proof
cofinality.
(for the case D
n C }.
Then by Proposition
A = [m]~, B = [m]m\( ~<~ n C ))
~ D~ for
and, by r e l a t i v i z i n g
~ < ~ < e,
to some
[s,S]
q ! P ~D
if necessary,
2.8 of D
is CR +
,
implies
q ~D ,
we may further assume
is empty.
Thus we may define
X(P),
The function X:[~] m + e
for pe[~] m
V~<8 ~(p)
to be the least ~ < e w i t h p ~ D
will have these properties: q i P ÷ X(P)
Define now
is
this is a theorem of
= df {p E [~]~l[p] ~ c
for each ~ < e ,
v~eD
F
that for any relation
{xlR is (strongly)
Then
By the m i n i m a l i t y
Happy Families
that
uni
function
Let e be any ordinal and.
see in particular
e must be of uncountable Now put
(strongly continuous)
Vp] y R(p,y),
families.
Let us therefore length less than J~ < e>
is (strongly)
R(q,F(q)).
(ZF + DCR + LSU)
to Galvin and Prikry
R c [m]~ x Power(u)
LU (LSU) is the assertion
For e countable,
Families
k e q,
CR +.
T H E O R E M 2.3.
PROOF.
and all
if there is some
such that
is called strongly
= tqn(k+l).
A relation
such that for all q ~ [s,p],
CR + FAMILIES.
F:[~] ~ + Power(m)
if there is a tree
sets of ~ such that for all q ~ [p]~
DEFINITION
OF
~ x(q),
V <s,q> ]p~[s,q]
= u{x(q)[q=p}
and
(X(P) >v).
¢(P) =n{x(q)]q=P}.
We assert
that V qc[~] To see that, put q = P0 Z Pl ~ P2 ~
v = ~(q),
w
]p c [q]~(~(q)
construct
< ~(p)).
sequences
"'" such that for each
n o < n l < n 2 < ... and
i,n i = min Pi' and for each
201
s _c n.+l~ , [s,Pi+l ] ~ D v, and If
p'z p, there is an
Pk+l ], By
so
p'\n k = Pink,
X(P') >v;
R
and
on
is continuous, [18],
12), if we define
since if
ml
so
<s,p>
with
R(sup, q), and so
Thus if for
~ < n to
we set
[0,p]
[0,p];
]q'~q ]p'=p(F(p')
and so by the KunenD~rtin
q E [~]w
To see that, take
CR relative relative to
and all
iff
= ~(p) >X(P)
page 99, Theorem 2G.2;
Notice that for any p R(q) > PR(SUp).
R is Borel,
R(p,q)
R(p,q),
theorem
see also page 114, footnote
pR(q) = U{PR(P)+lJp c [p]~ and
some n less than
q~F(p),
function
$(F(p)) >~(p)).
x(q) e ep(q) = qb(q') > q ~ ( F ( p ' ) ) > ~ ( p ' )
(Moschovakis
consequently ~(p) >v.
[p]m by setting
will be wellfounded
F
p' e [p'n(nk+l),
p ~ [m]m and a strongly continuous
Define the relation
Since
so
such that
Vp e [p]m(F (p) ! P
= q'):R
p = {niJi < m}.
n k such that
p' c D , and
LSU, there is a
F:[p] ~ + [ p ] ~
set
F(p) = q}, then for
pR(q) < n.
sup ! p, there is a q = su(F(p)\Jsl):
q c Is,p]
with
then sup~p
and
pR(q) > PR(SUp)
E~ = {p c [p]~JpR(p)
= ~}, each
But then since ~ is countable
• but that is absurd,
E~
u E ~
as
[0,p] =
will be is
CR
~
'
u E . ~
DEFINITION 2.4. EP is the statement of Theorem 2.3, that the intersection of any wellordered collection of CR + sets if CR +. PROPOSITION PROOF. or
If
[s,x]nC~ = 0}
Let
P §3.
2.5 (ZF)
•
be a member:
EP implies
(1.7).
is as in (1.7), let
Then each
D~
is
then P satisfies
D
= {xJVs ! n x
CR+: by
EP, v<~ n D
is not empty.
the conclusion of (1.7).
THE PERSISTENCE OF PARTITION PROPERTIES.
PROPOSITION 3.0 (ZF + EP)
Let ~ be an ordinal and
an invariant function.
Then ~ is constant on some
PROOF.
C
so by
(Is,x] i C
For
~
= {pjv ~ ~(p)}.
1.7, which by 2.5 follows from
relative to
[p]m,
each
C
is either
[p]~.
Then each
EP, there is a CR +
or
~:[~]m ÷ [K]K
CR
C
is invariant, p ~ [m] ~ such that,
:
put
202
D
= {qe [p]~l[q] ~ ! C } in the first case and
in the second. Let
p
Then each
Dv
is
be in the intersection,
for each
qe[p]m
and ~ <~,
~ ~(P) ÷ P e C
CR +
on
invariant
The above Proposition
: for example,
PROPOSITION
3.2
< K, K ÷ (K)%~, [~]w>
£
is a
p* ~ [~]m
for each
(ZF + LU)
that
PROOF.
For each
as in
that
0 < ~ = ~X ~ K,
to be
~(p)
define
~p:[K] % + p.
function
pp:[~]~
for
÷ ~
and that Then there
~:[p,]m + [<]K
is homogeneous
2 ~
~:[w] ~ ÷ [K] <
of partitions
pp(X) where
and
may fail if ~ is not required
there is a surjection
p,
Then
÷ ~(q)
Suppose
and an invariant
p ~ [p*]~,
{Dvlv
if ~ is an injection.
is a collection
<~plP
EP, of
~ ~ ~(p),
÷ ~ ~ ~(q),
~ ~(p) ÷ p ¢ C~ + D n C ~ = 0 + q ~ C REMARK 3.1.
by
iff
+ Dv c C v + q e C
= 0}
[p]~.
nonempty
~ e ~(q)
= {q~ [ p ] m I [ q ] ~ C
such that
#p.
by
= ~p( x ),
Supercontinuity
[8]
,
x
={ o
x(m~+n) l~ < %}:
note
nE~
that as
w% = k, ~x
is in
[K] ~
Hp = {qc [~]~I~(q) By
LU
there
is a
q ~ [p*]~,
~(F(q))
For each
q ~ [p*]~,
ordinal
greater
least ordinal [q,~)
p*
and an
define
than all
The regularity
x
is homogeneous
is. for
for
pp.
B(q')(O) setting
for ~
at least one element
q' ~ q;
of each
the soundness
B(q')
C(q) = C ( q ' ) w h e n e v e r
q ~ q'
Now put
~(q) = ~C(q).
form
for some y E [B(q)] %, and hence
Hence
~(q)
of
Then ~ is invariant; B(q),
is homogeneous
~q
C(q)(~)
for
of any
be the
the interval
q' ~ q.
of
Note that
{q'lq'
~ q},
and
: so that C is invariant. any
~q(X)
is independent for
be the least
of this definition.
therefore
~y
let
all
= ~(F(q)).
C(q)(0)
= v{C(q)(~')I~'
does not rely on an enumeration
by the homogeneity
B(q)
let
this definition that
pp}.
Write
C(q) ¢ [K] < thus:
of < ensures
Set
F:[p*] ~ ÷ [~]~ such that for
is homogeneous
~ such that
contains
whenever
x ~ [~(q)]~
= ~q~y) y
as required.
is of the
= pq~),
and therefore
which, of
x.
203
THEOREM
3.3
(ZF + EP + LU)
Suppose
and that there is a surjection extension, PROOF.
0 <~ = ~
~:[m]m ÷ ÷
[<]<
~ <, 2~p<~,<
÷ (K)l~,
Then in the Hausdorff
K ÷ (<)~.
Note first that by
quently
2.5
no new subsets
[<]% is the same whether
the extension,
interpreted
that
conse
in the ground model or in
and thus may be written without
Note secondly
of < are added; ambiguity.
~ ÷ (<)kp+l. ^
Suppose
P0
such that
I~ f:[<]% ÷ PP2
For each
I~ f
We shall find a
is constant
p e [p0 ]~,
define
~p(A)
= ~
3.2,
there is a
assert
that
If not,
^
I~ A
^
~ g and
~q. §4.
otherwise.
#(p)
^
~ ~): ~q.
on
for
by
function
~:[pl ]m ÷ [<3 K
for
[p2 ]~.
Zp.
Set
By 3.1
A = ~(p2 ). We
f.
q e [p2 ]m, ~ < ~ < p
so
But
A ~ [<]<
~
is homogeneous
~ constant
D,E c [A]~, (~)
for
~q(D)
= $, ~q(E)
A = ~(p2 ) = ~(q)
with
= ~, which
and thus is homogeneous
[ PARTITION
CARDINALS
An important ordinal
= p
is homogeneous
f
A is inhomogeneous for
P l~ f(A)
p e [pl ]~,
there will be
q l~ (f(D)
if
P2 E [pl ]~ with P2
~ :[<]~ ÷ ~+I P
P l e [P0 ]~ and an invariant
such that for each there is some
P2 ~ [P0 ]~ and an
[A]~.
a partition
~p(A) By
on
> 0
LEMMA
ordinal
onto which 4.0.
same whether
WITHOUT
in the study of
Power(m)
AD
is
~,
the least
cannot be mapped.
If the Hausdorff
calculated
DETERMINACY.
extension
is barren,
then 6 is the
in the ground model or in the extension. ^
PROOF.
Suppose
PIP
f:[~]~ ÷ ÷ 8. ^
put ~(q,r)
= ~ if
q I~ f(r)
is a surjection PROPOSITION K ÷ (<)Ip, PROOF. ments
then
of
4.1
For each pair
~ ~,
and
~(q,r)
[~]~ x [~]0,i onto
(AD
+
V
=
L[R])
If
e, 0
By a theorem of Kleinberg [13],
< < 8 ;
[14],
<
hence,
= 0
otherwise.
Then
a contradiction.
< ~l=
K + (K)%p in the Hausdorff
to be found in
(q,r) with q\p finite,
^
~ ~<
,
2_<~
< <
and
extension. is measurable,
so by argu
by a theorem of Moschovakis
204
([18], 7D.19, page 442), there is a surjection
~:[m]m ÷ +[K] < .
DC is provable in
ZF + AD + V = L[R], by Kechris
by Theorem 2.2 of
[16];
by Theorem
Thus all the hypothesis of THEOREM 4.2
If
AD
V~<@ ~ K(~ ~ m < @
If
AD
Again, by As
holds,
[i0],
m ÷ (m)m,
hold,
so is
LSU,
EP then holds.
so the conclusion follows.
is consistent,
so is
DC +
and K is a strong partition cardinal) + "there is
a Ramsey ultrafilter on PROOF.
3.3
[i0];
2.3 above,
m". it stays true in
DC holds,
so
DC
(cf Theorem 2.2 of
the results of Kechris,
L[R], as strategies are reals.
holds in the Hausdorff extension. [16]),
Kleinberg,
the extension is barren. By
Moschovakis and Woodin
Eli],
there
are arbitrarily large strong partition cardinals below @ in the ground model,
so by
4.0
and
4.1, the same holds in the extension.
A remark
in the introduction completes the proof. REMARK 4.3.
The significance of
4.2
is that as the existence of an
ultrafilter on ~ implies the failure of
AD, the hypothesis
is an essential ingredient in the KechrisWoodin
[12]
V = L[R]
derivation of
AD
from the existence of arbitrary large strong partition cardinals below 6. REMARK 4.4.
The arguments of section 1 generalize with little change
to obtain a new proof of Henle's result
[7]
that Spector forcing
[20]
at a strong partition cardinal K is barren. REMARK 4.5. measurable,
If and
AD
holds and
~ + (~)m,
Ramsey ultrafilter on
V = L[R],
then
~i
and
so in the Hausdorff extension,
~, and
ml
and
~2
~2
are
there is a
are still measurable: we
may say that there are in this model two and a half contiguous measurables. A model for that statement may also be found assuming something presumably much weaker than model of
ZFC
Con(AD): by an unpublished result of Woodin, a
in which < is
%supercompact and
admits a Boolean extension in which measurable,
and
~ ÷ (~)~;
K = el'
~ > < is measurable
~ = ~2' <
and ~ are still
in the Hausdorff extension of that model,
there will be a Ramsey ultrafilter on
~,
while the measures on K and
will remain measures as no new subsets of either will be added. Several open problems are related to PROBLEM 4.6.
Can
m ÷ (m)m
[16]
and the present work:
be deduced from
m ÷ [mien
for any
n ~ 3? PROBLEM 4.7.
Does
AD
imply that
ml
is huge?
Is there a huge
205
measure on
[~2]~I?
PROBLEM 4.8. DC + ~ + (~)~
Is it a theorem of
that there are no
PROBLEM 4.9.
Is there a
that there cannot exist
k
k
k c ~ such that it is a theorem of
if
AD
is consistent,
might be quite small, though:
guous large cardinals, PROBLEM 4.10.
see Apter
Is
EP
The least 8 for which PROBLEM 4.11.
[I]
and
a theorem of
2.3
k
ZF + DC
must be greater
Bull
[2].
ZF + DC + ~ ÷ (~)~ ?
fails must be measurable.
If < and % are strong partition cardinals with
which are studied in
[83
,
~ < %,
~, or vice versa?
The similarities between strong partition cardinals and ÷ (~)~,
ZF +
for limitations on conti
will < remain strong in the Spector extension for
above, suggest
or of
~?
contiguous measurables?
By a result of Kechris, than three,
ZF + DC + ~ i ~ 2 ~ °
MAD families coded on
w,
when
and the result mentioned in 4.4
that SDector forcin~ should preserve somethin~ more than
plain measurability.
But there are limits:
is replaced by a strong partition cardinal PROBLEM 4.12.
Does
ADR
the analogue of 2.3, when + fails for e = < .
K,
imply that every set of reals is Souslin?
A theorem of Woodin states that A D R is provable in
ZF + AD + "every
set of reals in Souslin". PROBLEM 4.13.
How strong is the theory
"8 is a regular limit cardinal"? every real
ZF + DC + V = L[R] +
Does it prove the existence of
The challenge in this question is to get 8 a limit cardinal: result of the Cabal that In O
[133,
~#
for
~?
ZF + DC + V = L[R]
Kechris and Woodin show that if
it is a
proves that 6 is regular. AD + V = L[R] holds,
then
cannot be weakly compact. PROBLEM 4.14.
It follows from the results of
is a strong partition cardinal)
follows from,
[I03,
that Con(~ I
e.g., Con(there are
arbitrarily large strong partition cardinals), but the proof goes via determinacy.
Is there a direct proof?
206
REFERENCES [ 13
Apter, A.W.
Some results on consecutive large cardinals, Annals of Pure and Applied Logic, 25(1983), 117.
[ 23
Bull, E.L.
Successive large cardinals, Annals of Mathematical Logic, 15(1978), 161191.
[ 33
Dordal, P.L.
Independence Results concerning Combinatorial Properties of the Continuum, Thesis, Harvard, (1982).
[ 4]
Ellentuck, E.
A new proof that analytic sets are Ramsey, J. Symbolic Logic 39(1974), 163165.
[ 5]
Hausdorff,F.
Summen von ~ i Mengen, Fundamenta Mathematicae, 26(1936), 241255.
[ 63
Galvin,F. & Prikry,K.
Borel sets and Ramsey's theorem, J.Symbolic Logic 38(1973), 193198.
[ 7]
Henle, J.M.
Spector forcing,
[ 8]
Hen!e, J.M. & Supercontinuity, Mathematical Proc. Cambridge ~thias,A.R.D. Philosophical Society 92(1982), 115.
[ 93
Jech, T.
On models for set theory without AC, Mathematical Reviews, 43 # 6078.
[I03
Kechris, A.S.
The axiom of determinacy implies dependent choices in L[R], J. Symbolic Logic 49(1984), 161173.
[i13
Kechris, A.S., The axiom of determinacy, strong Kleinberg'E'M''ties and nonsingular measures, Moschovakis,Y. N. & Woodin,W. 7779, Springer Lecture Notes in H. Volume 839, ed. by A.S. Kechris,
J.Symbolic Logic 49(1984), 542554.
partition properCabal Seminar Mathematics, D.A.Martin and
Y.N. Moschovakis, 7599. [123
Kechris,A.S. & Equivalence of partition properties and deterWoodin,W.H. minacy, Proc. Natl.Acad. Sci. United States of America, 80(1983), 17831786.
[133
Kechris,A.S.& Woodin,W.H.
On the size of 0 in
L[R], (To appear).
[143
Kleinberg,E.M. Infinitary Combinatorics and the Axiom of Determinateness, SpringerVerlag Lectures Notes in Mathematics, Vol. 612(1977).
[15]
Mathias,A.R.D. Happy Families, Annals of Mathematical Logic, 12(1977),59111.
207
[16]
Mathias,A.R.D. A notion of forcing: some history and some applications,(To appear).
[17]
Monro, G. P.
[18]
Mos chovakis, Y.N.
[19]
Oxtoby, J. C.
Models of ZF with the same sets of sets of ordinals, Fundamenta Mathematicae, 80(1973),105110. Descriptive Set Theory, North Holland, Studies in Logic, vol. 100(1980). Measure and Category. Heidelberg,
SpringerVerlag,
New York,
Graduate Texts in Mathematics, vol. 2
Second Edition (1971). [20]
Spector,M.
[21]
Vopenka,P. & Balcar,K.
V
A measurable cardinal with a nonwellfounded ultrapower, J.Symbolic Logic 45(1980),623628. On complete models, Bull. l'Acad~mie Polonaise des Sciences, S~r.Sci.Math.Astron. et Phys. 15(1967), 839841.
PROOF FUNCTIONAL E.G.K.
CONNECTIVES
LopezEscobar
University of Maryland Department
of Mathematics
College Park, MD. 20742 U.S.A. A characteristic
of the classical propositional
are truthfunctional, tic connectives
is that they
that is,the truth of a sentence depends only on
the truth of its prime components. notions.
connectives
On the other hand the intuitionis
are supposed to be much less dependent on semantical
Consequently,
one avoids
saying that a "sentence
is true",
rather one tends to say: "the construction Nevertheless,
c
proves
some aspects
(or justifies)
the sentence A".
of truthfunctionality
can still be found in
that: (I) a construction
c
(2) the conditions
either proves a sentence
for a construction
given in terms of the conditions for example,
c
proves
of constructions
or it does not
,
to prove a compound sentence is for the proof of the components;
the conjunction
A
such that
(A 0 A Ai)
co
proves
iff A0
C
is a pair
and
cI
proves
Ai• Thus for a conjunction
to be provable
that the conjuncts be provable. different
constructions
traditional
it is necessary
In other words,
(and sufficient)
the fact that quite
may prove a given formula is not exploited
in
intuitionism.
G. Pottinger conjunction",
[1980]
introduced a conjunction,
w h i c h he called "strong
w h i c h requires more than the existence of constructions
proving the conjuncts.
According
to Pottinger:
^
"The intuitive meaning of ^ assert
A ~ B
~
can be explained by saying that to
is to assert that one has a reason for asserting
A which
is also a reason for asserting B"° Hence ~ is one of the first, ly prooffunctional. a prooffunctional
if not the first,
connective w h i c h
This paper is an introduction connective;
is tru
to the "logic" of
in fact for the most part we shall con
sider the extension of the positive
intuitionistic
calculus
of impli
209
A
cation obtained by the addition of strongconjunction
~.
As with any "new" logic, there are five questions that immediately come to mind: (I) Is there a reasonable,
intuitive concept of validity for the senten
ces of the language? (2) Is there a formal concept of validity for the sentences of the language? (3) Is there a formal concept of derivation for the sentences of the language? (4) What are the relations between (I), (2)and (3)? (5) How does the new connective affect familiar mathematical theories (for example
: elementary number theory)?
The paper is broken down into 5 sections corresponding to the above 5 questions. §1.INTUITIVE VALIDITY. I.i THE LANGUAGE ~.
The sentential language ~
is to have the
following symbols: p,q,r ....
for the sentential variables, for the conditional connective, for Pottinger's strong conjunction,
( , )
for auxiliary symbols.
1.2 AN INTUITIVE INTERPRETATION FOR THE INTUITIONISTIC CONNECTIVES. Let
HA(C)
express the (decidable) predicate
proves the sentence
A"
dicate "the construction Finally if to
c,d
and c
~(c,O(x))
"the construction
c
express the (decidable) pre
proves the freevariable formula e(x)"
are constructions then
d'c
is result of applying
d
c,
The BrouwerKreisel interpretation of the intuitionistic conditional is that: (*)
~A=B(
And although
(*)
iff
~(c,(~A(X)
~ ~B(d'x))).
is not universally accepted, it certainly is, and was,
a good point of departure for an intuitive interpretation for the intuitionistic conditional. 1.3 AN INTUITIVE INTERPRETATION FOR STRONG CONJUNCTION. Kreisel interpretation for ordinary conjunction a is ~AAB(
iff
~A(C) A ~B(d).
The Brouwer
210
The latter
suggests
the following
~A~B(
iff
~A(C)
are equivalent
interpretation
for strong conjunction
^ ~B(C).
interpretations
(in an appropriate
theory
of constructions) ~A~B(C)
iff
~A~B(
~
iff
~A(C)
is an equality
1.3 AN INTUITIVE A
of
~
that
~A(C)
A ~B(d)
relation
CONCEPT
is intuitively
^ 7~B(C) , ^ c ~ d,
on constructions.
OF VALIDITY
valid
FOR SENTENCES
OF ~.
iff there is a construction
A sentence c
such
~A(C).
The following
sentences
(I)
A ~ (A~B) ~ B
(2)
(A=B)
(3)
(A=B~C)
(4)
(A=C)
(5) (6) (7) (8) (9) (lO) (ll)
~ (A=C)
are easily
seen to be intuitively
valid:
= (A=B~C)
= (A=B)
~ (A~C)
= (A~B=C)
A ~ B = (A=B) A = (B=C) = (A~B=C) A~B=A A=A~A A ~ B = B ~ A ((A~B)~C) {(A=B~C)
We let
IVAL~
§2.FORMAL
= A ~ (BgC) = (A=B)
Z
~ {(A=B)
be the set of intuitively
~ (A=C)
valid sentences
ALGEBRAS
AND CURRY ALGEBRAS.
is to be of the algebraic
type.
However
algebras
of open sets, we plan to use algebras
binatory
logic.
2.1~i where
DEFINITION~
o is a binary
Given an applicative we will write
= (A=B~C)} of ~.
VALINITY.
2.1 APPLICATIVE for
~ (A=C)}
operation
on M.
algebra
<M,o>,
"(a,b)".
from indeterminates
An applicative
The formal instead
related
algebra
of using
to Curry's
A polynomial
in
<M,o> of
M
the com
is an algebra M = <M,o>,
then instead of writing
x,y,z,...,elements
semantics
"o(a,b)",
is a term built up and the application
operator. 2.1.2
DEFINITION.
A combinatory
algebra
is an applicative
alge
211
bra
<M,o>
which
and which
is nontrivial
is combinatory
h(x,y,...,z)
in
<M,o>
2.1.3
DEFINITION. <M,o> #
..... z)
k
VxVy[((k,x),y)
(3)
VxYyVz[(((s,x),y),z)
=
are combinatory
assignment
is an algebra
algebra
such that:
C
<M,o,k,s>
=
and
x], =
((x,z),(y,z))]. on combinatory
SATISFACTION
IN CURRY ALGEBRAS. C
that
=
is a function We extend
logic
that the Curry
complete.
Assume sets of M.
a ~M
s,
It was one of the first theorems
2.2
an element
h(x,y ..... z)]
A curry slgebra
(2)
algebras
=
is an applicative
(I)
at least two elements),
that is, to each polynomial
there corresponds
VxVy...Vz[(..~(a,x),y)
such that
(i e. contains
complete;
<M,o,k,s>
pa
pa
is a Curry algebra.
which maps
the sentential
to act on all sentences
of
A
Cproof
variables ~
to sub
by requiring
that: pa(A=B)
=
{m ~M!
pa(A~B)
=
pa(A)
The elements
in
Curry algebra Loosely pa(A)
#
~
logic,
n pa(B). will be called
the "paproofs
of
A
~
is to be formally
for every Curry algebra
C
and
because
quired uniformity
a sentence
A
of the constructive
Cproof
(intuitionistic)
degree of uniformity
(in the
is required.
valid iff
assignment
pa.
charscter
of the
We achieve
by using terms of a first order language
for both the Curry algebras ~C
c pa(B)]}
of
a certain
cally,
÷ (m,n)
C)".
spesking,
However,
pa(A)
Vn[n Epa(A)
and the proofassignments.
is the firstorder
language
~C
Or more
(with equality:
the resuitable specifi
m ) which also
contains the individual the binary
Next, pa
constants: function
the unary relation
symbols:
given a Curry algebra
we form the firstorder
C[pa]
K, S,
(infix)
=
<M,o,k,s,pa(p),pa(q)
associated
to the language
C
symbol:
=
,
P,Q,R .... <M,o,k,s>
and a
Cproof
assignment
C[pa]
is a structure
structure: .... >. ~C"
Clearly
212
In order to relate each formula
A
pa(A)
of
~
to satisfaction in
a formula
~A[X]
if
A
is the sentential variable
if
A
=
(B~C), then
if
A
=
(B=C),
~A[X] =
then ~A[X] =
of
C[pa], we associate to ~C as follows:
p,
then
~A[X] = P(x),~..,
~B[X] ^~c[X], VY[~B[y] =~c[Xy]]
A routine induction then gives us that for every sentence (*)
m
satisfies
~A[X]
in the structure
Finally, given a closed term a Curry algebra
C,
tC
t
C[pa]
iff
A
t
~:
mepa(A).
of the firstorder language
is the denotation of
of
~C
and
in the algebra
C.
We are now ready for the definition of formal validity: 2.2.1
DEFINITION.
A sentence
A
of
~
is formally valid
there is a closed term
t
of the firstorder language
for all Curry algebras
C
and all
iff
L~ such that proofassignments ~ pa:t C e pa(A).
In view of the remark (*), the condition "t C ~pa(A)" can be replaced by: (**)
The firstorder sentence
We let FVAL~ 2.3
~Art] is true in the structure
be the set of sentences of
~
which are formally valid.
AXIOMATIZABILITY OF THE FORMALLY V A L I D S E N T E N C E S Let CA
be the following set of sentences of
IK ~
C[pa].
OF
~.
~C:
S,
VxVy[((Kx)y) ~ x],
~x~y~zE(((Sx) .y) .z) ~ ((xz) (y.z)) ] and
THMcA
be the set of logical consequences of
CA.
[email protected]'a completeness theorem and (**) gives us that: 2.3.1
THEOREM.
is a closed term
t
A formula of
~C
A
of
such that
~
is formally valid iff there ~A It] E THMcAo
An immediate consequence of the above theorem is that recursively enumerable set. following: 2.3.3
THEOREM.
FVAL~
is a
A variant of Craig's lenmm gives us the
There is a recursive axiomatization for
FVAL~.
PROOF. Let S0,S 1 ..... Sn,... be an enumeration of FVAL~ by a (primitive) recursive function. Then for axioms take the following
213
set of sentences
of ~:
{(S0&S0),((SI~SI)&SI),(((S2~S2)&S2)~S2) Clearly allows
the set is recursive. one to conclude
An application conjunction
A
As rule'of
from
of the concept of formal validity
is indeed different pa(A)
p = (q=p~q)
(2)
(p~q) ~ ((p~r)
(3)
(p~q=r)
(4)
(p=p)
3.1
A
requirements
we place on
~
A, for all sentences
of ~, then
A
and
(M)
if
r
~
A
then
in F also occurs
A,
F ~
A ~
B
A
whenever
~
F ~
sequence
of formal
r of sentences
are:
B,
every
sentence
that occurs
in A.
since the conditional
we also require
be a relation
and a finite
A
~
is the intuitionistic
conditional
that
F, A I" B 3.2
Let r ~ A
a sentence
F
validity
pa
A:
~ (p~q&r))
REQUIREMENTS.
between
if
(D)
sentences
DERIVABILITY.
(T)
Furthermore,
and proofassignments
it can
& (p=(q=p)).
of ~.Then the minimal (R)
C
For example
= (p=(q=r))
MINIMAL
derivability
take the rule that
is to show that strong
from conjunction.
is empty for the following
(i)
§3. FORMAL
inference
(((A~A)~...)&A).
be shown that there are Curry algebras such that
.... }
iff
SEMANTICAL
(see 2.2.1),
r ~ (A=B).
CONSEQUENCE. we propose
In view of the definition the following
definition
of formal
for semanti
cal consequence: 3.2.1
DEFINITION.
of the sequence iffthere C
if
B I .... ,B r
is a term
and all
The sentence
blC pa(B I)
of
assignments
'''"
~C
BI,..o,B r ~ A,
such that for all Curry algebras
then '
t
tC bl'"
validates Xl,...,x r
<,A>.
in symbols:
consequence
pa:
,b r Epa(B r)
We shall say that the term
is a semantical
of sentences,
txl ..... Xr
Cproof
A
.. b c pa(A). ' r
the pair
214
Call a pair
a sequent.
Using the formulae 3.2.2 3.3 Let
LEMMA.
A RECURSIVE AXIOMATIZATION
by a (primitive)
following
VALS = {
VALS is a recursively
<
VALS
Define
~A[X] of section 2.2
A}.
one can then show that: enumerable
set.
FOR DERIVABILITY.
<
recursive
function.
Then let
AXMS
of
be the
set of sequents:
{<<(B00~B00) ..... (B0r0~B0r0)>,
(A0~A0)>,<<((BI0gBI0)~BI0)
....
((B!r ~Blr )~Blr )>,((AI~AI)~AI )> .... } Again,
it is clear that
Then let
~
AX~S
is a recursive
set of sequents.
be the smallest relation containing
a sentence of
~}
and closed under
(T),
AXMS
u
{<
is a bijection of
in the usual sense, on (2) is, by definition, F: C
~ Set
B.
onto an equivalence
relation,
C, a diagram
an equivalence relation iff for all functors
preserving
valance relation,
A
In an arbitrary small category finite limits,
in the previous sense,
F
takes
(2) into an equi
in Set:
F(q) F(A)
:> F(B)
F(q') is an equivalence relation in Set. It turns out finite limits,
(see the next section)
the notion of equivalence
that, in a category with relation can be
defined by referring to finite limit diagrams;
'internally'
in particular,
between categories with finite limits, preserving
a functor
finite limits, will
take an equivalence relation in the domain category into one in the codomain category. Let (I) be an equivalence (ordinary)
equivalence B/R
relation in Set, let R c B×B
induced by
tion above.
Let
B ~ B/R the diagram
be the map that takes
(I) on
B
be the set of equivalence classes of
.........
A
is a coequalizer diagram,
q
b
be the
as explained in the definiR,
and let
into its equivalence class.
Then
)
q'
) B
,+ B/R
as it is easily checked.
For this reason,
a
234
coequalizer
of a pair of morphisms
(in any category)
that form an equivalence
is called a quotient of the equivalence
Now, part of the definition
of 'pretopos'
relation
relation.
can be put as follows.
A pretcpcs is a category which has finite
limits,
(colimit of the empty diagram),
of any two objects,
finally,
coequalizers
coproduct
of equivalence
completed by imposing conditions have as basic operations operations: particular
relations;
the fully defined finite
product operations,
above),the
operations.
limit operations object,
of coequalizer
and
~.
G'
logical
is:
~ .).
diagrams
the three
relations
(in this operation,
A morphism of pretoposes,
functor in EMR], and an elementary
functor that preserves
(or:
and the binary co
and finally the nonfull
operation whose domain is the class of equivalence graph consists
is Thus we
the following pretopos
initial
both fully defined,
and
the definition
on the mentioned
in any pretopos
ones mentioned
an initial object
functor
and whose G is +*,
also called a
in [MI],
all the listed pretoposoperations
[M2],
is a
(the meaning
of a functor preserving
an operation was explained
embedding of pretoposes
is a pretopos morphism which is conservative:
if it takes a morphism
in the domainpretopos
the original morphism is an isomorphism The rest of the definition tions put on the pretopos the pretopos
operations
than
of some condi
in properties
of
of the notion of pretopos will
theorem will be proved.
(GSdelDeli~neJoyal
Every small pretopos has a pretopos of
consists
all originating
the definition
and the following
Theorem 1.2.1.
into an isomorphism,
in Set.
In the next section, be completed
A pretopos
too.
of 'pretopos'
operations,
above).
representation
theorem).
embedding into a Cartesian power
Set. This theorem can be read as a definition:
egory having the pretopos
operations
into a Cartesian power of Set. theorem because essentially
is a catembedding
G~del's name is in the name of the
the theorem is closely related to, and in fact,
equivalent
ship is explained equivalent
a pretopos
and having a pretopos
to, G6del's
in detail
completeness
in [MR~.
instead of pretoposes. (as a set of exercises)
This relation
Deligne proved an essentially
theorem in ESGA4J, Exp. VI.,
formulated
A suitably modified
be given in the next section.
theorem.
for coherent
form of Deligne's
The concept of pretopos
in ESGA4, Exp. VI.~.
first advocated the concept of pretopos cal first order logic.
appears
toposes
proof will in passing
It was Andr~ Joyal who
as the basic notion of categori
235
We now consider the nonstrong Let
T
be a pretopos.
the partial
ordering
For
A
of subobjects
will see in the next section, Boolean
complement
~' c Sub(A) minimal
elements,
then
F(¢')
~
in
and
A Boolean pretopos object
A.
i.e.,
appropriate properties;
Properties are freely
is the Boolean in
morphism,
T'
in which Boolean
of pretoposes
complements
algebra for every
between
Boolean pretoposes
B
in our general setting:
+ A
a diagram
for suitable
categories
up to a unique
Semantical
B
~ A ~
is clearly
assured.
in Set.
of certain
in any category
with
(such as pretoposes)
isomorphism
operations
it
B'
limits and colimits
to much of the theory of logical operations.
interchangeable,
an example,
~'
is a pretopos
is a Boolean
of interchangeability
Set are fundamental
The Boolean
it is p r e s e r v e d by
F(~)
is an operation
of uniqueness 1.3.
are the
Sub(A)).
that there is no new notion of m o r p h i s m of
to any m o n o m o r p h i s m
the condition
of
The
is an element
structure.
"Boolean complement" assigns
Sub(A)
morphisms
the additional
if
+ T'
is a pretopos
It is important
Boolean pretoposes: preserve
F: T
in w h i c h
of
denote As we
lattice.
if exists,
moreover,
is the Boolean complement
always exist,
Sub(A)
~ v 9' = 1A (0A, 1A
respectively,
As a consequence, T,
let
(see V.7 in [CWM]).
is a distributive
(if it exists);
latticehomomorphisms. of
A
of Boolean complement. T,
~ ~ Sub(A),
such that ~ ^ #' = 0A, is unique
complement
of
Sub(A)
of an element
and maximal
complement
operation
an object of
(c.f.
in
Limits
IX. 2 in [CWM]).
As
if
ei
f. i
__
Ai
.
, ~ Ci
Bi
(I)
gi in an equalizer
diagram for
i = 1 and 2, then
elx e 2
fl x f2
AIXA 2
> BI×B 2
>
> CI×C 2 gl × g2
is again one. The interchangeability (c.f.
loc. cit.)
equalizer
of finite
limits and filtered colimits
is an important property
diagrams
a directed partial
of Set.
(I) in Set are given for all order,
connecting morphisms
and if, moreover,
~..: A. iJ
i
~ A. j
with
Thus, i~I,
we have,
if the
and
I
for each
is, say, isj,
236
ajkOaij and
similar o n e s
8ij' Yij'
= aik
~ j ~ k)
(i
for the B's and C's such that the diagram ei
A.
+ B. 1
c~ij
iBij
A. j
and
the
similar
ones
with
is again an equalizer Finite colimits Of course,
the
f's
and
products
all
colim C. 1 i
relations
with colimits.
are interchangeable
but not n e c e s s a r i l y
the only positive
features.
Coequalizers with
with more general in general.
(finite
limits;
even
For binary co
fact we can state is that they are interof products
and directed colimits,
ultraproducts.
i¢I. which way.
1
colim gi i
in Set offer less interchangeability
changeable with certain combinations the
then
~i
1
this much cannot be said about coequalizers products,
commute,
diagram.
of equivalence
or infinite)
g's
~ colim B. i 1
they are interchangeable
(quotients)
j
colim
colim e. l 1
colim A. i 1
+ B.
e. J
Let
U
Let
U °p
be an ultrafilter
P ~ Q
be the partial iff
Consider
Q c p,
on a set I, let
and consider
be a set for each
U °p
a category
set
U,
in
in the usual
the diagram U °p
P
Q = P
+ Set
r
' ~ A. ieP z
A.
canon.pro~.
~icQ 1 the colimit
Ai
ordering with underlying
of this diagram
is the ultraproduct
In shorthand:
H Ai/U = colim ~ A.. i~l p~uOP icP z
~ ~ A i , ; , iEP "
of the sets
Ai, iEI.
237
When we choose the standard representation limits, we obtain the usual definition maybe)
of ultraproduct
for the fact that under the present
consider vectors and not just of the
Ai
Moreover, product
i~ P of elements
P = I;
of products
of sets
definition,
aicA i
and co(except,
one has to
indexed by arbitrary
but, when one thinks of the possibility
P~U,
of some
being empty, one sees that ours is the right definitionl). with the standard choices made, we may consider
the ultra
(with given I and U) a functor Set I
+ Set
where the action of the functor on arrows properties we put
of products
and colimits:
is given by the universal
given
fi: Ai
~ Bi
for
i~I,
H fi/U = colim ~ f.. iEI pcuOP icP 1 It is a basic fact that coproducts products:
(disjoint
sums) commute with ultra
if
C. 1
AY 1
is a coproduct
1
diagram in Set for all
icI,
then
~c./u i l
~A./U i i is again one. ultraproduct ultraproducts
~ ~ B i / U I
The reader is invited to check this directly. of copies of the empty set is again empty.
In other words,
commute with finite coproducts.
Since ultraproducts
are combinations
limits, what we said above implies and quotients
Also, the
of equivalence
of products
and directed
co
that they commute with finite limits,
relations.
We therefore
conclude:
238
Theorem 1.3.1.
(Los's theorem)
Any u l t r a p r o d u c t f u n c t o r
()/U: Set I
~ Set
icl
i s a morphism of p r e t o p o s e s . The relation of the last theorem to the 'fundamental property of ultraproducts' Part 2. 1.4.
(Los's theorem in the usual sense) will be explained in
Generating diagrams, and the statement of the main theorem.
Composites of pretopos operations will commute with ultraproducts as a direct consequence of 1.3.1. operation on Set of taking
Such a composite is, e.g., the
(A×B)~ (C×D), whose graph can be construed
as the class of all diagrams
EilY F G
(1)
2
B
D
with (~I,~2), ( ~ , ~ ½ ) being products, (il,i 2) being a coproduct; the domain of the operation is the collection of all tuples (A,B,C,D) of objects. The general concept of a composite operation will be approached in a somewhat abstract manner.
There is a free Boolean pretopos F(G)
on any graph G; e.g. on four objects, A0, B0, CO, DO, as generators. This is defined by a universal property similar (but not identical) to the one used for free algebras in ordinary algebraic situations.
The
above composite operation can be performed in every pretopos, hence in particular in the free Boolean pretopos on
(A0,B0,C0,D0) , and we can
consider its result on the particular argument
having
say E0,F0,G 0 as additional objects and appropriate arrows. Then, the operation in question can be performed in any Boolean pretopos T in the following manner.
Say, the arguments are
the essentially unique pretoposmorphism A 0 to A, etc;
~
~:
A, B, C, D; F(G)
~ T
one finds that takes
will pick out the desired diagram(l) as the values
F(E0), F(F0) , F(G0) and appropriate ones for arrows. In other words, the operation is entirely described by two diagrams, DO: G ~ F(G),
239
and an extension domain,
G'
of it,
D~: G'
> F(G);
here
G
is the type of the
the type of the graph of the operation.
Moreover,
conversely,
every object and morphism
tained as part of a "composite be made clear by explicitly
Boolean pretopos
constructing
F(G)
creasing union of larger and larger diagrams every operation obtained Boolean pretopos
operation",
meaning of the latter. Boolean pretopos
from a
DO
and a
in
F(G)
operation";
as made up of an in
of such composites. D~
Thus,
as above is a "composite
if we have in mind some direct,
Instead, we take the description
as the first approximation
is ob
this could
syntactical
using the free
definition of composite
of a
operation. However,
the free pretopos
the nonfull pretopos
operation
The domain of this operation
construction of quotient
consists
additional
The free Boolean pretopos,
form (2) will not
conditions,
'contain' on
F that
operation will be available. of a pretopos
T
by imposing
a class of morphisms procedure
in
of identifying
the form of passing relation of
T.
relation
morphisms
in it.
conditions
elements
But, if we
'impose'
relation,
the
the
of forming a
analogous
in a (universal) ~
simply because
quotient
on it in the form of inverting
This is entirely to
~/E,
to the algebraic
algebra,
which takes
with
a congruence
E
~.
We will adopt the definition through'
in question,
There is a process
that a composite pretopos
with domain of type the finite graph
G,
a pretopos which is a quotient
pretoposes
to (2) being an equivalence
say F, on a fixed diagram of the
(2) be an equivalence
from the algebra
relation.
(2)
amounting
the operation
(2) will not be an equivalence further condition
of an equivalence
B
)
satisfying
Consider
of diagrams
A
relation.
is not enough.
T, T', Pretop(T,T') T + T';
denotes
means one that
of the free pretopos
to
on
G.
For
the category of pretopos
it is the full subcategory
of all functors from T Mod(T) = Pretop(T,Set).
operation,
'factors
T', with objects
of
(T,T')
the category
the pretopos morphisms.
240
DefinitionProposition Let
G
(i)
1.4.1.
be a small graph.
The free pretopos
over
G
together with a diagram
Fpt (G))
the following
universal
is a pretopos
9: G
property:
+ T
T (= F(G) =
(~ = ~G = ~ t )
for any pretopos
()o~: Pretop (T,T')
T'
with
the functor
.........(G,T')
defined by composition: F: T
> T' }
+ Fo~: G
is an equivalence
of categories.
and is determined
up to an equivalence
(ii)
* T'
The free pretopos
The free Boolean pretopos
over over
G
T',
is faithful,
full on isomorphisms,
free Boolean pretopos equivalence
over
isomorphism
F
over
F: C
g: FC
Ff = g.
~ FC'
surjective.
The
and is determined up to an
is full on isomorphisms there is an isomorphism
if for every f: C
such that we'll
FC
is isomorphic
indicate
to
in part
~ C'
such
D ~ Ob(D) D).
the proof of the proposition;
of the free objects will use formulas.
the weaker statement
(ii) is the presence
The reason
for
of a nonstrong
Boolean complement.
These definitions wellknown
of free objects
diagrammatical structures.
Boolean pretopos, as a consequence there is
and essentially
exists,
~ D
In the next section,
operation,
G
+ (G,T')
is essentially surjective if for all
C c Oh(C)
the construction
algebraic
for any Boolean pre
G.
(A functor
there is
exists,
the functor ()o~: Pretop(T,T')
that
G
is a Boolean pretopos
T (= F(G) = FBpt(G)) together with a diagram Bpt~ = ~G ) with the following universal property: topos
over
G.
M
definitions
of the definition:
of the
of free groups and other free
In either the case of pretopos
we have the following
such that
are natural versions
or that of
form of the universal
given
T'
and any
F: G
property ~ T',
241
G
~
,
F(c)
v T'
commutes up to isomorphism:
there is an i s o m o r p h i s m
~: F
.......
Mo~:
~
and if
then there is a unique i s o m o r p h i s m
v: M .... ~
~ M'
such that
/vocp
M'o~
commutes.
The e s s e n t i a l l y unique
be induced by F,
in
O.
E.g.,
all iso's in
f F
T.
F: C ...... ~ D, in
C
for which if
1.4.2
(i)
Let
~
Ff
for any pretopos
is an i s o m o r p h i s m
Inv(F)
~ T",
Q: T
= Inv(Idc)
(invertible)
= the class of
is said to be obtained
if we have the following universal
~ Pretop (T ,T")
induces an e q u i v a l e n c e of Pretop(T',T") c o n s i s t i n g of those
~ T'
the functor
()oO: Pretop(T',T")
Pretop(T,T")
denotes the c o l l e c t i o n of all
be a c o l l e c t i o n of arrows in the
A pretopos m o r p h i s m
by inverting the morphism8 in property:
M F
Inv(F)
is conservative
is said to
C.
Definition pretopos
with the said p r o p e r t y
and may be denoted by
For a functor those morphisms
M
G: T
onto the full s u b c a t e ~ o r y of ~ r"
for w h i c h
~ c Inv(G).
242
(ii)
~ T', a pretopos morphism, is a quotient morphism
Q: T
(a quotient) if it is obtained by inverting the morphisms in Q
is a finite quotient if there is a finite subset
such that
Q
is obtained by inverting the morphisms in
(iii) Let
T
be a Boolean pretopos.
(finitely) generates morphism
E
T
of
Inv(Q). Inv(Q)
E.
A diagram F: G
~ T
(as a Boolean pretopos) if the pretopos
MF: FNp t(G)
,,~ T
induced by
F
is a finite quotient
morphism. If and
M
is a (universal) algebra, in a given equational class, say,
is an arbitrary subset of IMIxlMI, then there is a universal
E
property similar to 1.4.2 (i) of a morphism
q: M
~ M' being
"obtained by identifying a with b", for all pairs c E: for any M", the map Hom(M',M") ~ Hom(M,M") defined by composition with q
induces a bijection onto the set of those
identify all pairs in quotient M/[E] of M E, with
q
the map taking an element of
class in [E].
M
~ M"
which do
E. Of course, such M' is nothing but the by the congruence relation [E] generated by IMI into its equivalence
(In equational classes, quotient morphism (those that
are "obtained by identifying those elements identified by the morphisms") are the same as regular epimorphisms.) Concerning (iii), note that in the context of an equational class, a map f: X > M generates M, i.e. f(X) generates M in the usual sense, iff the map by
f
F(X)
~ M
from the free algebra on
X
induced
is a quotient. Definition 1.4.3.
operation (ACBPO)
An abstract composite Boolean pretopos
is given by a commutative triangle G
Do
i=inclusion~
> T
/ ~
G' where
G, G'
are finite graphs,
T
is a Boolean pretopos, and both
D O and D~ finitely generate T (see 1.4.2). In any pretopos S~ e.g. S = Set, an ACBPU as shown defines the composite Boolean pretopos
operation of type
(G,G')
whose graph is
K' = Iso{MoD~: M c Pretop(T,S)}.
243
Remark. operation with
C
S
C, and whose
notation
of 1.4.3,
objects
define
in 1.4.3 from an ACBPU
(G,G').
To see this,
morphisms
let
of
is indeed
let us write
C
whose
are the isomorphisms
in
similarly.
the functor
i*:
K'.
Let
(K') is°
is the graph of an operation
i
are With the
of
(G',S) is°
D' e K'},
induces
> K is°.
of type
C.
K = {D'oi:
The functor
an C is°,
objects in
(K') is° be the full subcategory
are the diagrams
K is° c (G,S)
(restriction) K'
as defined of type
any category, for the subcategory
those of whose
K'
in
and
by composition
Note that to say that
(G,G')
is to say that
i
is
full^ and faithful. With F = FBpt(G), F' = FBpt(G') , D = MD0' ~' = MDu" i = M we have that the given ACBPU induces the diagram ( Bpt oi)' F
~ T
F'
commuting
up to an isomorphism.
Passing
to the induced
diagram
^.
D
Pretop(F,S) Is° .
Pretop(T,s)iSo
Pretop (F' ,S) is° we note that ~.
D
^
and
being
(K') is°
image of
(D')*
Since both ^
are full and faithful
quotients,
(K') Is°,
the full replete
.
(D')
it follows
that
is full and faithful.
Definition
1.4.3'
one of the same type
(i)
as a direct
consequence
of
D, D'
i*I(K') is°, the restriction
of
i* to
It follows
that
An operation
i
is full and faithful.
is a restriction of another
if the graph of the first
is contained
in the
graph of the second. (ii)
An operation
in Set of type
commutes w£th ultraproducts filter on I, D.: G' D'
i where
[U]
and Set I
if
the
D': ~' .+ S e t I P r ° ] i > Set (iel)
is the ultraproduct n ()/U: ieI
(G,G')
following has
in
having
when
its
components
all
K',
functor Set I
the graph
holds~
> Set.
then
U
is
K' an ultra
[U~ o D' e K',
244
Theorem
1.4.4.
ultraproducts
Every
Part 2. 2.1.
finitary
is a restriction
operation
of a composite
in Set commuting Boolean
Categories,,,,,for the working
pretopos
with operation.
logicli,an
Sites Let
C
be a category with
finite
topology on
C, J, is a collection
(abbreviated
as (A i
domain
~ A)i)
(such a family
to satisfy
Every
of families
of morphisms
closure
isomorphism
{A i + A: i~I} C with a fixed co
in
a Jcovering of A), with
is called
the following
(i)
A (Grothendieck)
limits.
J
required
conditions:
A' =+ A
is a 1element
covering:
{A' ~+ A} ~ J. (ii)
(Stability
under pullbacks).
Whenever
A.×A iB for all (Closure
(Aik
Aik
.....
then
B
(Ai×A
for all
> Ai
) A
i,
then,
If
(A i
i
through
(A i ..... > A)j
then
It is important of one object, the codomain collection precise
A.
definition
With
If
X
with
The reading
> A
j
with
and
denoting
the
we have
Ai
(A i ~ ~ A
A)j
is
factoring
c J.
A
since
more precisely, a (possibly
of the closure
in the empty set have
empty)
conditions
J
as a
set of morphisms under
the more
should be clear.
of topologies
is jointly
is regarded
Aik
arrows,
Therefore,
we should,
C = Set,
the family
with
> A) i e J
that we may allow the empty family as a covering
is not given,
Examples
is
but not of another.
of pairs
with codomain
there
(A i
> A) i ~ J, and
a family so that for all > A,
If
of the given
~ A)i,k c J. (iv) (Monotonicity). A~j
) B)i E J.
under composition).
(Aik + Ai)ke K E J composite
i,
and
[
I
(iii)
> A) i ~ J
A
A.
is a pullback
(A i
let
J
are the following. be defined by:
surjective,
a site, we always is a topological
(fi: A
i.e. U{Im(fi):
~ A)i~i
ieI} = A.
~ J
When
iff Set
mean this eanonival topology. space,
ordering
(under inclusion)
category
in the usual way, we let
and
C = 0(X)
of the open sets of J
X,
is the partial regarded
be the set of all
as a
(U i ~ U)i such
245
that
U U. = U. i i
The Grothendieck ordinary
A site a topolegy functor
topology
topology
on
(C,J) J
J
X
obtained
in contexts
is a category
on
C
with finite
A morphism of sites
C.
F: C + P
in this way may replace
preserving
the
like sheaf theory.
finite
limits,
(C,J)
limits
together with
~ ~D,J')
is a
and taking Jcovers
into
J'covers: (fi: Ai + A)i c J implies (Ffi: FA i  + FA)i a J'. We also talk about a (J,J')continuous functor, or even a Jcontinuous one, if
J'
is understood. Given any collection
with a fixed codomain a least topology
generated by is a site, covers
J
on
J0'
J0
[such C
or by
J'covers
then the functor
[C,J0).
in
C,
J0"
J, or
(C,J)
each one has
is said to be
Also note that if, in addition,
C + ~
preserving
[we may consider
is a morphism
of morphisms
may be called a pretopology],
containing
and the functor
into
of families
J0
finite
limits
it a morphism
of sites:
(P,J')
takes
(C,J0) + (D,J')],
(C,J) + [D,J').
A finitary topology, or site, is one which is generated by a finitary pretopology, i.e. one in which every covering is finite. that a topology there
J
is finitary
is a finite
Theorem
2.1.1
(Deligne's
small
finitary
there
is a Jcontinuous
Set
which
J0 J
site,
and
Suppose
A = ~A i + A)i
C
by all morphisms J
is a small
C
on
which
using
Let
taking
A
Then in
topology).
One may consider
limits,
J0 o
and
the collection
into canonical
The completeness by
J.
be a
into a family
category with finite
generated
(C,J)
theorem
coverings asserts
that
The theorem will be
the following
Proposition
2.1.2.
I, the terminal
the empty family (C,J) + Set.
C.
c J.
any family not in
are carried
[C,J0) + Set.
is the same as the topology
proved
that
in
C + Set
Note
[A i + A)i~ I ~ J
theorem).
(in the canonical
is a finitary pretopology of all coverings
(A i + A)ici,
completeness
functor
is not a covering
Remark.
just in case for all
such that
I' a I
J0
in
Suppose
object J.
of
(C,J) C,
is a small
is not empty,
finitary
site such
i.e. not covered by
Then there is at least one morphism
246
P~oq,~,:
Let
preserving
LexCC,Set) finite
denote
limits;
the c a t e g o r y of all functors M e Lex(C,Set) coverings.
which,
category is
C + Set.
in addition,
able functor is in
functors
functors
M
of
(C,Set),
into canonical
as a suitable directed co
(C c C).
Since each represent
and directed colimits
Lex(C,Set)
C + S e t
subcategory
carries Jcoverings
¢(C,)
Lex(C,Set),
of members of
of all
a full
We have to construct
The construction will give
limit of r e p r e s e n t a b l e
(C,Set))
the
Lex(C,Set)
are again in
(in the sense of
Lex(C,Set)
[this is a
consequence of d i r e c t e d colimits commuting w i t h finite limits in Set], the c o n s t r u c t i o n will,
at least, ensure that
a separate matter to ensure that Let
S = (S,~)
and let
> CoP
S~
+D
s ~ t 1
....
be a diagram of type
in
C °p .
the Yoneda embedding:
Dt
~ Ds
Let us consider the composite
~ coP
D
M(D)
S ~ts:
~
S
S .....
y
It will be
the coverings as well.
be a directed partial ordering,
D: S .....
(with
M e Lex(C,Set).
M 'respect'
Y
C[
~ (C,Set)
~ C(C,)),
and its colimit:
= colim C(Ds, ) e Lex(C,Set). seS
fi Let
,4 = (A i
~ A)ie i e J.
For
M
to carry
A
into a canonical
covering in Set, it is n e c e s s a r y and sufficient that for every and every
g: D s
~ A,
(*) there are
i ~ I, t ~ s
D
and an arrow
g
commute.
is aaptured for
Let us call a d i a g r a m of the objects diagram
Ds
D': S'
restriction of
D'
to
with S.
A
When condition
in the diagram
(*) holds,
D
by
of the above type consistent
D
A continuation of
is empty. , C
making the diagram
, A i
(This is a matter of inspection.) g
g'
÷A
D t ..... g '
we will say that
s c S
the following hold:
S
a subposet of
D S',
g'
if none
is a d i r e c t e d and
D
the
The main step in the proof is the following
247
Lemma
2.1.3.
Suppose
A = (fi: Ai consistent
D
) A)iE I
is a consistent
is a finite Jcovering
continuation
of
D
in which
Once the lemma is proved, straightforward. in a diagram,
Note that,
it remains D
g
g: D
of
> A, for
A.
the proof of the proposition
diagrams,
is a continuation
with
~
ordinal,
D ,
is
for a covering
in any continuation.
of
and
A. Then there is a
is captured
once an arrow is captured
so captured
is a system of consistent < ~' < B,
diagram,
If
(D)~< B
such that for
then we have an obvious
colimit, U D, ~<~ a consistent start
diagram,
with,
we h a v e
( D O ) ( * ) = 1 e C. the
lemma,
construct
a continuation the
Now,
and using
the
diagram
each
diagram
any
colimit
D . DO ,
consistent
continuation
with
of
m o r p h i s m of sites It remains
to
S = (*), by repeatedly
using
we c a n
in which every arrow with domain in
for every Jcovering.
As we said above,
(C,J)
~ti' gti
that,
An utype co
so constructed will give us a continuation in
this means
D, that
D of
is captured M(D)
is a
~ Set.
to prove
Let us choose, morphisms
also
diagrams,
in which every arrow, with domain any object
for all Jcoverings.
Note
diagram,
construction
is captured
limit of continuations DO
consistent given
a consistent
the original
of
the ]emma.
for any
t e s,
and
to form a pullback ~ts
Dt
.....
) Ds
i e I,
the object
Dti
and
diagram g
~A
T, Dti t
and
*" A .
gti
Then,
for any
u
~uti
making the following
such that commute:
i
s ~ t ~ u
we have a unique
arrow
248 ~LIS J Du
6u t
6t s ' ~
) Dt
)
D
)
S
A
(1)
'If i Duz ~
i
~" A .
Dti
gti
z
j
Ul
If
Dti
is
i e I
all
is n o n  e m p t y , such that
i ~ I
there
is
finite,
and s i n c e
S
for all
i ~ I.
Thus,
(~ti:
Dti
by a x i o m
consistent.
Dt
so is
u ~ t,
is n o n  e m p t y
t I• > s
Dti
is e m p t y
is empty,
Dti i
there
is a c o v e r i n g ,
Du~'z
We c l a i m that
for all
such that
is d i r e c t e d ,
) Dt)ic i (iii),
and Dti
is
t ~ s. is empty.
t ~ s
there
Otherwise, Since
s u c h that
I
is
t ~ ti
for all
i e I.
by a x i o m
(ii) of t o p o l o g y .
contradicting
for
But,
the a s s u m p t i o n
Hence,
that
D
is
The c l a i m is thus shown.
Let us fix
i
as in the claim.
We are r e a d y to c o n s t r u c t
the d e s i r e d
continuation
D'.
The
^
underlying D t' = D t we have
p o s e t w i l l be
for
t e S,
ensured
that
is g i v e n as follows:
S' = S ~
and D'
{t: t~s}
D t' = Dti
for
is c o n s i s t e n t .
for any
= S u {t: t~s}.
t > s.
Thus,
The p a r t i a l
We put
by the claim,
ordering
on
S'
x,y e S',
x < y in S' ~=* e i t h e r
x, y c S and x < y in S ^
One v e r i f i e s The r e m a i n i n g
easily
or
x = t, y = fl and t < u in S
or
x E S, y = t and x < t in S.
that this
is,
data of the d i a g r a m
indeed, D'
6 '^ = fit ~uti ~t
= ~ti
a directed partial
ordering.
are g i v e n as follows:
(t < u) ° ~uti
(t ~ u)
(see d i a g r a m ( 1 ) ) . The c o m m u t a t i v i t y o f t h e l e f t s q u a r e in (1) i s used t o show t h a t D' so c o n s t i t u t e d i s , i n f a c t , a f u n c t o r . Now, l o o k i n g a t ( 1 ) , we see t h a t g i s c a p t u r e d f o r A in D' by
gti'
c o m p l e t i n g t h e p r o o f o f t h e 1emma.
• 2.1.3. •
2.1.2.
249
Theorem Lemma 2.1.3. collection
2.1.1 will now be reduced to Proposition Let
J
be a topology on
of objects of
C.
Let
J'
for which there in
X.
Then
Let
X'
and an extension
A' = (A i i e I'  I.
1
e X'
Paoof:
for all
It is immediate
Conversely,
it suffices
satisfying
the condition
A ~ X. A'
+ A) iE I e J'
A.
X
A
~ A
generated by J'
of
A
into an object
that
such
satisfying
is a topology;
A' A I' = I
A' e J
the condition
to show that the collection
J
can he
of all objects
if and only if there is
÷ A)ie I
that any
2.1.2.
an arbitrary
Then
be the collection
is at least one arrow
A = (A i
and
be the topology
together with the empty cover for each described as follows.
C,
and
is in
J'
of the families
this is an easy verification. m 2.1.3.
Lemma 2.1.4. domain
Suppose
the family
I, the terminal
topology
J'
(Ai N
object of
generated by
J
÷ l)iE I
with the joint co
C, is not a Jcovering.
and the empty coverings
Then in the
for the objects
A i, 1 is nonempty.
Proo,,f:
Suppose
Jcovering
Bj
I is empty in
(Bj~
~ Ai. J
> l)j
to some
J'.
Then, by Lemma 2.1.3,
such that for each
i. e I. J
j,
there
is a
there is an arrow
Since
A.
Bj
I
commutes
(I being terminal),
covering,
contrary
by axiom
(iv)
(A i
>1) i
isaJ
to the assumption. 12.1.4.
Proof of Theo,rem 2.1.1.: We consider the comma category C/A (denoted in [CWM]). C/A has finite limits; the terminal object in C/A
C+A
is the arrow
id: A
~ A.
We have the functor
250
F:
C/A CxA
C Ct
1~2 A C×A_
C fll
+ fxid A
C' C'xA
embedding
C
into
C/A;
topology J/A on C/A
Bk
F
F
preserves
"~A
kcK
is a morphism of sites
always a pullback
finite limits.
We define the
as follows:
(C,J)
diagram in
'~ B)kc K c J
e J/A ~=~ (Bk
~ (C/A, J/A).
The following
is
J/A:
A' .,
+
A'×A
(z) + AxA ~=All the above facts can be verified easily,
if not already known.
Now, assume the hypotheses
The family
of the theorem. fi
I is not a covering J' on fi: Ai
C/A
on
IC/A
generated by ~ A e Ob(C/A),
Ai
.~ A   A
in J/A
J/A.
/ie I By Lemma 2.1.4,
in the topology
and the empty covering of each
the terminal object
Ic/A
is not empty.
By
251
Proposition
2.1.2, there is a m o r p h i s m of sites M':
By the definition M = M'oF: fixed
C
of
M
A' = Ai,
into a pullback
~ Set.
J', M'(f i) = 0,
+ Set.
i • I,
(¢/A,J')
for all
is a m o r p h i s m f = fi'
in Set by M',
M' (fi)
i c I.
(C,J)
Now,
~ Set.
the pullback
diagram
let
With any
(2) is taken
i.e. the diagram
,
M(A i)
,
j
M(f i)
, M (A)
1 = M' (1C/A) M'(~)
is a pullback of
M(A)
in Set.
Since
picked out by
i e I, i.e.,
M'(f i) = 0, this means
M'(4)
that the element
is not in the image of any
the image of the family
A
under
M
M(fi) ,
is not a canonical
covering. 2.1.1. 2.2.
Coherent Let
C
categories. be a category with finite
Then the subobjects denoted
Sub(A).
(see [CWM])
In fact,
meet of the subobjects being represented by
Sub(A)
represented D .... ~ S
f: A'
f*FB
A
an object
in
(^) semilattice, B
C.
ordered set, > A,
C
the ~ A
diagram
~ A
~ A,
f*: S u b A is defined by pullback:
is a meet
by the monos
from a pullback
B
Given an arrow
limits,
of A form a partially
the map
(i)
~ Sub A' ~ A] = [B'
~ A']
where
252
is a pullback,
f*
operation, and each
~ A'
that
"C
Sub A (A e Oh(C))
other words,
element
~f(~)
~f(~)
preserving
for
~ ~
iff
~ ~ f*(~)
extremal
the minimal denoted
of
for all
is the least subobject
~
Sub A'
3f(iA, ).
element
of
To say that
this circumstance
epimorphism.
element.
3f,
of
, e Sub A; A
Im(f)
= 1A
A
unless
of
~f,
in
such that by
IA,,
means
let us
that
f
for all
f,
f
is an
is equivalent
to saying that every m o r p h i s m can be factored as the composition extremal
epi followed by a mono.
isomorphism any
g: B
Beakcondition ÷ A
for
~
means
of an
that a m o r p h i s m
just in case it is both a mono and an extremal
The socalled above,
It is immediate
does
the mono is an
is referred to by saying that
The existence
means
A, mf(~), n e c e s s a r i l y
the maximal
not factor through any mono with codomain isomorphism;
(having a join
(sup of empty family of subobjects),
~ e Sub A' there is a subobject
Denoting
Im(f)
finite sups of subobjects"
is a lattice
(I) of posets having a left adjoint,
such that
~ f*(~).
of meet semilattices.
has stable
is a lattice h o m o m o r p h i s m
The map
write
B'
v) with a minimal f*
that for every unique,
~A
is a h o m o m o r p h i s m
By the condition we mean that each
B
is an
epi.
that for any
f
as
and a pullback A'
f
+ A
(2) B' the
+B
h
diagram Sub A'
~f
+ Sub A
Sub B'
,~ S u b
B
3h
commutes. f
This
is equivalent
is an extremal
epi,
the Beckcondition,
to saying
then so is
we say that
h. C
that whenever If
3f
"has stable
in the pullback
always exists images".
(2)
and satisfies
253
A category suitable for finitary coherent
simply: T
(geometric) logic,
a coherent category (in [MR]: "logical category")
or
is a category
with finite limits having stable finite sups of subobjects and stable
images. Let
T
be a coherent category.
collection of all families that some finite
I c I'
Define
(fi: Ai
÷ A)iei
is a topology on
T.
to be the
of morphisms of
T such
we have V Im(f i) i~I
J
J (= JT)
=
1 A.
Indeed, axiom (ii) follows by the stability
conditions on sups and images. following two formulas:
Axiom (iii) easily follows from the
for morphisms
A"
g
A'
f
+ A
we have
~fg (¢) = ~f(~g(¢))
(~ c Sub A")
and for f
A'
+A
we have
3f(iYl~i) = iYl3f(~i )
(I finite, ~i ~ Sub A').
The two formulas are, in turn, easily checked.
Axioms
(i) and (iv) are
immediate. Note that a monomorphism is a covering in
J
(as a singleton)
iff
it is an isomorphism. For a functor
F: T
~ T'
serving finite limits, saying that (T,J T)
~ (T',JT,)
between coherent F
T
and
F
preserves finite
sups and images (this latter condition means, naturally,
the order preserving map induced by
that for
~ SUbT,(FA) F, FA
is a lattice homomorphism
preserving minimal elements, and that whenever then
pre
is morphism of sites
is equivalent to saying that
FA: Sub T A
T'
f: A'
~ A
is in
T,
254 Fa (3f(q~))
= 3Ff(FA(¢) )
(~ e Sub A ) ) .
The truth of this assertion is quite clear. naturally,
a coherent functor,
in ~MR]).
The category of all coherent
Coh(T,T');
it is a full subcategory of
T
, Set is a model of A functor
T;
if
Ff
have finite limits and
or coherent morphism T
, T'
(T,T').
Coh(T,Set)
is denoted
being an isomorphism implies that
implies that
F
F
preserves
f
F
may be called,
("logical functor", is denoted
A coherent functor
reflects isomorphisms
F: T + T'
T'
Such an
Mod(T). (or is conservative)
is an isomorphism.
them, then
If T,
F being conservative
is faithful.
One can apply Theorem 2.1.1 to conclude that for any small coherent category isomorphism,
T,
and any monomorphism in
there is a coherent
T
T
which is not an
, Set which takes the given mono
morphism into a mono which is, again, not an isomorphism. several such coherent
functors together,
Putting
one for each mono in
T
that
is not an isomorphism, we obtain Theorem 2.2.2.
(Completeness
for finitary coherent logic).
Any small
coherent category has a conservative coherent embedding into a small Cartesian power of Set. • 2.2.2. One should note the easily seen facts that if coherent and conservative, T
as well;
T',
moreover,
T,
is
if its
then the original diagram is a limit similar reflection properties hold
for sups and images.
This means that any
objects and morphisms
involving commutation of diagrams,
sups of subobjects
+ T'
then, for any finite diagram in
Fimage is a limit diagram in diagram in
F: T
'diagrammatic property'
of
finite limits,
and images that holds throughout in Set, and hence
in any Cartesian power of Set, will hold in any coherent category as well.
This fact expresses
'coherent category' Definition
2.2.3.
(fi: Ai + A)iei
for the
the completeness
of the defining axioms for
'standard coherent logic', that of Set.
In a category with finite limits, a family is said to be an effective epimorphic family
following holds, with
if the
255
A Ai A J a pullback diagram for all for any object
i,j e I:
and any system (gi: Ai , B)ie i of morphisms, if gifij = gj f!. z3 for all i and j in I, then there is a unique morphism h: A + B such that hfi = gi for all i in
B
I.
pr0position 2.2.4.
The topology
JT
defined above in a coherent
T
coincides with the socalled precanonical topology, i.e. the one in which a family is a covering iff it contains a finite effective epimorphic family. The fact that any family in for
T = Set,
general
T,
JT
is effective epimorphic is seen,
by a careful inspection.
Then the same fact, for a
is essentially a consequence of completeness,
2.2.2.
The
detailed proof of 2.2.4 is postponed until Section 2.5. 2.3.
The quotientconservative
factorization.
The definition of a quotientmorphism is given in Section 1.4 in the context of pretoposes.
The definition has a general character;
can be repeated in other, similar, situations.
Such a situation is
given by a (concrete)
certain categories
2category
"objects" of the 2category),
(c.f. [CWM]):
it (the
certain functors between them (the
"morphisms" of the 2category), and certain natural transformations between those functors
(the "2cells" of the 2category).
is the 2category of pretoposes, pretoposmorphisms, transformations between such.
E.g., Pretop
and all natural
Similarly, we may talk about Lex, the
2category of categories with finite limits, functors preserving them ("Lexmorphisms") and all natural transformations between such. The 2category of coherent categories is
Coh.
The 'standard' 2category is Cat, the 2category of categories, functors and natural transformations. We want also of
Set,
the category of sets to be an object of
Lex, Pretop, etc.
theoretical universes:
[email protected] !
Cat, and
Therefore, we have in mind three setfor
i = ~, i, 2,
with
@0 < @i < 82
256
inaccessible
cardinals.
Ob(Set)
= [email protected] , Set e Ob(Cat)
c [email protected],
Cat e [email protected]
. Small categories are those in [email protected] 2 Note the easily seen but important fact that a m o r p h i s m which
a quotient
and conservative
at the same time is an equivalence;
is true as a direct consequence
of the definition,
and holds
is
this
in any of
our 2categories. Another I: T any
obvious
~ T'
of the definition
is a quotient morphism
T" ~ D,
()oi: D(T',T")
result essentially T" = Set,
consequence
if
I: T
in [MR]
+ T'
is full and faithful,
in the 2category
+~(T,T")
contained I
is that for
placing
then,
for
D = Pretop,
A and
I* = ()of: Mod T' + Mod T
is a quotient
First, we will be concerned with Lex; concept of a q u o t i e n t  m o r p h i s m
D,
is full and faithful.
is such that
then
is that if
(cf.
§ 3.1
below).
as we said above,
the
is defined as in 1.4.2, with Lex re
Pretop.
Proposition
2.3.1.
Any Lexmorphism
> T'
F: T
can be factorized
into Lexmorphisms: F
T
+T'
T"
F = FoQ,
so that
Paoo~:
Let
objects
as
Q
is a quotient,
z = Inv(F) T.
c Morph(T).
and
r
is conservative.
We define
T"
to have
the same
The morphisms A ..........÷ B
of
T"
will be equivalence
classes
of pairs
(f,s) as shown:
C
with varying relation:
C
and with
s ~ Z, under the following
equivalence
257
(f,s) ~
(f',s') ~=~ (Ff)(Fs) I = (Ff')(Fs') I
(remember
that for
s e E, Fs
is an isomorphism).
To have a simpler
notation,
for the right hand side of the last defining
equivalence
we
rather write fs I = T' and read:
"fs I equals
f's 'I
f,(s,) I,
in
T'".
Thus,
(f,s) ~ (f',s') ~=~ fs I = T' This clearly defines f e T(C,A), For
(f,s)/~,
The letters arrows
in
an equivalence
s ~ T(C,B)
n ~,
the m o r p h i s m s, t, u always
the definition
f's 'I
relation on the pairs
with fixed
A
represented by
is:
and
B
(f,s) with
but varying
(f,s), we'll write
stand for m o r p h i s m in
E;
C.
rfsl~.
in diagrams,
E are indicated by double arrows.
To define
composition,
what we need to show is that,
in the
situation
the morphism Consider
gtlfs I
is equal,
in T',
to one of the form
hu I.
the extension
with the pullback
indicated;
this diagram is in
T,
except
for the
258
dotted arrows.
Reading
it in
in
follows
is an iso in
that
t'
T',
T',
it is commutative
we also have the dotted arrows;
and the pullback T',
since
remains
a pullback.
t
thus,
is;
It
t' e Z.
Now, it is clear that
(gt 1)(fs 1) = ( g f ' ) ( s t ' )  i T'
Therefore,
we define: rgtl~
o
rfsln
= r(gf,) (st,)l~.
T" This definition equivalence
is legitimate;
classes
again be equivalent, in
if different
are chosen,
simply because
of the
righthandsides
will
they will be the same morphism
T'
The identity morphisms
in
A
T'
are given in the obvious way:
,~°°°°~
A
So far, we have defined the category as the identity on objects, The functor F takes objects), and rfsl~ into F
representatives
the resulting
is welldefined,
and takes
it is faithful,
if
fs I
becomes
is an iso in
T',
then
f
f ¢ Z;
the inverse of
f: A
The functor ~ B
and
F = FoQ.
Q
T'
we can form
acts
Q, i.e.
(since rsfl~,
on clearly,
is also
an isomorphism under
is an iso in
therefore,
Q
into rf(idA )I~
A into F(A) (so that F = FoQ fs I in T', i.e. (Ff)(Fs)I;
conservative: T'), hence
T".
s
fs I
is an iso in
and it is clearly
rfsl'.
Let us check that
T"
has equalizers,
First of all, we observe that,
and that
given any two parallel
F
preserves
them.
morphisms
,.ooo,oo°o~ ,oo°°°o°°°~
in with
T", the
we may represent same
s.
In
fact,
them as consider
rfsln
and
rgsl~
simultaneously
259
t'
S
It is clear that, with rgtl~ = r(gs,)u I".
u = st' = ts',
Given two parallel let
e
morphisms
be the equalizer
equalizers•
and
s
of
f
~
and
is an iso in
=
we have
rfsl~
rfsl~
~
,
g (in T);
T',
es
=
since
= r(ft')u I~,
rgsll F
is the equalizer
in
T"
,
preserves of
fs I
and gs I in T' Therefore, we are left with the task of showing that se = rse(id) I~ is an equalizer of rfsl~, rgsl~ in T". To do this, Consider
we consider
the following
a morphism diagram
in
y = rhu I~
satisfying
~¥ = ~¥.
T:
j\
(the dotted pullback
arrows
with
the composite
s ~
appear only in and is
that they are equal
h;
e'
T'
r(fh')(us') I~,
that
T").
~y
is
s' of
e'
gh'
and fh'
h' and
r(gh')(us') I~.
is to say that they are equal in fh' = T'
in other words,
and
is the equalizer
form a gh'
T', i.e. that
;
is an isomorphism
in
T'
i.e
that
Now,
TO say
e' ~
260
There is a unique arrow commute,
i
by the definition
is commutative.
in
T
of
making the square containing
e.
The whole diagram,
se
is a monomorphism
is easily seen), completes
in
the uniqueness
the verification
The verification
T',
T'
o se.
and hence in
T"
of the factorization
of equalizers
of the terminal
in
as well
(as it
also follows.
This
T".
object and products
well as the remaining parts of the facts that morphisms
when read in
Hence = rhu I~ = ri(us'e')l'~
Since
it
Q
and
Y
in
T",
as
are Lex
are left to the reader.
It is clear that
Inv(Q)
= Inv(F).
Since
doing the bare uinimum to invert the morphisms intuitively rigorously
clear that
Q
is a quotient;
T" in
is constructed by Inv(Q),
it is
it is not hard to prove it
either. •
Proposition
2.3.1,.
essentially
unique.
The quotientconservative More particularly,
factorization
in
2.3.1.
Lex
is
if in
T'
Q, Q
are quotients,
r, r
are conservative, ^
and we have an isomorphism
A
r Q= r Q, then there are an equivalence
functor
r~rE
whose composite
E
(i) and isomorphisms
261
is
t h e one g i v e n i n
Proof:
(I).
Straightforward. •
Let us work in a fixed category
graph of a morphism
f: A ~
B
T
with finite
is the subobject
limits.
2.3.1'
The
represented
by the
monomorphism:
A functional subobject monomorphism
X
~
~ AxB
A
of
A×B
is an isomorphism. in
by a
~ AxB
, A
(Verify that these concepts
mean the expected
Set.)
Mapping morphism,
T(A,B)
into
Sub(AxB)
we obtain a bijection
subobjects by
is one that is represented
for which the composite X
things
> AxB.
of
i: X
AxB.
> A×B,
In fact,
> B: the canonical
Proposition
2.3.2.
g: A'
~ A
in
T
graph is
subobject i
is
represented
~2i(~li)i
projection). A Lexmorphism
if and only if the following f: X
onto the set of functional
given a functional
(Andrew Pitts)
morphism of the form
by taking the graph of the
T(A,B)
the m o r p h i s m whose
(~2: AxB
a quotient
of
> FA
in
and an isomorphism f
X
F: T
condition holds: T',
there
X m FA'
> T'
is
for any
is a m o r p h i s m such that
~ FA
commutes. Paoo~:
Note that the particular
Q: T
+ T",
the first
factor
ization of any Lexmorphism, uniqueness
in 2.3.1 of the quotient
in the quotientconservative
satisfies
of this factorization
the condition holds
construction
the condition.
(2.3.1'.),
factor
Now, by the
one easily deduces
for any quotientmorphism.
that
262
Conversely,
assume
quotientconservative
that the condition holds,
and consider the
factorization F
T
+T'
K/ T"
First, we may easily verify that as the one assumed for an equivalence.
such that
of
f.
+ rB
Let
h: A f
rh = f,
is that
T'.
is a monomorphism,
and
hence
in
the same condition
rg g
+ B
there
and
represents
r
rg
g: C
have the same graph,
in
T
is
Suppose
> A×B
represents
hence,
subobject
whose
in
the graph
subobject,
a functional
be the m o r p h i s m
r
its graph
is
a functional
represents
is full.
Consider
By the condition,
r is conservative,
A×B.
But then rh rg;
£g
In particular,
since of
f: rA
~ FA×rB ~ £(A×B).
T"
£ satisfies
Next, we use this fact to show that
The only thing to prove
we have a morphism £A
F.
[g]
graph is
[g].
namely the one represented by
as required. • 2.3.2.
Proposition a coherent
2.3.3.
Suppose
functor.
Then,
ization of
F
in
Lex
T, T'
T"
It follows
categories,
F
and the functors
~ T'
Q
and r
are,
that the Lex quotientconservative
coherent m o r p h i s m
~ T'
F: T factor
from 2.3.1: T
we find that
are coherent
taking the quotientconservative
is already a q u o t i e n t  c o n s e r v a t i v e
in fact,
coherent.
factorization
of a
factorization
in Coh.
Proof:
Let us note first that
QA: Sub(A)
> Sub(QA)
every subobject Qf: QA'
> QA.
~
of
Q
induced by QA
Consider
is full on subobjects: Q
is surjective.
is represented the imagemono
the function
Indeed, by 2.3.2,
by a mono of the form factorization
of
f
in
263
T: A'
q ~ A"
i ÷ A,
mono as well.
Since also,
morphism.
Since
morphism.
But then
thus
F
f: A
preserves
Fq
Qi
Ff m FQf
is an extremal
represents
and
is a mono, epi,
Fq
is a
Fq
is an iso
Fq m FQq, Qq
is an iso
the same subobject
of
QA
as
Qf;
as required. ~ B,
extremal
Since
is conservative,
# = QA([i]), Let
iq = f.
and
~ ~ Sub(A),
be in
T.
epis, by the subobjectfullness
To verify
of
Q,
that
Q
it suffices
to show the equivalence
Q(~f(¢)) But, applying preserves
r
Since that
to both sides,
extremal
the equivalence T"
Q
~ Q(~) "=~ Q(¢)
epis;
~ Q(f*(~))
(~ E Sub B).
the equivalence
since
F
becomes
is conservative,
true since
it follows
F
that
itself holds. is, also,
has images
heSS of
F,
pullback
of images
essentially
and that
it easily
F
follows
them.
that the property
is reflected
Similar arguments
surjective
preserves from
T'
to
(see 2.3.2)
it follows
By the conservativeof stability under
T".
apply to sups of subobjects. • 2.3.3.
Corollar Z 2.3.4.
If a C o h  m o r p h i s m
is a Lexquotient,
then it is a
Cohquotient. m 2.3.4. Corollar[
2.3.5.
on subobjects
Proof:
The
'if' part, 2.3.4.
A Cohmorphism
and essentially
'only if' part assume that
it suffices
satisfies Let X m FA';
f: X
~ FA
follows
Coh
iff it is full
from the proof of 2.3.3.
F: T + T' F
satisfies
For the
the conditions.
is a Lexquotient,
i.e.,
be given.
Find
A'
and an isomorphism
the composite FA' m X
f', the subobject
:
FA'
~ FA'xFA
FA'×FA,
for some mono
of
f ~ FA.
FA'×FA
represented by
is represented by i: A" ........~..A'xA.
By
that it
of 2.3.2.
f': The graph of
in
surjective.
to show that
the condition consider
is a quotient
Fi: FA" + F(A'xA)
But then the diagram
264
Fi
F~ 2
FA"
÷ F(A'xA)
~ FA
F~ 1
X commutes,
and
so does +
FA
FA" as
required. • 2.3.5.
Corollary
2.3.6.
then
is Boolean as well.
T'
Proof:
If
F: T + T'
Straightforward
is a Cohquotient,
and
T
is Boolean,
from 2.3.5. i 2.3.6.
2.4.
Pretoposes. Every category
the context
in this section has
is not explicitly mentioned,
(at least)
finite
limits.
If
we work in a fixed category
with finite limits. The ith projection
AlX'''XAn + Ai
is denoted by
~i'
and
for the morphism
<~il' we write Definition
• ..
, ~ i k >: A l X ' ' ' x A n
~ Ail
×.
"'×Aik
~il...ik" 2.4.1.
An equivalence relation is a pair of parallel
morphisms )
R
such that
i =:
R
+ AxA
+
A
is a monomorphism,
and such that if
265
we also write
R
for the subobject
of
A×A
represented by
i,
then
we have the following: AA s R
(reflexivity)
=21*(R)
= R
(~21: AxA
=I2*(R)^~23*(R)
> A;
symmetry)
~ ~I3*(R)
(~12' ~23' ~13: A x A x A   ~ One immediately in 1.2.
notes that in
A×A;
Set,
It is clear that an equivalence
transitivity).
this is the definition relation
given
in the sense of the
last definition
is taken into another one by any functor preserving
finite limits;
hence the condition of 2.4.1 implies
condition
in 1.2.
conservative relation
if a pair of parallel
functor preserving
according
relation too. limits',
Also,
to 2.4.1,
It follows
the special
two definitions
the defining
morphisms
is taken by a
finite limits into an equivalence
then the original pair is an equivalence
(by the
'completeness
of the logic of finite
case of 2.1.1 for the trivial
topology)
that the
are equivalent.
Next note that the kernel pair of any morphism is always an equivalence
relation,
equivalence
relation
equivalence
relations
since this is true in Set.
could be defined by the condition
taken into a kernel pair by some conservative limits. It is easy to see that if a morphism pair of morphisms, effective 2.2.3.
it is a coequalizer
is called an effective epi:
morphism
epi iff
{f}
2.4.2.
an initial object, equivalence (i) codomain (2)
a morphism
epimorphic
finite
of some f
Such a is an
family according
epis are extremal
to
epis.
A pretopos
is a category with finite limits having
coproducts
of pairs of objects,
relations,
and satisfying
The initial object O
that they are
is a coequalizer
clearly,
is an effective
(checkl),
functor preserving
of its own kernel pair.
It is also easily seen that effective
Definition
Since in Set, any
is the kernel pair of its own coequalizer
O
the following
is strict,
is an isomorphism; If
C
A
B
and coequalizers additional
of
conditions:
i.e. every morphism with
266
is a coproduct [i], by
diagram,
then
[i'] they represent, O
are
i, i'
monos,
[i] ^ [i'3 = O C
and
for
the
(0 c = subobject
subobjects represented
* C); If i n
(3)
C A
B
p.b.
p.b.
^
A
i)
i I ^
Lemma Let
form a coproduct
i, i'
then
also
diagram,
Any pullback
(5)
Any equivalence
2.4.3.
universal
A
and
B
are pullbacks
as shown,
form a coproduct;
(4)
of an effective relation
In a pretopos,
s: A
B
C
let a morphism
be the coequalizer
property
epi is again
the coequalizer,
an effective
epi,
is a kernel pair of its coequalizer. f: A
~ B
be given.
of the kernel pair of we have a unique
i: C
f; ~
by the B such
that A
f
,,), B
C
commutes.
Then
effective
s
and
(hence extremal)
We know that mono
is postponed
Proposition (ii)
i
s
A functor
epi followed
is an effective
until
2.4.4.
give a factorization
(i)
between
of
f
into an
by a mono. epi.
The proof
that
i
is a
later. Every pretopos pretoposes
is a coherent
is a Cohmorphism
category. iff it is a
Pretopmorphism.
Proof:
(i).
the supposed
Let
T
stability
It is immediate
be a pretopos. of effective that
OA,
T
has stable
images by 2.4.3 and
epis under pullback
represented
by
O
(2.4.2.(4)). ) A,
is the
267
minimal
subobject
of
A;
its p u l l b a c k
along
any
A'

~ A
is 0A,,
Let
two s u b 
by 2.4.2(1). Here objects
is
of
A
the
construction
be represented
f o r m the c o p r o d u c t C
~ B~C.
of
B~C,
We h a v e
joins
by the
with
of
subobjects.
monos
B + A,
canonical
a unique
arrow
C
injections
Bn C
. + A
~ A.
L e t us
B ....., B ~ C, making
the
following
commute:
B
A.
Now,
let us
f o r m the
effective
epi
 mono
B~C
We
claim
that
E ......... ~ A it,
then
D
, A
is any m o n o there
tive
diagram
Now,
if we
of
B~C
~ A:
÷ A
represents
such
that
is a u n i q u e
B~C
f o r m the
factorization
effective
both
epi
the
desired
B + A,
...... ~ E
 mono
B~C
\o/
such
join. C
that we h a v e
factorization
~ E
Indeed,
.~ A
of
if
factor
through
the c o m m u t a 
B~C
.....+ E,
268
then,
since
E
D'
> E
) A
B~C
, A,
means that
) A
is a mono,
B~C
form an effective
~
D
~ > A
factors
The stability
2.4.2(3)
and
and the composite
epi  mono factorization
which has to be isomorphic
proved.
D'
through
to the one under
E
of
(I).
This
) A, which was to he
of sups under p u l l b a c k
is a consequence
of
(4).
(ii).
The construction
of images and joins
it clear that any P r e t o p  m o r p h i s m For the converse,
what we need is a C o h  c h a r a c t e r i z a t i o n
effective
epis and coproducts
category,
and hence
in a pretopos.
in a pretopos,
makes of
Note that in a coherent
a m o r p h i s m is an effective
it is an extremal
epi, as a special
in a pretopos
a diagram
T
in a pretopos
is a Cohmorphism.
case of 2.2.4.
epi iff
Also, by 2.4.2(2),
C
A is a coproduct
diagram iff
meet of the subobjects morphisms
(a) A
~ C, B + C
represented by them is
form an effective
same diagram is a coproduct B + C
B
form a covering
epimorphic
family.
diagram iff
in
JT"
are monos, OC,
and
Hence,
(a) and
(b)
(c)
by 2.2.4,
(b) hold,
and
Thus, we have the required
the
the two the
A + C, Coh
characterizations. Given a coherent m o r p h i s m T'
we now see that
it remains
F
to see that
relations.
Let,
F: T + T'
preserves F
finite
preserves
between pretoposes
T
and
limits and finite coproducts
coequalizers
of equivalence
in f
R
A
h
÷B,
g h
be a coequalizer
of the pair
(f,g), an equivalence
by the above, F preserves effective epis, Fh hence a coequalizer of any of its kernel pairs. of the definition
of pretopos,
(Ff, Fg)
is a kernel pair of
equalizer
of
(Ff, Fg)
relation.
(f,g) is a kernel pair of
h;
Fh.
is a co
We conclude
that
Fh
hence
as desired. •
Proposition
2.4.5.
For any coherent
pretopo8 aompletion of
Since,
is an effective epi, By the last clause(5)
T
category,
we have the
2.4.4.
(free)
269
YT: T ~ + with
P(T)
universal
a pretopos, property:
¥ = ¥T
P(T)
a Cohmorphism,
for any pretopos
T',
the functor
) Coh(T,T')
( ) o ¥: Pretop(P(T),T') is an equivalence Moreover, full),
of categories. y
is conservative,
and "every object of
object
X
of
the form Proe~:
~(r)
(Y(Ai)
P(T)
full on subobjects
is covered by
there is a finite effective
It resembles
It is convenient relations
that the final result
of the quotientfield sums;
(Cohds for short),
satisfying
conditions
is a Cohmorphism
(4) in 2.4.2 holds.
(1),
Cohqe(T)
as follows.
A Cohdscat
preserving
for short,
is a coherent
We prove two propositions.
is also a Cohdscategory,
Let us put ourselves
and
One is our
the other with Pretop has the addendum that
provided
T
is a Cohdscategory.
imply the one to be proved.
To carry out the first construction,
let us make some preliminary
into a Cohdscategory.
Morphisms
the form
between
finite disjoint
tuples of the form
a
the initial
relation has a coequalizer,
The second proposition
~A. i i
of
A coherent category with quotients of
It is clear that the two propositions observations.
of
It turns out
(2) and (3) in 2.4.2;
with Pretop replaced by Cohds,
replaced by Cohqe.
In the
completion.
(automatically
in which every equivalence
proposition,
domain.
into two steps.
category having an initial object and coproducts
equivalence relations, a Cohqecategory condition
of an integral
we define coherent categories with disjoint
object and binary coproducts). category
in algebra of the
in the second, we adjoin quotients
is the desired pretopos
sums, and their 2category
Cohdsmorphism
family of
long when written
to the result of the first step.
To make this precise, egory is a coherent
although
to break the construction
first, we adjoin disjoint
any two objects,
epimorphic
any one of a number of proofs
type of the construction
equivalence
(hence also
¥", i.e. for every
~ X)i~i.
The proof is quite straightforward,
out fully.
and the following
÷ ~ B. j j
sums are in onetoone
correspondence
with
of
270
* ~. B j> i •
1
Moreover,
J
a morphism of the form
gives rise to
f: A
fj,
j
from the pullback f
A
B.
J
i. J
J f.
J
A.
+B.
3 and in fact, the ~j disjoint fj: Aj for all
form a disjoint
sum representation ~ Bj,
3 sum diagram,
(Aj ~ A)j
one gets a unique
f
of
Conversely,
A,
given a
and morphisms
making the last diagram commute
j.
This suggests the following definition are. finite (possibly empty) a tuple is denoted =~A." i
tuples (Ai) i A morphism
Cohds(T). Ai
Its objects
of
T;
such
1 r~A
i
is given by:
for
of objects
a disjoint
~


+
i
r~B j
sum representation
~ j
(Aij
....~.. Ai) j
of each
Ai, and a morphism Aij ~ Bj for each i and j. Two such morphisms, one as above, the other with primed items, are identified if and only if there is a system of isomorphisms
A i m A!I, A.Ij ~ A~.Ij,
such that all of the following commute:
1
1
T
T
A..
~
1J
A.~ .
13
B
To define composition, a pair of morphisms ~A.
i
I
we return to a Cohds category,
, LB.
j
J
~ ~C
k
k.
and in it,
271
Consider the following rfA. i i
~ T[B. 1
IS   +
Bj T 'B
,
~
k Ck
p.b. j
k
~
f
+ Ck •
Aik
We are interested in deducing, for the sake of a later definition, the data A. 1
Ck for the composite morphism, out of the ones A.
A i j ~ ~
Bj
B.
Bj k
Ck
for the factors. We form the pullbacks Aij k. It is clear that the arrows (Aijk , Aik)j form a disjoint sum diagram. The desired morphism Aik , Ck is, therefore, given by the universal property
272 of this last disjoint Now,
sum,
from the components
it is clear how to formulate
The functor
y~ohds: T
~ Cohds(T)
We leave the details construction
the subobject
R
of verifying
one.
the required properties
(p,q)
of morphisms,
represented by the mono
relation.
~ A/R.
analyze
is the obvious
> C k.
of the
Cohqe(T).
Instead of a pair
PR: A
> Bjk
of composition.
to the reader.
Let us turn now to
equivalence
Aij k
the definition
For the quotient
Let us put ourselves
an arbitrary m o r p h i s m PR
A
let us refer rather to
:
of
R,
> AxA
into a Cohqe category,
f: A/R
> B/S. f
÷ A/R
as the
Consider
and let us
the pullback
B/S
X and the subobject
R
let us write
B
represented
<~,~>:
by
AxB;
X
let us denote the
^
latter subobject AxB
consisting
the following
by
X.
If our category
of those
implications
aRa'
is Set,
for which
X
f(a/R)
is the subset of Therefore,
= b/S.
are true:
& aXb & a ' X b '
~ bSb'
(2)
T r u e ~ 3b aRb with
a, a' ranging
subset
of
AxA
over
A, b, b'
over
B.
such that the implications
Conversely,
(2) are true,
if
X
is a
then there
is
^
a unique
f: A/R
> B/S
the same as the given In an arbitrary
such that
X
deduced from
Cobcategory
T,
1A
with the projections
in the first inequality
AxAxBxB.
(2) and
in
T
< 3#1 (~.) referring
(3) mean the same in Set;
iff for all Cohmorphisms
A, B, R, etc. meaning
as above
is
we write
"~12"(R) ^ ~13"(X) ^ "~24"(:X) < #34"(S)
Certainly,
f
X.
M: T
M(A), M(B), M(R),
> Set, etc.
l
(3)
]
to the product and
(3) holds
(2) holds with all of
273
We make, Cohqe(T)
therefore,
the following
are given by pairs
equivalence
relation on
A;
definition.
(A,R), with we write
A
The objects
an object of
rA/R"
for
T,
(A,R).
of
R
an
A morphism
A
rA/R"
~ rB/S ~
B') satisfying
is given by an
X ~ Sub(AxB)
(a 'relation
from A to
(3).
As for the composite
of two morphisms,
note that if we have a
pair of morphisms A/R   . and
X, Y
deduced and
are the relations
from their composite
Y,
which,
B/S
C/U
+
deduced from them, then the relation is the relational
product
X~
of
in Set, is defined as
aCX[~)c ~=~ ~bcB (aXb & bYc). The official
definition
is
~ with the projections
=
3
713 (~ 1 2 *(~)
referring
^ ~2 3 *(Y))
to the product
Notice that we have to show that if appropriate
data),
so, it suffices
then
X[ST satisfies
A×B×C.
X, Y satisfy
(3) with
R
(3) (with the
and
U.
For doing
to verify the same fact in Set, and appeal to
completeness. We leave the remaining
details
to the reader. • 2.4.5.
Corollary
2.4.5'
If
T
is a Boolean category,
then
P(T)
is a
Boolean pretopos. The proof is an instructive of 2.4.5.
exercise,
using the
'moreover'
part
• 2.4.5'. Let us call a Cohmorphism objects, statement that,
quotientlike if it is full on sub
and every object in its codomain of the
'moreover'
for a Pretopmorphism,
a quotient.
is covered by it (see the
part of the last proposition). being quotientlike
But first, we prove
We will show
is equivalent
to being
274
Lemma 2.4.6.
A quotientlike
conservative
Pretopmorphism
is an
equivalence.
Proof:
Let
F: T ~
T'
We have an effective
satisfy the conditions,
epi family
(F(Ai)
A
=
X ~ 0b(T'). with
~A.
i we therefore have a single
and let
~ X) i (one in JT);
1
effective
epi
f: F(A) .........~..... X.
Let
P .....
Y
,, + F (A) p'
be the kernel pair of by
:
Y
ed by: equivalence
f;
let
, FAxFA. B
~ AxA,
relation,
Y
be the subobject
By assumption, say, whose
and
F
Fimage
+
g'
relation.
AxA
represented
is
Y.
Since
represent
Y
is an
is conservative,
B
is an equivalence
of
we have a subobject,
Let
A
h: A
C
be its quotient.
But
then the two diagrams P
.4~
Y p'
)
Fg
.
F(B)
f
F (A)
F(A)
Fh
~ X
~ F(C)
Fg'
are isomorphic
since each of
and second diagram, that
F
is essentially
consequence
f
and
respectively: surjective.
of being conservative
Fh
is a coequalizer
in particular F
in the first
X ~ F(C).
is full and faithful
This shows as a
and subobjectfull. • 2.4.6.
Proposition
2.4.7.
an isomorphism, a conservative
(i) Every Pretop m o r p h i s m can be factorized,
as the composition (Pretop)morphism.
of a (Pretop)quotient
up to
followed by
275
(ii)
A Pretopmorphism
on subobjects,
is a quotient
and every object
if and only if it is full
in its codomain
is covered by it (it
is quotientlike). (iii)
If the domain
of a Pretop
quotient
is Boolean,
so is its
codomain.
Proof:
(i)
it being
Given the morphism
in Coh,
and factorize
F: T
÷ T'
in Pretop,
it in a quotient
we consider
and a conservative
one
in Coh:
T
F
,,
~ T'
\/ T"
Now,
form the pretopos
clearly
completion
P (T")
y: T"
of
T".
We
have a diagram F
T'
T"
\
~
F
P (T")
in which
Fy m r.
The universal
=T'"
properties
of
Q
and
y
immediately
^
imply that
Q = Qy
is a quotient
like, by 2.3.5 applied to
y.
Using
to
Q,
in Pretop.
Moreover,
and the "moreover"
the latter property
of
it is quotient
part of 2.4.5 applied
Q, we now show that
r
is
conservative. Let
X + Y
isomorphism.
be a monomorphism
Consider
the pullbacks,
a finite
one for each
in
covering
T'"
taken by
family
(QA i
r
into an
~ Y)i'
and
i: X
~
T
Z. i
 . . . . . . .
Y
I
QA.. 1 ^
^By
Q
being subobjectfull,
Zi ^
> QA i is isomorphic
to some
^
Qfi: QBi + QAi' with fi a mono. r phism (as a pullback of an isomorphism).
takes Since
Qfi
into an isomor
F m pQ,
and
F
is
276
conservative,^ (Zi X
Qfi
itself is an isomorphism, for every
i.
But then
> QAi .. ~.. Y)i is a covering, hence, since it 'factors through' ~ Y, the monomorphism X ~ Y must be an isomorphism. It is a simple general fact that a Cohmorphism conservative with
respect to monomorphisms in the domain category is conservative. follows immediately when one reflects that a morphism an isomorphism iff the mono representing the subobject mono
A
) C,
f: A
This ) B
3f(l A)
is
and the
in a factorization A
> C
/p,q>
A×A with
(p,q) (ii)
the kernel pair of
f,
are both isomorphisms.
By the proof of (i), and 2.4.6.
(iii) By 2.3.6. and 2.4.5'. 112.4.7.
The final group of results in this section should, logically, be the first ones in the study of concepts related to the quotientconservative factorization. Pff0position 2.4.8. In any one of the 2categories Lex, Coh, Pretop, the following is true. If T is any object of the 2category, N is any set of morphisms in
T,
then there is an arrow Q: T
~ T[~ I ]
in the 2category so that "Q is obtained by inverting the morphisms in ~" in the sense of Definition 1.4.2.(i) (with Pretop replaced by the relevant 2category).
T[~I]
and
Q
are determined up to equivalence,
resp. up to isomorphism. The proof will be discussed (although not given) in the next section. 2.5.
Relations with symbolic logic. Let us start with a fundamental observation that goes a long way
towards explaining the relation of model theory and categories. be a small graph) or category, and consider a diagram (functor)
Let
C
277
M: C
> Set,
language
Then
given by
each object of prets
each
morphism >
C:
interprets
C,
say
operation
f: A , In fact,
are precisely
empty partial
> Set
each sort of the language,
say A, as a 'partial
> M(B).
Set
possibly
C,
M
(sorted unary)
of
M(f): M(A) C
is the same as a structure for the m a n y  s o r t e d
M
C.
domain'
symbol
as a corresponding
if
C
is a graph,
domains.
If
are those Lstructures,
then the diagrams
is a category,
for
L
that satisfy a certain set of identities: ............ h
i.e.,
kind of operation
for the language
C
i.e.
and it inter
of the language,
B,
the structures
A
M(A);
C,
with
the functors
the underlying
graph of C
whenever
> C
B
commutes,
we have the identity Va ~ A g(f(a))
= h(a);
also,
(i)
Va e A idA(a) for the identity m o r p h i s m Moreover,
natural
idA: A
= a
> A.
transformations
C
between
diagrams
(functors)
+ Set
are seen as the same as homomorphisms in the usual sense between structures. In first order logic over the language of which has a definite 3x
is meant
(¥x c X), that
sort,
an object of
to range over a fixed sort;
(~x e X)
with
X
we have variables A quantifier
to emphasize
the sort of
x.
Yx,
each or
this, we write
We have an equality
sign
is sorted in the sense that only terms of the same sort are allowed
to fill in the places equality
of it (equivalently,
sign for each sort).
unary operation
symbols
we have all the Boolean
there could be a separate
The only nonlogical
(the arrows
the natural way as illustrated L,
L, L.
under
in
L);
(i).
connectives,
symbols
are sorted
their use is regulated In full first order
in
logic over
and the two quantifiers.
278
Although manysorted seem to lack expressive ical logic, attention
languages with unary operation
power,
it turns out,
existential Let
by using
~
be a coherent
category,
a string of variables ÷x.
If
is a functor
~ Set,
{a c M(X): M If
M: T ~ ~[a]},
~
subobject,
M
i.e.
a subset,
of
of
formulas.
Proposition
(i)
M c Mod(T).
~
For
T
M(X).
of
graph,
be the sort of M~(~)
~(~)
in
of and
if
M
=
M,
a subset of
then
The next proposition
M(~)
is a
tells us that
logic is at least as great
formula
category,
~ a
x i,
in particular
a Cohfunctor,
categorical
= M~(~)
~(~)
there
over
(the
is a unique
subobject
of
(
~ Sub M(X))
is called the canonical interpretation of [~:~].
~
over
The proof of the existence the complexity
M
a Boolean category,
any first order formula
sections,
and
such that
it may be denoted by (ii)
X.i
For any coherent
M(¢)
~(~);
X,
graph of) T, a coherent
[5] = X
let
then we may consider
as that of coherent
for any
disjunction
its underlying
the interpretation
power of coherent
2.5.1.
L
is any Lstructure,
is a subobject
the expressive
underlying
formula is one that is built conjunction,
such that each free variable
~ = <Xl''" . 'Xn> ,
If
X = [5] = XIX...×X n.
of
(finite)
quantification. T
in
M(X).
of categor
to such languages.
up from atomic formulas
formula,
as a biproduct
may
that in fact there is no loss of generality when we restrict
A coherent, or positive existential,
occurs
symbols
~,
there were
the same conclusion
for
T. of
~,
is an instructive implicit
holds
examples
proceeding
by an induction
exercise.
In fact,
of constructions
on
in previous
of canonical
interpretations. • 2.5.1.
The last p r o p o s i t i o n Proposition
2.2.4.
can p r o f i t a b l y
Suppose
show that it is an effective 2.2.3,
assume we have
gjf'ij
(i,j
c I).
gi: Ai
Consider
be used in the proof of
(fi: Ai
÷ A)ic I c JT;
epimorphic
family.
~ B
(i c I)
the following
such that
formula:
V (3x c A i ) ( f i ( x ) = a A gi(x) i
we would
like to
Using the notation
= b)
gifij
=
of
279
with two free variables is that,
a,b of respective
function
A
Now,
~ B, let
¢ = [ab:v]. morphisms
as immediately
# e Sub(A×B)
If
in Set for which
requirements; infer,
in fact,
at least,
that
M e Mod(T).
object of is
~.
A×B
M ~ Mod(T):
then
M
in
M(h)
v, and
Let
hM
h
hold,
satisfying
Mab(~) , i.e.
subobject ¢
of
the
M(¢).
We
M(A)×M(B)
is a functional
for
sub
be the m o r p h i s m whose
follow,
hMOM(fi)
and as we know
> M(B)
is
h: A + B
and
of
interpretation
hM: M(A)
clearly,
hfi = gi
= hM,
The uniqueness
of
M(¢) is a functional
T.
The idea
takes the data to objects
the same hypotheses function
By completeness,
B.
seen.
the graph of
The equalities
A and
the graph of the desired
be the canonical
M e Mod(T),
already there is a unique
all
sorts
if we are in Set, then this defines
graph
since they are true in any
= M(gi).
can be deduced in the same manner. • 2.2.4.
Note that for any subobject is a formula canonical
;(x)
¢ ~ Sub(X),
with one free variable
interpretation
is
~.
If
¢
in any fixed
x
of sort
X
T,
there
whose
is represented by
m: A
÷ X,
^
then not
~(x)
can be taken to be
uniquely
determined
by
(3a ~ A)(m(a)
¢,
we w i l l
= x).
Although
;(x)
is
the formula representing
call
it
Set
are
E.g.,
it
¢. O n c e we a c c e p t directly
interface
a natural T
with
functors model
transformation
h:
Set
is
an elementary
consequence
of
2.5.1.:
h:
to
that
M
~ N
doxical
as Let
remember
topos
is it
us
at
is a coherent
prove that such that,
i
T
proof
stage,
holds
in
Boolean,
of
makes
to
two the
sense
functors following this
say
that
M, N f r o m
M, N e M o d ( T ) ,
M(s)
In
not
yet
immediate then
not
this
proof,
established
any
that
para
we h a v e that
to
a pre
With the notation of 2.4.3, we want to Let
M: T
is surjective.
÷
Set
Then,
be a Lexfunctor
in Set, M(f)
and by taking into account
and M(s)
the meaning
of
in Set, we see that
M.
(Va,
a'~A)
If(a)
This
immediately
= f(a')
¢=~ s ( a )
implies
that
= s(a')]
M(i)
In order to be able to apply completeness, J
we c a n
(Why' i s
2.4.3.
we h a v e
category.
have the same kernelpair, pullbacks
is
Note
embedding.
is a monomorphism.
in addition,
between
structures,
first?)
the
this
........ ~ N
embedding.
if
complete
that,
theory. M
an elementary
may s o u n d
into
generated by the single
covering
{s}
is a mono in Set. consider
of the object
the topology C.
Since
s
280
is an effective
epi,
effective
(a condition on pretoposes),
epis are stable under pullback
effective
epis are closed under composition and since the pullback composite contains which
epis are extremal,
(exercise; or see 3.1.5 in EMR]),
it follows
that every covering
a morphism that is an extremal
is not an isomorphism
general
completeness
T
of a composite m o r p h i s m can be formed as the
of two pullbacks,
A + C
in
extremal
epi.
It follows
in
that a mono
is not a covering by itself in
theorem
2.1.1,
if
i
J
is not a mono,
J.
By the
i.e.
the mono
in the factorization
A
.....
>
C
A×A
with
(p,q)
M: T
the kernel pair of
~ Set,
contradiction. Although
M Jcontinuous,
i,
is not an iso, then there
such that
M(i)
is
is not a mono,
a
this proof is not even shorter than the direct proof
(see the proof of Theorem 1.52 in ETTJ),
I find it preferable
because
it is more comprehensible. Let us now look at the interaction egorical
formulations
categorical
conditions
in the opposite
is a class of structures wellknown diagrams this:
in
sense.
the Coboperations)
elementary
classes.
E.g.,
are strict
type;
of the particular
Set
is a product
our
for the operation
limit and colimit
of binary products,
a diagram
A
thus,
is verified by writing out the
C
in
of
Note that each such graph
over a finite similarity The assertion
characterizations Set.
the expressibility
is that the graphs of all the Boolean pre
(and of course,
(finitely axiomatizable) statement makes
direction:
and cat
by symbolic means.
The first observation topos operations
of the symbolical
B
diagram iff the following
is true:
we have
281
(VaEA)(VbeB)(~ccC)(fc=a (Vc~C)(Vc'eC)((fc=fc' It is instructive diagrams
Next,
disjunction;
in fact,
the underlying diagram in
that expresses
T).
with symbols (2) in
see (i) above.
a Pretopmorphism
T
~ Set
we write
in that diagram,
is a product
diagram, we
We also add axioms ensuring By the definition
of what
> Set, it is clear that the resultexactly the class
Let us discuss ultraproducts the categorical
(c.f.V.3
for this
and for each in
appearing
T
(3) to our axiom system.
ing axiom system will axiomatize
C
operation
T,
(over the
which is in the graph of that operation,
sentence,
that we have a functor;
C
class
In fact, the axioms
for each pretopos
E.g., if we find that the diagram
With
class.
that the diagram would be the given kind were it in Set.
add the sentence constitutes
it is not an elementary
is an elementary
graph of
T
down the firstorder
is defined
we see that, for any (small) pretopos
Mod(T)
class are the following: dividual
(3)
+ c=c')
on the domain of the operation)
as a consequence,
the class of objects of language
gc=gc')
A
to see that the class of all coequalizer
(without restriction
by an infinitary
A
fc=b)
A
of structures,
Ob(Mod T). i.e. functors,
from
point of view. a (small)
graph,
the category
has limits and colimits in [CWM]).
Therefore,
(C,Set)
of all diagrams
that can be computed pointwise
if we define ultraproducts
in (C,Set)
by the same formula as in Set: n Mi/U = colim n Mi , icl peuOP icP and similarly
for morphisms
in
(C,Set),
then,
for
A
c Ob(T),
we
have ( H i~I and
a similar
tion
given
C
> Set
formula
here
for
coincides
Mi/U )(A) = morphisms with
the
~ ieI in
usual
M i(A)/U T.
This
one,
(4) shows
considering
that
defini
diagrams
as structures.
Now, let
T
be a small pretopos,
and let
C
be its underlying
graph. Then, if in the above the M i again in Mod T. This is an immediate
are in Mod(T) then consequence of 1.3.1
called Los's theorem)
(4).
from the fact that theory,
the
the
and the formulas
Mod(T)
is an elementary
and the usual formulation
Of course,
~Mi/U is (which we
it also follows
class in the sense of model
of Los's theorem.
282
Let us turn to the construction free Boolean pretopos,
of the free pretopos,
needed for proving the assertions
fact, the constructions
pretopos
structures.
operations,
finite diagrams.
proof of the existence
composites
It would also be possible
to give the
versions
of the adjoint
functor theorem.
symbolic
logic provides
a simple and intuitive procedure.
Boolean
However,
the free coherent
category on a graph
analogous
G.
of the
and they would take the form
of the free objects by appropriate
First, we construct
In
of the
free groups and other free
The "words" would be literal
(Boolean pretopos)
of labelled
in 1.4.1.
could be given by a direct application
"method of words" used for constructing algebraic
and the
(2categorical)
in these two cases
category,
and the free
They satisfy universal properties
to (i) and (ii), respectively,
in 1.4.1. +
For FCoh(G) , we take as objects all the pairs (x,~) with a finite tuple of distinct variables, ~ a coherent formula with free variables we mean,
all included really,
in
x.
"the set of
We write ~
E~:~] with
~
and
~
disjoint, ++
such that and
among
with
meaning unique existence
first order logic.
(x,~), since
Given two objects
formulas
~
with free
and s a t i s f y i n g
and
Such formulas
~
between the same objects
equivalent
formulas.
functional
relations.
are not necessarily
A morphism
,,[2, referring
disjoint
Coh
~G this defines
to provability
Now, two provably
functional
if they are provably
is an equivalence
class of provably
tuples,
in the given objects,
and
is left to the reader to form
of composition
: G   +
~
of morphisms,
and that
FCoh(G]...
the free Cohcategory
over
G
completely,
of course more work is to be done to verify the required properties. This will not be done here; The construction
in
are called provably functional
are identified
The case when,
ulate, as well as the definition of the Gdiagram
Although
~".
coherent
relations from the one object to the other. relations
instead of
[~:¢]
we consider
variables
3~
xy,
[~:~]
for details,
see [MR].
of the free Boolean category
over
G,
283
Boole ~G : G ulas over
G
+ FBoole(G )
is similar,
as language,
in defining both the objects
but uses all first order formand morphisms.
The free pretopos, Pt = ~G : G
~ Fpt(G)
is given as the composite
G
Fcoh(G )  
........
* P(Fcoh(G)) Y
with the pretopos
completion
taken from 2.4.5.
property
is straightforward,
required universal of
~
and
~oole
y.
The free Boolean pretopos
in place of Finally,
~;
one uses
the proposition egories
of categorybased
conditions".
Since
(see however
some more concrete lectures
I am not quite
category theoretical
e.g.
case of Lex, with 2.3.1 for Lex,
terminology.
the internal
Let me first
described,
of fraction"),
are closely but unless
even for the case of Lex,
is not
Let me add that, of course,
the
case of what we want for the
with the notation of that proposition. of the factorization
(Gentzensequents)
R e Coh,
and
are not.)
Z c Morph(R),
R (see p. 128 in [MR]).
expressing
that each
o ~ Z
= b
(o: A ÷ B
in
in a consider
Add the following should be
invertible: True ~ (3~a e A)o(a)
in
con
the special properties
for our purposes,
for Coh and Pretop are given in [MR] Given
in
for the case of Lex and Coh
facts alluded to above,
theory of
facts
I point out
These constructions
the mere existence
crucial
The constructions
axioms
these general
and the ones for Coh and Pretop are, of course,
of the general
in
"2catalgebraic
Kelly and myself),
("calculus
is a special
z = Inv(F),
of the factorizations, different
terms.
in [GZ].
in 2.3.1
(Note also that although sequences
ready to describe
it, the construction,
found in full generality construction
asserted
facts concerning
of the desired quotients.
in [GZ]
with
just as in the
the mere existence
detailed constructions
related to constructions I have missed
Again,
that Professor Jean Benabou recently
in Montreal,
in elegant
2.4.8.
defined by essentially
paper by G.M.
constructions
on the basis of those
is obtained similarly,
of general
structures
a forthcoming
of all mention
TR,
just discussed,
are consequences
of the
2.4.5'.
let us turn to Proposition
case of the free objects
The verification
Z).
284
Let the theory so obtained be denoted by
Consider
T'.
the categor
of T' (see Theorem 8.1.3., p. 239 in [MR]), and call ization RT, it R[~I]. The canonical interpretation of Y R in T', and that of T'
in
induces
RT,
the required Cohfunctor
R Given the properties
of
TR
have the required universal
~ R[~I].
and
RT, ,
it is easy to verify that we
property.
The case of Pretop can be handled as it was with respect to free objects:
first, we carry out the construction
in Coh, then we take the pretopos
Part 3. 3.1.
The statement Consider
any coherent
Conceptual c0mpleteness.
the notion of the pretopos T,
morphism
T ÷ P(T)
S = Set)
an equivalence
in particular,
completion.
and proof of conceptual
category
on the given pretopos
one associated
that induces,
completion
(see 2.4.5.):
a pretopos
Coh(P(T),S)
of the categories
with
P(T), and a Coh
for any other pretopos
of categories
an equivalence
completenpss.
S (e.g., for
+ Coh(T,S),
hence,
of models:
Mod P(T) + Mod T. We interpret category
a 'theory', pretopos
this situation by saying that the notion of coherent
is conceptually T
by the concepts
completion
of concepts extension
in
incomplete:
T
'theory'
when extending
(abstract
by definable
equivalence
"behave~semantically
for any (pre)topos
By contrast,
relations),
or even,
More precisely,
Theorem 3.1.1. I: T + T'
the following completeness
of small pretoposes
is sufficient categories
(Conceptual
in its
in T, quotients the resulting T
itself",
the same category of S
S.
the notion of pretopos
is conceptually
the sense that it does not admit any extension to models.
category,
sets) inherent
'concepts'
in the same way as
by having the same category of models, valued models,
definable
(finite disjoint sums of
a coherent
in
with respect
theorem holds.
of pretoposes).
to be an equivalence
that I induce an equivalence
complete
conservative
For a morphism
of categories,
I*: Mod T' + Mod T
it
on the
of models.
This is Theorem an essentially
7.1.8 of [MR].
different
version of the theorem.
Recently,
proof yielding
Andrew Pitts has found
a stronger result,
a constructive
285
On the basis of this theorem, "real" or "complete" P(T).
E.g.,
if we ever
try to recover
Mod T, we should expect to get not be any difference P(T)
it seems natural
form of the theory P(T)
in principle
is its pretopos
T
in an abstract way from
instead of
T
in this area
completion
since there should
between concepts
from the point of view of the category
ation is borne out by results
to say that the
T
in
T
of models.
and those in This expect
(see [M3],
[M4],
[L], [MI],
[M2]). The proof of Theorem so stated in [MR].
3.1.1 in [MR]
Note that
I: T ÷ T'
means two things:
first, that
full and faithful,
and second,
those
T + S
that
tions being required condition,
and even that for
morphism
(Strong conceptual
I: T + T'
"I* is faithful",
I;
The strong form of the first
for pretoposes).
to be a quotient,
it is
I*: Mod T' + Mod T be full and faith
7.1.4.
in [MR] says that
I* being essentially
is conservative,
"I* is full" implies
Theorem 7.1.6 says that
surjective
on objects
clearly
the strong version of conceptual
completeness
analog of) conceptual
to note that in the "doctrine"
completeness
holds,
(a classical
Lex,
specific
the 2category
of all pretoposes.
for Boolean pretoposes
in the common sense of model theory, more natural
see [GU])
them as
Being interested
(corresponding
to theories
in full first order logic),
it is
to consider the category BPretop of Boolean pretoposes,
with morphisms to consider
(the obvious
result,
but its strong form does not (see [MP]). Speaking about Boolean pretoposes, we have considered in Pretop,
implies
version.
It is interesting
in results
For a
implies that "every object in T' is subcovered by I".
implies the original
objects
onto
both condi
3.1.2 follows by 2.4.7(ii).)
Since I
is
surjective
to require
completeness
"I is full with respect to subobjects",
Therefore,
not
only.
of small pretoposes
(In fact, Theorem
that
inverted by
S ~ Pretop.
S = Set
result,
~ Pretop(T,S)
is essentially
says that it suffices
sufficient that the induced ful.
that
I*
to hold for all
completeness
Theorem 3.1.2.
I*: Pretop(T',S) that
invert all morphisms
conceptual
gives a stronger
(in Pretop) being a quotient
the Pretop morphisms.
BPretop
Furthermore,
it is more natural
to be Groupoidenriched rather than Catenriched;
286
this means
that BPretop(T,T')
morphisms
T + T',
isomorphisms. are those of
Writing C
should be the groupoid of all BPretop
with only those natural C is°
and whose morphisms
are now saying that we should put
without
understanding (ii)).
Obviously,
concepts
involved,
(see Section
Therefore,
we
Gpdcategory), Even
the reader can see here is intimately
defining Boolean pretoposes
I.i).
in the context of BPretop,
a quotientmorphism
C,
had to be defined
the fact we are considering
related to the fact that not all operations are strong
of
as 2cells.
this plainly on the way the free Boolean pretopos (1.4.1
that are
whose objects
= Pretop(T,T') is°
algebraic'
transformations
the general
C
in this way has good general
(it is an 'essentially
but not if we put in all natural
of
are the isomorphisms
BPretop(T,T')
The reason is that BPretop construed algebraic properties
transformations
for the subcategory
the natural definition
of
is the following
Definition 3.1.3. A morphism I: T + T' in BPretop is a quotient if Iis°: (Mod T') is° + (Mod T) is° is full and faithful and essentially surjective by I.
onto those models of
Note that, by 2.4.7 Pretop
is automatically
I: T + T'
in BPretop
which
invert all morphisms
(iii), any quotient
Boolean.
of a Boolean pretopos
completeness
holds
specified.
the question
if an
for Boolean pretoposes.
answer is "yes" with a curious qualification: involved should be countable objects and arrows).
in
in the sense of 3.1.3 is the
in Pretop in the sense previously
3.1.3 is still useful since it suggests
analog of conceptual
inverted
It follows easily that, for a morphism
to be a quotient
same as to be a quotient Definition
T
The
the Boolean pretoposes
(meaning that they have countably many
Theorem 3 . 1 . 4 . ( S t r o n g c o n c e p t u a l c o m p l e t e n e s s f q r c o u n t a b l e Bo,olean pret0poses) For a P r e t o p  m o r p h i s m I : T + T' o f c o u n t a b l e Boolean pretoposes T, T' to be a quotient, it is sufficient that Iis°: (Mod T') is° ~ (Mod T) is° induced by I be full and faithful. This result is essentially Abraham Robinson memorial although
say that a Cohmorphism (see before
In 1975, at the
Gaifman announced a result which,
stated in a language not employing
related to Theorem 3.1.4. like
due to Haim Gaifman.
conference,
categories,
A version closer to Gaif~an's I
between Boolean
2.4.6) provided
I is°
is closely result would
(Coh)categories
is full and faithful.
is quotient
287
In this section, will
we will
apply it to prove
applicable
to the proof
The proofs
first prove
a stronger
Theorem
3.1.4,
and then we
form of it, which will be directly
of the main result,
Theorem
in this section will use various
1.4.4.
methods
of model
theory. The first
lemma,
Lemma
It is analogous to Theorem different. As above, with denotes the functor (Mod T') is°. Lemma 3 . 1 . 5 .
3.1.5.,
obtained
Assume
from
I:
that
able Boolean categories
of Gaifman's
theorem•
I*: Mod T' + Mod T restricted
T + T '
X
in
T'
to
is a Cohmorphism between count
I is°
such that
"subcovers" every object
is the heart
7.1.6 in EMR], but its proof is very I: T ~ T', Iis°: (Mod T') is° ~ (Mod T) is°
is full and faithful.
Then I
in the sense that there is a
finite family of diagrams m. 1
X. I
IA.
1
(i c I)
T',
in
(pi)ie
I
Proof:
Recall
language
~(~)
L
c ~(~)
of types each x
finite ¢(~)
M
¢(~)
and
means
that would
then we have an obvious
realized
in all models
countable model
in
of
"M
simply
A type
T
(i.e.
for all (~ i(xi):
has a model in
~(~)
realizes
i~I}
M ~
omitting
(matching
simultan
~(~):.
consistent
If
implies
any isolated
type must be
T.
to the notation Mod T'
with
is no tuple
s(~)
T u (@}
converse:
is called
family
T
all
case, we may say
T ~ @),
~
of Lformulas
appropriate).
that there
satisfy @,
~(~)
÷ o(~))
then
Given a
in a countable
is consistent
T ~V~(¢(~)
(for all Lsentence
Let us return
(see [CK]).
a set
given a countable
~(~)"
in the opposite
is complete
if
is nonisolated,
"M omits in
set),
type" may be more
says that,
each of which
for sorts)
(OTT)
tuple of free variables
"incomplete
OTT
theorem
is a countable
is consistent) •
~i(~i).
eously;
types
(having at least one model)
is isolated by
T w {~x¢(~)} a(~)
T
(Morph(L)
(although
such that the family
epimorphic.
the omitting
theory
with a fixed
a type
being monomorphisms,
i
is effective
consistent o(~)
m.
with the
(each
of the Lemma. M(X),
Let
X ~ Ob(T'),
M
be any
is countable);
T
288
let us fix L'
M
for a while.
(= the underlying
(written
graph of
Consider
graph of
simply
a)
the language
T') by adding
for each
T) and for each
A e Ob(L)
a e M(IA).
Let
L
obtained
a new individual
from
constant
(L = the underlying
i: L' ÷ L
denote
the
^
inclusion. We have the structure language L whose L'reduct i*M be the theory in
M;
T Now,
various single
of
M
let us consider
element
b
such that
isolated. ZA(XA)
L:
the set of all sentences
complete
MX, X e 0b(L')
any finite
~
in
M, i.e.
We now also consider, the requirement
ZA(XA)
in
M;
L
hence,
of elements
T',
product
the set
true
~(~)
to a
We claim that the
sort).
of all Lformulas Z~(~)
A < 0b(L),
is a variable
is non
the type
of sort
"x A ¢ A" implicitly). by a remark
of the
we could pass
Suppose
for each
(x A
ZX(XA) contains
~
in
is isolated.
= {x A = a: a ~ M(A)} omitted
tuple
(using products
M ~ s[~],
in
Let
(see above).
of an appropriate
(complete) type of c(~)
in
is clearly
M = (M,a)aciA,A~Ob(L) of the is M and for which (a)M = a.
above,
IA:
hence,
Of course,
each
ZA(XA)
each is non
^
isolated.
Therefore,
by 0TT, we have a model
÷ ~(~) and every ZA(XA) as well. Consider of M', and consider the Lstructures I'M,
hA:
a ~
make up an isomorphism omits
each
a model that
~A'
of
h
hA
However,
I*h' = h,
in particular, h
cannot
same type in there
M'
as in
~
does in
M'
having
We have constructed the claim is, therefore, We conclude finite
tuple
say that show) if
~
that,
that any countable
~
and
~
~
reason
have
(I*M')(A); of
would
~
M'
is
implying
preserving
to an isomorphism
h', h'(~)
M'
since
I*M"
is structure
clearly
and
h': M ÷ M' have the
A
since
M'
the same type as
M,
but,
~.
omits
~(~),
to the fullness
of
Iis°;
proved by contradiction. still with the fixed countable has an isolated
It is wellknown atomic model
type.
(c.f.
to
c.
M
of
In model
[CK])
T, each theory,
In our case,
we
(and not hard to
is homogeneous in the sense
have the same type in it, then there
of the model mapping and
h
a counterexample
of elements
is atomic.
M
onto
^
is nothing
omitting
by the fact that
on elements
be extended
since with any such
^
T
M' = i*M' , the L'reduct I*M'. Then the maps
indeed,
is surjective
is "L'elementary
is elementary,
reflecting. with
hA
of
(a ~ M(IA))
h: I*M ÷ I*M':
each
T,
~ M'(a)
M'
that
is an automorphism
this implies
the same type in
M,
they are identical:
is that an automorphism
of
M
that if
~ = ~.
The
is the same as an automorphism
h
289
and by the faithfulness of of M such that I*h = Idi,M, therefore, the only automorphism of M is the identity. X ¢ Ob(L')
We now see that for any L'formula various
~(x,y)
sorts
matching
~
with
IA
with
x
of sort
A e Oh(L),
such that
and
X,
~
I is°,
b c M(X),
a tuple
there
and there are elements
M ~ ¢[b,~]
and
is an
of variables a
of
of
M
M = Vxx'(¢(x,a)A¢(x',a)+x=x
).
^
Namely, i f
~(x,$)
M ~ ¢[b,a], in
M,
isolates
and i f
hence
M ~ ~[b',~],
in fact,
an a p p r o p r i a t e
make
product
We c a n r e  w r i t e
b
then
b
in
M,
then of course
and
b'
have the
same t y p e
~
sort
and
in
~
into
singletons
by p a s s i n g
to
L.
our last
conclusion
this
way:
for the
fixed
M
X, M ~ ¥x
V
~y(¢(x,y)
Here in the infinite form
type of
b = b'
We c a n ,
and
the
¢(x,y)
with
of the form formula
IA,
holds
whose models
^ Yxx,(¢(x,y)
disjunction x
ranges
a free variable
with
A e Oh(L)
in all countable are exactly
L~wenheimSkolem
~
^ ¢(x',y)
of
over all formulas X,
depending
models
the objects
M
y a variable on
of
of
and the compactness
r~ vx V
~.
T,
Since
with
Mod T',
theorems,
that there are finitely many formulas the ¢ above such that
+ x = x')).
(1)
of the
of a sort the displayed
T
the L'theory
by the downward
it is easy to conclude
¢o(X,Yo),...,¢n_l(X,Yn_l)
~i(x)
as
(2)
i
~i(x)
from
~
formed
from
~i
as the formula
"V"
is formed
in (i).
It should now be clear that I.
after
Namely,
let
i < n
(2) implies
and consider
that
the canonical
X
is subcovered
interpretation
by
in T'
of the formula
~(yi ) (remember,
T'
the sort of Pi
= ¥xx'(¢i(x,Yi) df
is Boolean).
yi);
let
be the morphism
[yi: ~(yi)]
mi: X i ÷ IA i
Xi + X
in
(it is clear from the definitions is indeed Pi
functional).
form an effective
the relation
of
T
to
^ ¢i(x',Yi)
Then
T'
is a subobject
represent whose
graph
family
is
T', ensuring,
translates in
T'
of
IA i (IA i
this subobject. [xix:
that the last subobject
(2) clearly
epimorphic
÷ x = x')
of
Xi x X
into saying
(see 2.2.4)
by completeness,
Let ~
¢(x,mi(xi))] that the
(here we use
that sentences
290
provable from
~ 'become true' in
T'). •
Lemma 3.1.6.
Suppose
categories such that
P~oof:
I: T ÷ T' I is°
3.1.5.
is a Cohmorphism between Boolean
is full.
Then
I
is full on subobjects.
This is essentially a direct consequence of Beth's definability
theorem.
Let us state Beth's theorem in a notation convenient for our
purposes. Assume a graphmap and let sort
L
~(a)
I(A).
models
and
L'
(diagram).
are graphs Let
~
be a formula of
L'
> I*(N),
M, N e Mod T,
Mfm)
incl.
Then there is an Lformula
, M(IA)
incl.
(= I*(M)(A))
~(a)
(: I*(N) (A)).
* N(IA)
such that
M(m) = I*(M)(~) for all
T.
Assume the hypotheses of the lemma. L
and
and let
T
L'
To apply Beth's theorem,
be the underlying graphs of
be the theory axiomatizing
rary subobject of ~. of
and any isomorphism
IhA
N(~)
let
is
with a single free variable a of
[f
~ Mod
> L'
A E Ob(L),
we have a commutative diagram
({acM(IA): MI= mEaD} =)
M
I: L
L', let
Assume that "~(a) is preserved by all Lisomorphisms of
T", i.e. that for any
h: I*(M)
(languages), and
be a theory over
I(A), and let
¢(a)
T
Mod T'
and
T',
Let
~
respectively, be an arbit
be the L'formula representing
We claim that ~(a) is preserved by all Lisomorphisms of models ~. indeed, given ~i,N and h as above, by I is° being full,
there is where
h: M ÷ N
such that
~ = [a: ~(a)]T, ,
I*h = h.
We can now take
the canonical iterpretation of
f = h~ ~
in
T'
Thus, the claim is shown.
By Beth's theorem, we have I([a: ~(a)]T) = ~
~
as stated;
it is clear that
as required. •
P~oo~ of 3.1.4: 2.4.7 (ii).
3.1.6.
The theorem follows directly from 3.1.5, 3.1.6, and • 3.1.4.
291
With
T
a pretopos,
Mod(T), we call
dense
~
~
a (not necessarily
if the evaluation
T 
full)
subcategory
of
functor
OM, Set)
(defined by
M AI
÷
M(A) M(A)
M
lhA N (A)
N
f
is conservative. objects and
Let's say that
'ultraclosed'
Theorem
3.1.7.
depend only on
Let
I: T + T'
able Boolean pretoposes, subcategory
of
Then
I
in
if the class of its
Mod(T).
Both
'dense'
ObOM). be a pretopos
Assume
(Mod T') is°
and faithful.
Lemma
is ultraclosed
~
is closed under taking ultraproducts
that
~
such that
is a quotient
m o r p h i s m between
is a dense Iis°~:
~
count
full ultraclosed
÷ (Mod T) is°
is full
morphism.
The proof will obviously be accomplished
by 3.1.4 and the following
3.1.8.
Then
Assume
the hypotheses
of 3.1.7.
I is°
is full and
faithful.
Proof:
Recall
that every
underlying
graph of
T',
underlying
graph of
T.
M ~ Mod T' and
The fact that the ultraclosed implies that for every
M ~ Mod T'
(elementary equivalence). without Eel,
free variables)
is a subobject
implies there
is
M c Mod T'
that if N ~ ~
of
Indeed,
~
N ~ ~
consider
I,
the terminal
object of
= 1(Is)
theorem",
Mod T'
4.1.11
T'.
subobject
i.e.
N ~¢.
the L,
M ~ N
(formula
(cf. 2.5.1(ii)), The density of of i), then Now,
let
and by the "ultraproduct in [CK],
the
is dense
such that
any L'sentence
(01 = the minimal N([¢~)
of
interpretation
by the last sentence
version of the compactness
L', over
there is
its canonical
[¢~ ¢ 01
over
is a structure
subclass
¢;
such that
be given;
is a structure
I*M e Mod T
since
~
is
292
ultraclosed, i.e.
we have
N • ~
satisfying
all
~
satisfied by
M,
N ~ M. Next,
recall
(small) families for all i • I, such that
the KeislerShelah
<Mi>ici, then there
(Mi)U m (Ni)U
with exponent
U,
for all
family.
The isomorphism theorem is a singleton.
by considering disjoint ever,
the composite
of the
<''" Mi M i.
of
M
of the constant from the special case
"''>i•I
over the
Another remark is that the logic;
how
and it can also be deduced
case.)
to the full subcategory
of
(Mod T') is°
to a member of
is closed under taking isomorphic
copies
in
~,
whose objects
we may assume
(Mod T') is°
OM
that
is
in (Mod T')is°).
As a consequence ever
the ultraproduct
in [CK], only for onesorted
are all those that are isomorphic 'replete'
Given two
in [CK] stated for the case
case has the same proof,
from the onesorted By passing
(M U,
case follows
structure
theorem is proved,
the general
theorem.
~<M: j•J>/U
is 6.1.15
The general
sum of the languages
isomorphism
i ~ I
is the ultraproduct
when
I
isomorphism
i• I of structures such that Mi ~ Ni is an ultrafilter U (over some set J)
<Mi>i• I
ultrafilter
of the preceding,
is a small U
we have the following.
family of models
(on some set)
such that
in
MU
Mod T', belongs
there
to
~
When
is an
for all
1
i • I. that
Namely, MU = NU i
for all
choose
i
Next,
'
consider
such that ~
M i ~ Ni;
is replete
the language
structure
morphism
I*M 1 ÷ I*M 2.
of
L',
A • Ob(L);
here
down a set structure I~N
T" N
consisting
(graph)
L",
In detail,
let
L"
XI, X 2
U
such
MY e 1
is a model of
in the language T"
L'" X e S
i: L" ÷ L'" L'"
in
which
M I ~f I~i*P,
whose models
is an isomorphism for
for each
denote the inclusion.
theory
hiA = P(×A )
L"
h: M 1 + M 2
in
L"
such that an L"
of
L"
are in
such that
(Of course,
of an
obtained by adding an
= Ob(L')  {IA: A • Oh(L)}. df Furthermore, let T'" be the
are exactly those L'"structures
M 2 ~f I~i*P
A • Oh(L)
X • Ob(L')
It is easy to write
M 1 = I~N and df are the components hA
edge
to
L"
for each
if and only if
Hod T', and the N(XA)
XX: X 1 + X 2
sum of two
(IA)I ÷ (IA) 2
If, 12: L' ÷ L".
type of a
MI, M2, and a homo
denote the two copies of
injections
of sentences
are in
the similarity
be the disjoint XA:
isomorphism h: I*M I ÷ I*M 2. Let us consider the extension
Let
then choose
and ultraclosed,
of two L'structures
together with a new edge
under the canonical
M 2 ~f
since
i • I.
composite copies
Ni c ~
(i e I)
Mod T',
h X = P(XX) T'"
P
and for which for
includes,
X E S,
for there and
among others,
293
axioms
saying that
×A = XA'
whenever
Now we see that the conclusion following
(*)
IA = IA').
of the lemma is equivalent
For every model P
of
T'"
Consider model of
~"
N
such that
the class
and
of
I~N,
T"
there
is exactZy one model
i*P = N.
~
of L"structures
I~N
belong to
of the lemma imply the following
~.
N
such that
Notice
For every T"
N
in
such that
that the hypotheses
~,
there
is exactly one model
definability
(*) from
(***), obtained by replacing let
N
theorem
such that
p
'exactly one' by
be a model of
(i*P 1 = i*P 2 = N)
T",
(I~N) U
are both expansions
by the uniqueness
part of
being distinct
Pl' P2
be expansions
to
of
which are models of
NU e ~
such that
L"
~.
Hence
assignment that if in
Suppose
is contained to
N
L".
L"
in
is a model of
Suppose
P (i.e.
~f(x,y) T",
P(f)(a)
of
N)
P(f) ~f
However,
N,
T'"
is automatically
of Beth's
the reduct
a model of
theorem are satisfied.
T", Thus
T'"
is a theory in
In this case,
there is an
f: X ÷ Y
L'"L"
in
the L"reduct
for
a ~ NX, L'"L"
in
(Usually,
contains
only a
too, and can
case.)
N = i*P
(***)
such
of ~f
of P);
b E NY.
case is w e l l  k n o w n
be proved in much the same way as the special Since in our situation,
does not contain
is the interpretation
in
the general
We use the
there is at most one
theorem is stated for the situation when
single new symbol.
Hence,
that
are two languages
L'"
also that
T'".
= b ~~ N ~ ~ f E a , b l
theorem.
L'"
but
to each
then
the interpretation
and
L'"
that is a model of
of an L"formula
P
other words, Beth's
of
Then
T'".
But then it follows
and we have that for any L"structure P
N U c ~.
N U
Pl(XX), P2(×X); N(XI) + N(X2) different, be (if you like because two parallel maps
form.
any new sort with respect expansion
of
is a first order property).
in the following
(graphs)
To show
Find an ultrafilter
It is at this point that we can invoke Beth's theorem
of (*), called
'at most one'
T'".
(**)' PIU = p~.
To this end,
(cf ~CK~).
both belong
P1 = P2: were the two maps so w o u l d their Uultrapowers Set
and let
which are models of
(I~N) U'
of
(**).
The first remark is that we have the w e a k e n i n g this,
P
i*P = N.
we will make use of Beth's
L'"
is a
statement:
We are left with the task of deducing
in
N
A
(**)
Pl'
to the
statement:
of any model
expresses
for each
X ~ S,
P
of
that the hypotheses we have a
294
~x[xl,x2),
an L"formula,
interpretation
of
~X
We can now prove Remember
that each
claim that each XX'
XX
N;
the proof.)
P
of
pU
P(×X)
CX
XX
of
T'",
is a model of N[XI)
N U.
T'"
P'
T'",
and
This completes
the proof.
NU
p __ pU = p,,
U
which
P
of course,
such that is a model
is the same as the P'
is nothing but the
is an expansion
is the interpretation
is a model of
also denoted
[This will,
of
CX' P'(Xx) which
We
obtained by letting
This means that P
~".
× N(X2).
there is an ultrafilter
of the
in
the
hence we can consider
of
N
is a model of
of a structure (X e S)
P
is an expansion
By the properties
interpretation
N
is an L"formula,
As before,
(* *), there
ultrapower
Suppose
it is a subset
we have that
By
T'"
of
is the graph of a map N(X I) ÷ N(X2),
complete Mo
P
P(×X ).
(*) as follows.
in
NU
which
is
and for the L"'expansion
P[×X ) = XX'
of
P
~x(xl,x2)
its interpretation by
such that for any model
in
of
~X
in
of
N,
N.
Since
is a model of
T"'
and for P'
as well. • 3.1.8. R 3.1.7.
Theorem
3.1.4.
throws some light on a situation
this discussion,
we use the terminology
be a locally finitely presentable a faithful
functor,
filtered
colimits
case
is l.f.p,
A
presentable then
F
A.
Proposition
itself.
objects
induces
in
8
pointed out in [MP] that F
is full.
I
I
To formulate T = F(C)
considerations 1.4.1.
induced by
F
Returning assertion
limits
the category of finitely
8 ~ LEX(C°P,Set)) (so that
is not necessarily
8
and
in [MPJ says that in this
this statement
over
C,
and
~ = Af.p
F m I*).
,
It is
a Lexquotient
we can still say that
precisely,
With
with
property
even if C
+ F(C)
generates
T
some analogs
a canonical
Lexmorphism,
that for any Boolean pretopos
The existence F: C
we introduce
in Lex, the free Boolean pre
is faithful,
like those in Section
is a quotient
C
~: C
+ Lex(C,r')
surjective.
F
which creates
Let
F: A ....... + 8
2.3.
situation,
For a Lexmorphism
we may say that
and
(for
on the level of Boolean pretoposes".
is defined by the universal and essentially
of [MP~).
category,
I: C +~
in Section 1.4.
()o~: Pretop(T,T')
in EMPI
C = 8f.p.,
(hence,
In the general
of definitions topos
If
a Lexmorphism
"generates D via
and notation
(l.f.p.)
full on isomorphisms,
in
considered
of
F(C)
can be proved by
2.5 used to prove ÷ T,
with
T',
full on isomorphisms,
T
the assertions
a Boolean pretopos,
if the P r e t o p  m o r p h i s m
F(C)
+ T
in Pretop.
to our situation
is that the composite
above,
the precise
formulation
of the
in
295
D
generates
F(C)
is easily
verified
provided
out that
definability
section,
is
treatment,
that
a consequence) in fact,
by A.M. P i t t s
is essentially
Craig interpolation
a result
has recently
and [P2].
It
the methods
i n [ J T ] he u s e s ,
ical"
proof
of all
3.2.
The pro o f of the main result.
results
In p r e p a r a t i o n fundamental, first,
S
and
that
will
theorem
played
context
(of which
a key r o l e
in this categorical
of intuitionistic
is possible
eventually
of this
that
lead
Pitts'
logic,
methods,
t o a (more)
and
"categor
paper.
for the proof of 1.4.4,
'trivial',
This claim
been given a purely
i n t h e more g e n e r a l
in [Pl]
countable.
3.1.4.
the
theorem,
~ V(C)
¢
by a p p l y i n g
L e t me p o i n t Beth's
C
÷
Zcategorical
I remind the reader of a
adjunction.
in fact could be any category.
There
Let
S
= Set;
at
is an adjunction between
the 2functors
Cat ° p ÷ making
•
C 
a left adjoint of

F
+
~ Cat
~=(,S) g.
In detail:
CI
(C,S)
= Cat(C,S)
C'l
(,F)
=
C'[
(H 1
()
t h e effect of
o F:
6
is this:
•, ( c , s)
(c',s)
F C
If
~ foid H
(H ¢ O b ( C ' , S ) ) .
,,)
F'
All
arrows
functor
identical
are meant in Cat,
Cat to
) Cat.
~,
and n o t
we h a v e a c o n t r a v a r i a n t functor
Cat
~ Cat,
is
g.
The adjunction
consists
of an isomorphism between two categories:
ec,v: ( c , ( ~ , s ) )  , which is natural the following
i n Cat °p,
as a contravariant
Y:
in both
C
and
+
D.
( v , ( c , s~)
In detail:
OC, D
maps
X
into
296
C
x
+ (D,S)
D
y
, (c,s)
)
(
,÷ X ( C ) ( D )
C
D I "'
YI With
f
C
D
g
a morphism
natural
>
, D ' I .....
C'l
> X'
in
, X(f) D
> x ( c ) ( g )
> [ C
~: X ..
....
Oc, D
(C,(D,S)),
l
].
associates
the
transformation
D I (of
course,
in
C, N
D
ranges
means
that
F
~ (CI over
., CC )
Ob(O),
C
over
Ob(C),
etc).
Naturality
given
~
F'
X
>
( H , S)
_~
x'
'
(H,,s)
+
D'
Sh
,
D
H'
under
the
isomorphism
corresponding H
D'
composites >
$h H' Having
between
the a p p r o p r i a t e
@C',O'
composites
>
the above
to the
in
(F, ~ . . . . .
aCiD (X)
P
go o v e r
(C,S)
~eC,D(~)
,.
l(f,S)
(c',S).
(F' ,S)
OC,D(X')
adjunction,
we can c o n s t r u c t
another,
this time
2functors G P r e t o p °p
Now,
G
(as a c o n t r a v a r i a n t
Cat
+
functor
Pretop
.
....... > Cat)
is g i v e n by:
297 G(T) = Mod(T) and otherwise
similar
(a full subcategory
formulas
as before.
m(c)
It is essential, topos,
of course,
for any category
=
of
(T,S))
Also,
(c,s).
that the functorcategory
C,
with operations
(C,S)
'inherited'
is a special case of the fact that limits and colimits egories
are computed pointwise,
We have, the
provided
for the rest of the effect of
'pointwise'
character
will ensure that
F(F),
the codomain ~,
of the pretopos for arbitrary
similar
F: C
S.
This
in functor catcategory has them.
formulas
operations
is a pre
from
as before;
in (C,S),
~ C',
(C',S)
as defined above,
will indeed be a Pretopmorphism. The adjunctionisomorphisms Below,
referring
certain restrictions appropriate (0 = @C,N
are defined as before.
to the last adjunction,
of
2category,
~(F),
or
G(F),
and LH~, ~H l
for some appropriate
for
we will write F* for
for a morphism
F
@(H),
respectively
eI(H),
in the
C,D).
We turn to the proof of 1.4.4. Let 0
0
be an operation
be of type
(G,G'),
denote the full subcategory similarly With
for
of
K',
and domain
(G',S) is°
K.
Below,
with the objects
in
let
K'
will
K';
K c (G,S) is°.
i: G 
topos completion, F
in Set commuting with ultraproducts;
with graph
* G'
the inclusion,
~ = Bpt,
F(G)
and similarly
denoting
the Boolean pre
for the primed items, we have
making G
~
, F (G)
(1) G' commute up to an isomorphism.
~'
,F(G')
We can now construct
the following
diagram:
298
K
incl
* (G,S) is°
p
÷
~
(Mod F(G)) is°
i*
K'
incl
, (G',S) is° +
F*
~
(23
(Mod F(G')) is°
Here, all the starred functors are defined by composition with ~; F* is in fact F(F), properly restricted. The righthandside square commutes up to an isomorphism inherited from (i). Moreover, by the universal properties of F(G), F(G'), ~* and ~'* are equivalences of categories. The lefthandside square commutes when p is defined as the restriction of i* to K'; in fact, Oh(K) is the image of Ob(K') under i*. By the definition of "operation in a category", p is full and faithful. Taking quasiinverses of ~*, ~'*, and composing them with the inclusion, we obtain the upper square in the lefthandside one of the following two diagrams: K
H
~ (Mod FG) is°
Pl K'
Note that
=IF* H'
~ (Mod FG') is°
H, H'
(K,S)
÷
LHJ
p * l _~ (K',S) ÷
F(G)
F LH'.J
F(G')
are full and faithful.
The construction of the rest of the diagrams proceeds as follows. First, we pass to the square on the righthandside, by the help of our adjunction. Second, Q and r form the quotientconservative factorization of LH'J. Finally, again by the adjunction, we return to the triangle on the left (and take appropriate restrictions.)
299
Claim
F(G)
3.2.1. To deduce K'
rr~
QF
this from 3.1.8,
is full and faithful
faithful,
and
is a quotient. let's
~ (Mod T) is°
rr~
quotient),
~ T
since
Q*rr~ m H'
consider
(QF)*
~ (Mod F(G)) is°
H', Q*
are
since it is isomorphic
to
Hp,
(Q*
is because
(QF)*rr 7
The composite
and both
Q
is a
is full and H
and
p
are full
and faithful. Let whose
be the image of rr~;
~
objects
morphisms
are of the form
rr~(6),
with
image is welldefined,
the subcategory
rr~(D'),
6: D'
~ D"
with in
K'.
(Mod T) is°
Since rr~
and we have an equivalence K'
of
D' ~ Ob(K'), K'
and if full, the
~~
such that
+M
(Mod T) iso commutes.
Under the adjunction,
ing to the inclusion definition
of
conservative, that
rr~
'dense'
orization,
above.
we conclude
and
that
is full and faithful. and quotients, G
and
G'
is dense in
~
is full in T(G)
and
F,
and
F
is
By the facts
and by the last fact
(Mod T) is°
By the constructions
correspond
in the
to
Mod T.
are full and faithful,
to show that
KI, K 2
or Mod T above) X: K 1
X: K 1
.....
~
and that (QF)*~
of the free Boolean pretopos T
are both countable,
is ultraclosed
~ K2
Thus, ÷ K2,
ing of an isomorphism
X
in
since
Mod T.
('preultrafunctor'
in which ultraproducts
is a functor
fied isomorphisms.
as follows:
~
it is clear that
A ufunctor
itself,
~ OM, ~ appearing
are finite.
It remains
categories
T
functor
Since it is isomorphic
that
(QF)*rF ~
we have
the functor
is the evaluation
that preserves consists
in [MI])
have been defined ultraproducts
of a functor,
with
(e.g.,
K'
up to speci
denoted by
X
together with a transitionstructure consist
[X,U]
for every ultrafilter
U
on any set
I
300
(KI)I
[U]
~ K1
(K2)I
[U]
~ K2
IX,U]: X o [U]KI
,~ [U]K2 o X I
(here the [U] are the ultraproduct functors on KI, K2). ufunctor is one in which the [X,U]'s are all identities. The ufunctors K1 ÷ K 2 form a category, utransformations as morphisms. A utransformation natural additional
transformation condition
between of
the
compatibility
functorparts with
of
IX,U],
(Kl)I
[U]
÷ K1
(K2)I
[U]
÷ K2
A strict
u(KI,K2),
with the
o:
X
÷ Y
X
and
Y
[Y,U]:
the
is
with
a the
diagram
gives a commutative diagram of 1cells and 2cells: X[U]
+ Y[U]
IX,U] 1
flY,U]
[U]X I
+ [U]Y I .
If, in particular, K 1 = K, K 2 = S, and in fact, the forgetful functor u ( K , S)
qK
then
u(K,S)
is a pretopos,
+ (K,S)
(forgetting the [X,U]) is a conservative Pretopmorphism; easy to see (using Los's theorem).
this is
301
If I: T 1 ~ T 2 is a Pretopmorphism, then I*: Mod T 2 ~ Mod T 1 is a strict ufunctor. Similarly, for a diagram DO: G , T, (Do)*: Mod T
, (G, ~
is a strict ufunctor.
Now, let us turn to the diagram under (2) and (3). (~')*
is a strict ufunctor;
is one.
also, the inclusion of
As we said,
K'
in (G', @ i s °
It is easy to see that the quasiinverse of the functorpart of
a ufunctor whose functorpart
is an equivalence can be made into a u
functor by endowing it with a transition structure
(such that, in fact
it will be a quasiinverse with isomorphisms at head and tail that are utransformations,
but we don't need this additional fact).
in (3) can be made into a ufunctor, also denoted by readily seen that
LH'J
F(G')
factors
..........
H'.
Thus,
H'
It is then
(exactly) in the form qK' ....... ~ ( K ' , 9 .
~ u(K',~
Let u(K',S)
~
F(6')
T be the quotientconservative
factorization of
conservative, and hence
is conservative, by the uniqueness of the
qr
H.
Since
q = qK'
is
qe factorization, we have (K',S)
÷
q,
u(K',S)
*
H
F(G')
T with rqF~ ^
L an equivalence of categories. Now, we verify directly that is a ufunctor. In fact, we may put
=
([q~] (Di)icI)A
:
[~ (A) 'U] (Di)i¢I
Remember that each is defined.
((qr)(A))(HD~/U) + H(q~(A))(D~)/U. i i
r(A) c Ob(u(K',S))
is a ufunctor, hence [~(A),U]
302
We have
that
rr~ m
L*=qr ~.
Thus,
up to isomorphisms,
the image
^
of
vr ~
is the same as that of
ultraclosed serves
since
them. This
Thus
is closed
~
argument
by replacing c.f.
K'
rq~.
The image
under ultraproducts
rqr~ and
is clearly rq~
pre
is ultraclosed.
could have been subsumed
Cat
of
by a 2category
in the basic
of categories
adjunction,
with ultraproducts;
[MI]. We have shown that
3.1.8.
is applicable
to conclude
the Claim. •
3.2.1.
Let us define D~ = Q ~':
G' ~ +
T
D O = D~ o i: = Q o ~' o i: G Since and
Q QF
is a quotient,
D~
is a quotient,
In other words,
T.
is a generating
DO
is a generating
we have the commutative DO
G
diagram.
Since
D O m QF~,
diagram. triangle
~T
(1)
i=inclusion~ G'
in which both D' ~ K',
DO
and
we have
that
MoD~ m D'.
It follows
1.4.4 would be proved (and
(i) gives
repair Lemma
3.2.2.
and
generate
that
G
Also,
K' ~ Iso(MoD~: composite
notice
that for every
and, as is easily
M ~ Mod T}.
DO, D~
Thus,
Theorem
finitely generate
Boolean pretopos
seen,
T
operation).
by proving
Given a commutative DO, D~
T.
= ~r~(D ') ~ Mod T, df
if we had that
an abstract
the situation
graphs,
D~ M
triangle
both generating D1
T,
[I), with
F +T I
G, G'
finite
we have a factorization > T
We
303
in which
each of
Note that, M • Mod T,
DI, Di
we have
K' c Iso{NoDl:
finitely
MoD~ m (MoF)
operation
Proof of 3.2.2:
The
We s t a r t
mimicked T
subobjects inverting
T,
of
and
S,
with
domain
follows
of
of
A
T in
the
has
the
flavor
£ c Morph(T).
T,
then
the
is
for
any
is
r
,
be
if can
R
and
is an arrow
r
and
morphism In
S
be mimicked s F:
fact,
~ in
for
R
is
the
T ~
~ T'
the assertion T,
A
T
Rr~S
in
can
constructing
there
monos Pretop
a theory
E.g.,
invertible.
T ¢ • Sub(A×B)
'abstract somewhat
pullback R
as shown;
to
R ~ S
that
representing Fo
i.e.
"axiom"
The reason that
being but
new axioms
morphisms,
iff
of
the Theorem.
remarks.
adding
from
of the abstract
proving
straightforward,
some
such
FR ~ FS
in
Di,
inverting
arrow.
respectively,
monomorphism
and
and
constructed T,
is a restriction D1
is
(universally)
out
a single
explicitly
T I.
hence
some preliminary
(universally)
~ T[£ 1]
are by
by
that
with
The process
K'
given by
lemma certainly
The proof
tedious.
o D 1';
N • Mod TI} , i.e.
Boolean pretopos
nonsense'.
generates
if the assertion of the lemma is true, we get that for
~ S
is clear.
To give another
then the axiom
"¢ is a functional
(universally) imposed on T by (universally) 91 ~ A ¢, ~ A×B (see before 2.3.2). To start the proof of the lemma, F
example,
D
if
subobject"
is
inverting the composite
let us consider
the triangle
~T
F'
induced by
(I)
(here we use the same notation
as after Definition
1.4.3).
^
We pretend that
i
is an inclusion
(i.e., we also write
simply
R
for
^
iR
with
R
in
F),
and similarly
for the canonical
~: G
, F,
^
~': G' + F'.
We will write
R
for
D(R)
^
D(R)
(R
in
F)
and for
^
(R
in
F').
Remember
that
D
and
D'
are quotients
in Pretop.
304
This means
that they are full on subobjects,
there are
A
2.4.7
in
F
(in
F')
(ii) and also by taking
mentioned
by 2.4.7
To each
an object
in
G',
we assign
~X ~ Sub(Ax×X )
~ ~
Rx e Sub(AxxAx)
for any
T (by
many objects
an object
AX
of
F
in
F'
in
( i n T).
be such that
F
(in T).
PX
be such t h a t
RX = (the s u b o b j e c t of Ix×A x the k e r n e l p a i r of PX Let,
in X
epi
$X = graph of
Let
X
p: A . ~ +
the sum of the finitely
PX: iX Let
epi
(ii)).
X,
and an effective
and for every
and an effective
g: X ......+ Y
in
G',
induced by) (in T).
S$ ~ Sub(AxxAy)
in
F
be such that
S~ = (PxXPy)l(graph(g)), i.e.,
there
is a pullback
A:xX?y A
Px×PY
Sg In addition, RX
is the trivial
were
introduced.
Since
G'
^ " XxY
+ graph(g).
let us make sure that
and
+
equivalence
is a finite
graph,
in case
X
relation
is in
G,
in the above
finitely many
We are going to define T 1 as a finite quotient universally impose on F' the following conditions: AI. A2.
"~X
is functional"
(X c Ob(G')).
"R X
is the kernel
pair of the morphism
graph is A3.
~X'"
then
AX = X
(equality).
of
qx: AX
F'
items
We
> X
whose
(X c Ob(G')).
"S~ is the pullback in A2." (~: X + Y
(qxxqy)l(graph(g)) in G').
with
qx' qY
as
305
Some explanations specification in
F'
whose
A2, A3. that ~X
~i
qx
of
by remarks
Let AxA
[aeA,
= : df
a'eA:
= R
is
under
Note
concerning
As promised,
are
R'
< R ,
R'
of
z3
the subobject
< R)
F: T + T'
FR' = FR.
how to impose
~ = ~X
The set Z2
an inequality
corresponding
in Pretop
q: A + X is obtained
of subobjects.
to A3 is left to the reader.
we define
QI: T + T 1
the canonical
are all satisfied
the statement
is done as
^ ¢(a',x)]
A2
T 1 = F ' E ( ~ 1 u ~2
A2, A3
Z2
and construct
that for any
if and only if
AI,
~2' note
exist only when
is the kernel pair of the morphism
F~
above
above,
(3xcX)(o(a,x)
(i.e.,
X c Ob(G').
The description
with
(it will
Consider
specified
The axioms
AI, FR
graph
F'
the subobjects
R'
satisfying
in
"axioms"
A2 and
of
F'
one for each
as an arrow
Concerning
The specification
R':
in
to the respective
above.
of arrows
A = A X.
R = RX
AxA
Zl' ~2' ~3
functional).
of
whose
is equivalent
sets
X c G'.
Fix
AxX,
finite
from remarks
does not exist
has become
First of all, we intend here the
definite
inversion is clear
follows.
of
are in order.
of certain
by
u ~3)
i
quotient.
D',
]
Since
clearly,
of the lemma exists.
F
It remains
the conditions
AI,
as in the diagram
in
to show that the
composite ^
F
is a finite quotient Let F
T2
I
quotient
"R X
A3'.
"The subobject
is an equivalence S~
A X /R X + Ay/Ry"
clear
finite
of
F
obtained by imposing
on
conditions:
A2'.
We intend
T1
as well.
be the finite
the following
Q1
~ F'
sets
(see 2.4.1).
the proof of 2.4.5
~2''
~3'
Concerning
relation"
of
AX×A Y
(X ~ Ob(G')).
gives
(~: X + Y
in
of morphisms ~3''
rise to a morphism G').
of
F.
the inequalities
show what we need here.
We let
~2' (2)
is pretty (or (3)) in
306
T 2 = FE(E2, with
Q2:
F + T 2
Next,
note
the canonical
u ~3,)
i ]
quotient.
that we have a triangle ^
F
i
~
F'
T 2
commuting
up
isomorphism,
to
G' given
with
Qi
induced by
~"
~' T 2
as
X
Q2Ax/Q2R X
............. la r°w' duc d Y Then,
there
Q2Ay/Q2Ry •
is an essentially
F
G: T 2 + T
unique
~
F'
Q2 =
in Pretop
such that in
Q1
Q½
(2)
m G
2 the righthandside
triangle
commutes
up to an isomorphism.
To obtain
^
G,
note
solution
that is
Q1 i Q2"
satisfies This gives
the conditions an essentially
A2', A3' unique
G
whose universal with an
isomorphism ^
GQ 2 m Q1 i . The concrete
definition
of
Qi
gives
GQ½~'
(3)
an isomorphism
m Q1 ~'
(4)
307
for
9'
the canonical
the isomorphism lefthandside
triangle
We claim that obtained by imposing AI'
diagram
GQ~ ~ Q1
G' ~ F'
induced
by
G is a finite quotient, on T 2 the conditions
property.
that composition
Let
with
G
T3
(2), Q1 and Q2 are quotients. H: T 2 + T 3 satisfies
in Pretop
hence
G
satisfies
G
certainly AI',
there
then
is
IG m H,
the correspond
pretopos.
a full and faithful
is a consequence
satisfies
AI, A2, A3, hence
IGQ2 m HQ2 '
arrow A X + Ax/Rx"
be an arbitrary
induces
Pretop(Ti,T 3) + Pretop(T2,T3)
that
(3).
in fact that it is
"~X is the graph of the canonical (X ~ O b ( G ' ) ) .
ing universal
of
in the
in (2) is the same as the isomorphism
To show the claim, we have to verify
Then
such that the pasting
(4) and the isomorphism
functor
of the facts
satisfies
AI'
(as inspection
I: T 1 + T 3
as required.
The fact that in
Finally, shows)
such that This
shows
if
HQ½
IQI ~ HQ~. the claim.
Q2 and G being finite quotients, their composite, is easily seen to he a finite quotient as well. But then iQl , being isomorphic to GQ2 , is a finite quotient as well. This
completes
the proof of the lemma. m3.2.2. Ii.4.4.
Let us note that the same proof, establishes Theorem
the following
3.2.3.
ultraproduct operation.
Any strong
functors
"Composite "composite replaced
Let me point
a set.
The versions
with
instead versions,
of the general
with
pretopos
in the same way as
the free Boolean
pretopos
1.4.4 and 3.2.3 express
morphisms
of the theorem
Set I + Set,
of 3.1.4,
(over a graph).
In fact, of the form
talking still
a
ultraproduct Set I + Set,
I
about aZZ Pretop morphisms
of just ultraproducts,
phenomenon
are 1.4.4 and 3.2.3 requires
2categories"
of a composite
of Set in Pretop".
than the original
formulation which
operation"
Pretop
instead
in Set commuting
is defined
out that Theorems
property
are particular
of the form weaker
operation"
pretopos
3.1.2
operation
is the restriction
pretopos
Boolean
using
of 1.4.4.
finitary
by the free pretopos
"completeness functors
version
although
seem to be nontrivial.
of completeness
the context
The
two instances
of "essentially
of
algebraic
308
References EMRI Lecture
M. Makkai
and G.E. Reyes,
Notes in Mathematics
EMPI
M. Makkai
First order categorical
no. 611, Springer Verlag,
and A. Pitts,
Some results
logic.
1977.
on finitely present
able categories. EGZ~
P. Gabriel
topy Theory, ECKI
and M. Zisman,
SpringerVerlag,
Calculus
of Fractions
and Homo
1967.
C.C. Chang and H.J. Keisler,,
Model Theory,
NorthHolland,
1973. ESGA41
Theorie
des Topos et Cohomologie
M. Artin, A. Grothendieck no.'s
269 and 270. ECHL~
and J.L. Verdier.
SpringerVerlag,
G. Cherlin,
Etale des Schemas, Lecture Notes
in Mathematics
1972.
L. Harrington
and A.H. Lachlan,
~0categorical
~0stable
structures.
EShl
S. Shelah,
Classification
EMIl
M. Makkai,
Stone duality for first order logic,
of the Herbrand Symposium,
ed.
Theory,
NorthHolland,
ed. J. Stern, NorthHolland,
1978. Proceedings
1982, pp. 217
232. EM2~ Advances
M. Makkai,
Stone duality
for first order logic,
to appear
in
in Mathematics.
ETTI
P.T. Johnstone,
EMRII
Topos Theory,
Academic
Press,
M. Makkai and G.E. Reyes, Model theoretic
theory of topoi and related categories,
Bull. Acad.
1977.
methods
Polon.
in the
Sci.
24,
pp. 379392. ECWM~
S. MacLane,
SpringerVerlag, EPI]
A.M.
Pitts, Amalgamation
of Heyting algebras. EP21
A.M.
Categories
of
Pitts, An application
EM31
29(1983),
A. Joyal and M. Tierney,
Grothendieck,
and interpolation
J. Pure and Applied Algebra
J. Pure and Applied Algebra EJT~
for the Working Mathematician.
1971. in the category
29(1983),
155165.
of open maps to categorical
logic.
313326. An extension
of the Galois
theory
Memoirs A.M.S.
M. Makkai,
859, SpringerVerlag,
The topos of types, pp. 157201.
Lecture Notes
in Math.,
no.
309
EM4] A.M.S.
M. Makkai,
269(1982), pp.
[LJ
D. Lascar,
Full continuous
On the category of models
J. Symbolic Logic 47(1982), pp. [GU]
of toposes,
Trans.
in Math., no.
of a complete theory,
249266.
P. Gabriel and F. Ulmer,
Lecture Notes
embeddings
167196.
Lokal p r ~ s e n t i e r b a r e
Kategorien.
221, SpringerVerlag.
"Added in proof. A.M. Pitts has recently
proved an important result which is a strong conceptual
completeness theorem for Heyting pretoposes (conceptualizing full first order intuitionistic logic). His proof uses results of [PIJ, [P2J and [JT]. Theorem 3.1.2 is essentially a special case of Pitts' theorem, in fact with the countability condition removed. After the fact, it is not hard to see that one can deduce the stronger version of 3.1.2, i.e. strong conceptual completeness for small Boolean pretoposes, from the countable case, i.e. 3.1.2 itself.
One uses a forcing argument
and an absoluteness argument, after suitable preparation."
PROBLEMS
IN TAXONOMY,
A FLOATING LOG
Jerome I. Malitz University
of Colorado,
Department Boulder,
Boulder
of Mathematics Colorado
80309
U.S.A.
ABSTRACT. Given a region partition of
R R
in an ndimensional into
k
pieces
Cl,C2,...,c k respectively we have
Ixcil
normed linear space,
RI,R j .... ,Rk with centers of mass
such that for each
~ Ixcjl
is there a
for all
j ~k.
i ~k
and each
xc Ri
The problem is open even for
n = 2 and k = 2 but some partial results are known. The problem is highly relevant diagnosis §i.
and classification INTRODUCTION.
to the foundations
The problem of classifying
diagnosis.
medical
in general.
ing to certain characteristics and medical
of taxonomy,
arises in several
individuals
accord
fields such as taxonomy
This problem has been treated by different
authors (see [4],[5],[6],[7] ). If the characteristics can be measured by some metric then the individuals are represented by points in some product
space.
A classification
a partition
{XI,X 2 ..... X k}
reflect relationships
of
Xi
We call such a partition
We consider two notions by A. Ehrenfeucht
Of course,
X
of these points is
the partition
A point
X E Xi
should
should be as
as it is to the members of
Xj
for
'good'
of closeness.
of the computer
biology department nition measures
X.
between the points.
close to the other members j ~i.
of
of the set
The first arose in a seminar led
science department
at the University
of Colorado
and L. Gold of the
at Boulder.
This defi
closeness by average distance and is discussed
in
§3.
The second definition
suggested by Matt Foreman and the author, measures
closeness by distance
from the centers of mass of the
paradigm method of classification definitions
an d is discussed
the definition based on average distances
§5
§4.
This is the For both
good partitions
For
need not exist.
from paradigms we have only partial
and the existence problem is open.
The notation and basic definitions In
X i s.
the main problem is the existence of good partitions.
For the definition based on distance results
in
a heuristic
argument
are presented
in
is given which suggests
§2. that good partitions
311
exist. §6 presents §2.
some open problems°
DEFINITIONS
space with set
Y
AND NOTATIONS.
the usual norm.
will be discussed
continuous
version°
D(p,Y)
ii)
= Ipcl
D#(p,Y)
=
iii)
D#(p,Y)
where
of
P(X) = {XI,X 2 ..... X k}
let
x E Xio
C(x),
in
version
Y
n
to a and a
is given
has uniform density.
is the center of mass of Y. :
= [/ylpyldy]/fy
Then
is a discrete
p
y cY}
for
Y
finite with n
the
Y  {p}.
Let
The goodness
c
Y
in Euclidian
of a point
case each point
case
~ E{Ipyl n
cardinality
of distance
and for each there
In the discrete
unit mass and in the continuous i)
We will be working
Two notions
dy for
Y
be a partition
the contentment
of the partition
P,
G(P)
bounded
of
of
X
and measurable.
with each
X i~0
x, is minj~i D(x,Xj)
is min C(x). xEX
and
 D(x,X i).
P is good
if
G(P) e 0. If we use the
D#
distance
instead of the
D
distance,
C # and
we get
G#o §3.
GOODNESS
that for
IN THE SENSE OF
n = I and
k arbitrary
be obtained
in polynomial
n = 2
k = 2
3.1.
and
EXAMPLE.Let
matter
(0,0)},
P
for arbitrary
(0,0),
and show
{(0,I),
We do not have a general points
and raised
a
proved
#good
the question
(unpublished)
P that can
for
n > i.
For
is no.
X = {(0,i),
to check each
P = {{(0,i),
time,
the answer
D #. A. Ehrenfeucht
there is always
(0,i),
G (P) < 0o
(i.i,0)}},
method
(i.I,0)}.
For example,
then
ing theorem and its proof indicate
in
if
C#((I.I,0))<0.
for constructing
4, or for partitions
It is an easy
counterexamples
k > 2 pieces.
the difficulty
with
The follow
of generalizing
the
above example. 3.2.
THEOP~M.
for any
X
For every
contained
there is a partition
P
PROOF.
Let ~ > ~ > 0o
X
X
into
satisfying
E > 0 and every
k
there is an
in the unit disc and having at least into Let
k f
parts with be a
N
such that N
elements
G(P) > s.
iI function
on a subset
A
of
312
and Let
a)
Ixf(x)[ < ~ for all
b)
ixyj e s
B = X  A.
be the number Notice
that
of radius contained For
x ~ A,
for all
x,y ~ A u F[A].
B = f[A] u K where
of points
in
A
and
Ixyj e ~ for
all x , y • K.
Let
k
of elements
in K.
k is bounded, in fact 6  centered at the members 2
in the disk of radius
the number
n
k(!)2~ < H(I + ~)2 since the disk 2 of x ~ K are pairwise disjoint and
1 + 2
x • B  K, =
C#(x)
1
~ I y~A
i
JxyJ
~ n+kI y~B
_ 1 ~ jx_y I ny•A
>
1 n+ki
Hence
for K
[Ixy~
~ y~A
 Ixf(y) 1]
k n+k+k~f e  ~ n
or
x ~ K,
> ~ for large enough
Hence
for
N
C#(x)
or
C#(x)
large enough
If the set
K
nonempty.
and easier argument
shows
n. (since
k
is bounded)
we have
good partition
an undesirable
G#(A,B)
~.
by separating
approach
from the
of taxonomy. X
of points
is not required
can be modified so that for each G(P) < k for all P. In the proof of 3.2 points cells of the partition. diagnosis
x • B  K and
a similar
that are close to each other,
standpoint
Jxyl
yEK
> ¢ for
x cA
Note that the proof gives an approximately points
~
1___!__ ~ jx_y I n+kI y~K
k n
large enough,
empty,
ix_f(y) i
1 n+kI
y•A
n~ n+kI
>
For
~
I n+ki
[xyJ
this makes
to be bounded,
then example
that are close together sense.
example.
3.1
we get a four point example with
are placed in different
From the point of view of taxonomy
little
focus by the following
k
This shortcoming
or medical
is brought
into
313
3.3
EXAMPLE.
Let
X 1 = {(d,ei)
i~ n} u {(l,l+s i) X ~
: i ~ n} u {(l,ci)
: i ~ n}
: i~ n}
>0
where
and let
X 2 = {(d,l+~i): Ee l• < ~ .
for each i and
For
Xi, C(x)
I~ + /l+(de) 2
>
nd+e
2 Hence
for any
d
partition of 1
2nI and large enough
X = X 1 u X 2.
P is
n,
In fact,
P =
for
{XI,X 2}
n = I,
is a
#  good
s I = 0 and
#  good
3 §4.
GOODNESS
measurable
sets
IN THE SENSE OF D.
X.
A cut
C
in
Here w e consider only bounded
n
space is a plane of dimension
n  I, and so for n = 2 a cut is a line. For
k = 2,
a good p a r t i t i o n has a very simple geometric
characteriza
tion. 4.1 THEOREM° and
c I ~ c 2.
Let Let
P = {XI,X 2} ClC 2
with
ci
the center of mass of
be the line segment joining
C be the cut that is the perpendicular bisector closed half space determined by ing statements i)
P
ii)
C
of
that contains
cI
and
ClC2 . c i.
c 2.
Let
Xi Let
Pi be the
Then the follow
are equivalent:
is good.
Xi ~ P i
The restriction
for that
i = 1,2. c I ~ c2
is needed.
unit disk centered at the origin and ed at the origin,
then
X1
For example,
the disk of radius
P = {XI,X  X l} is good.
of taxonomy or c l a s s i f i c a t i o n
theory,
if
X
is the 1/2 center
From the point of view
good partitions with
c I = c2
are of no interest. Let
C
be a cut that partitions
of mass of I II So
X i.
Say that
if C bisects
Property
I
into
{XI,X 2} with
has property:
ClC 2
if C perpendicular
{XI,X 2} is good if
L
X
C
to
ClC 2.
has properties
is easily satisfied.
I and
II.
ci
the center
314
4.2.
THEOREM.
that satisfies PROOF. If
For every line
L
there is a cut
C
normal
to
L
I.
Consider
L
to be the
C(x) partitions
X
xaxis and let C(x) be the cut at x.
into X I and
X 2 with centers of mass c I and c 2
we let Ul(X) be the distance between c I and C(x) and u2(x) be the distance b e t w e e n
c 2 and
as x increases some
x,f(x)
In general, property
C(x).
Let
f(x) = Ul(X)/U2(X).
over the domain of f,
is continuous
I.
We conjectured
one cut perpendicular
that if
X
ly proved by M. Elgueta
(unpublished). L
and a line
that satisfies
to a given line w h i c h have
is convex,
then there is only
to a given line that satisfies
bounded
I.
This was recent
I do not have an example of a
with infinitely many cuts perpendicular
tained by Matt Foreman.
His beautiful
only in odd dimensional
L
Euclidean
space since it depends on the fact that
function
fixed point
x = f(x) or an antipodal point x
THEOREM
f
II was ob
proof, which we give below, works
a continuous reflection of
to
I.
The next result giving the existence of some cut satisfying
4.3.
and
Hence for
= I as needed. there may be many cuts normal
X
f
f(x) goes from 0 to ~.
on the surface of the sphere
S 2nI
x = f(x)
has a
(here x is the
through the origin). (Matt Foreman).
in an odd dimensional
Euclidean
Let
X
space.
be a bounded measurable Then
X
region
has a cut with property
II. PROOF.
We may suppose that the center of mass
of a sphere c
normal
S 2nI = X.
to the vector
w i t h centers of mass
For each c,s.
Cl(S),
So
S 2nI
the same side of s*
C
C
and
C(s), =
f
as
divide s.
of
X
is the center
C(s) be the cut through X
into regions
XI,X 2
One of the
c's, say
The line from
c
through Cl(S)
f(s).
is continuous.
f cannot map
f(s*).
c
let
respectively.
at the point we call
f : s2nI ÷ s2nI
fixed point
Let c2(s)
Cl(S ) is on the same side of intersects
soS 2nI
s
But this means
Since
onto that
s.
s
Hence
C(s*)
f(s) are on f
has a
is perpendicular
to the line extending
C,Cl(S)
since c = mlc I + m2c 2
where
mi
argument
for
2space giving a cut with property II
than
two
We have a different but for
n
even and greater
and so is perpendicular
and
is the mass of
to
Cl(S),C2(S)
Xi.
we do not know how to get such a
cut. Of course,
for a cut to be good. it has to have both properties
! and II.
315
As yet we do not see how to prove the existence of such cuts even for regions
X
§5.
in
2space.
A FLOATING LOG, HEURISTICS.
strongly suggests X in the plane,
Here is a physical model which
to us that good cuts exist.
We will consider a region
although the argument can be given for 3space and is a
bit simpler. Cylindrify levels.
X
in
3space and truncate
This results
the cylinder at two different
in a 3dimensional
object like a section cut from
a tree limb, a wooden log.
Float the log in liquid with its faces
a copy of
to the level of the liquid.
X) perpendicular
until stable position surface of the water, C with centers of mass
is reached. XI
the region above
c I and
By increasing or decreasing log so as to maintain where
C
At such a position,
c2,
C
we have
and
(each
Rotate the log with
X2
C
the
the region below
c I c 2 perpendicular
to C.
the density of the liquid while rotating
stability
(c~
satisfies both conditions
the
i C) a position will be attained I and
II.
We have not been able to convert this heuristic
argument into a mathe
matical proof. §6. PROBLEMS. Of course, the main problem is to show that a bound..... ed measurable region in Euclidean nspace has at least one good cut. In fact, we believe
that there must be several.
At the moment, we do not even see that regions one good cut.
Nor do we see that regions
in
2space have at least
in even dimensions
2n > 2 have
cuts with property II. For discrete point masses the questions C
are also open even if we allow
to cut through some points partitioning
the mass of these points
as convenient. The proof of 3.2 suggests and of measure > 0.
the following problem.
Is there a partition
{XI,X 2}
X I and X 2 have the same measure and i)
IXllpqld q ~ Ix21Prldr
for all
p c X I.
ii)
IXI IPqldq ~ I X21pqldq
for all
p c X 2.
Let
X
be bounded
of
X
such that
316
Postcript.
Shortly after the conference we obtained a proof for
k = 2
and all odd n. Fixed point theorems are used in this proof. Recently, A. Ehrenfeucht settled the question positively for all n and all k using methods that are not only simpler but more constructive.
However,
related problems in taxonomy that are settled by the fixed point methods do not yield to his methods.
REFERENCES [i]
Ehrenfeucht, A.
Classification Theory
[2]
Ehrenfeucht, A. & Malitz, J.
Problems in mathematical taxonomy (in preparation).
[3]
Elgueta, M.
(unpublished).
Unicity of cuts equalizing distances to centers of mass in convex regions
[4]
Chen, C.
Statistical pattern recognition. (197~ee
[5]
Murtagh, A.
Mauldin, R.D.
Hayden Books,
Chapter VIII in particularS.
Survey of recent advances in hierarchical clustering algorithms,
[6]
(unpublished).
Comp.J., 26,n ~ 4 (1983) pp.354359.
ed., The Scottish Book.
Birkhauser (1981)(See
problem 19, p. 90 in particular). [7]
Steinhaus, H.
Mathematical Snapshots. (1969)(See p. 50).
Oxford University Press
COUNTING PROBLEMS J.Paris Department Manchester
IN BOUNDED ARITHMETIC and A.Wilkie of Mathematics University
Manchester MI3 9PL, England
§
INTRODUCTION. In this paper we shall consider
P R O B L E M I.
Let
A
be a
the following problems.
A 0 subset of
~.
Then does
~.
Then for
{I m = IA n n i} have a
A0
definition?.
P R O B L E M 2.
Let
A
{i have a
A IAnn i =
Does
IA 0 ~ AoPHP
does
i mod k}
where
AoPHP
(A 0 pigeon hole prin
~x
~z <x
0(Y'z)AVYI>Y2~xVz < X(0(Y2'Z)
^ 0(Y2'Z)
÷ Yl = Y2 )
0A0?
There are several reasons is possible, ningful,
for studying these problems.
even tempting,
feasible
to view the
collections
induction)
as the self evident
standpoint
it is very natural
the
A0
~
as the mea
fragment of Peano's
axioms.
IA 0 (A 0From this sets
the size of sets is unique. reasons
for w a n t i n g
to know how
sets are and in how weak a fragment of Peano's Axioms
we can still prove the A0
One is that it
subsets of
to enquire w h e t h e r we can enumerate
And apart from this there are practical extensive
A0
of natural numbers and to view
in increasing order and whether
every
k ~
is the schema
@x [ V y for
i
A 0 subset of
A 0 definition?.
P R O B L E M 3. ciple)
be a
subset of
~
PHP.
For example Ritchie
is in Linear Space.
[9] has shown that
The general
feeling how
ever is that these classes are not equal and since A E Lin. Space
~
{
i m =
I A n n l } c Lin.Space
a way to confirm this belief w o u l d be to give a negative
answer to
p r o b l e m I. Problems
1,2,3 are all open and our intention in this paper will be
318
to survey
the area
(as we know it) and to prove
some new results
of
our o~v~fl. NOTATION AND DISCUSSION= in [7], the
[8].
A0
Most of our notation
When we employ expressions
definition
of exponentiation
will be to base 2 and in expressions be assumed
Throughout
of
A ~A~
xy = z
like
13].
log(x),
xe
as
we shall be using All logarithms it will always
that we mean the integer part of these quantities. this paper
and we identify subsets
like
given in
is standard,
M
a cM
M
with
defined by
will
stand for a countable A M0
{xIM ~ x < a } . A0
formulae,
model
of
is the collection
with parameters
I^ 0
of
from M.
For
let A (~) = {I m = I A n n l } , A (k) = {n[
]A nn]
It is easy to see that {[i
= 0 mod k}.
A(k)¢A~
A I A n n I = i mod k} ~ A ~
1 (2) is equivalent A e A~
is
CkX < t
(where
CkX
additional
t
0(x) ~
CkA 0
problem
ed by such formulae.
meaning
by in
M
PROBLEM Vx,y[<x,
M.
Let
Then problem A~
However
=
y> e B
Let
is
A~
in problems
i, 2 when we replace
these problems
Is there
counting,
~
defin
to:
B c A0
IA n n i = m
such that,
in
M
> 0 ^ <xl,yl> ~ B)
~' A ^ <xl, y> c B ) } ] ? or,
the standard may have no
to:
~> {(x = 0 ^ y = 0) v (x > 0 ^ x  i ~ A A y
closed under
of
this
?
since the expression
A ~A .
but with
be the subsets
2 is equivalent
v(x > 0 ^ x  I i.e.
CkA ~
CkA~
we must generalize I'.
LA) be the quantifier
formed as for A 0 formulae
adjoined.
We shall also be interested ~
2 is as follows.
] { x < t i 0 ( x ) } i = 0 mod k,
Is
model
(A (k) ~A~ ?)
is a term of
be the formulae
quantifier
to :
A ( ~ ) ¢ A~ ?
Another way of expressing Let
just if
(*)
to put it another way,
can
319
we count in
M?
PROBLEM Vx,y[<x,
2' . Let
y> ~ B <   > { ( x
= 0Ay
Is there
such that,
V(X>0AXI~AAy
kl>
M A0
=
0^<XI,
~' A A < x  l , y >
tive answer for all such can exhibit a replaced by
40
M
cB)}]?
theorem p r o b l e m i' (say) has an affirma
just if given any
formula
~(x,y)
A0
and prove (*)
formula in
14 0
0(x)
we
(with xI ~ A
e (xl) etc) .
We remark here that the general solutions.
feeling is that problems
Indeed we suspect that problems
A ~ h~ , for example
Finally,
~ B)
closed under counting mod k, or, can we count mod k in M?
Notice that by the compactness
negative
in M,
= 0) yl> cB)
is
simple
B •h
V(X>0AXI~AA0
V(X>0AXI ioe.
A• 40 .
i, 2 fail for quite
for the set of primes.
in connection w i t h p r o b l e m 3, notice 1h 0 + V x, 2 x exist
so the p r o b l e m amounts
1,2,3 all have
that by the standard proof
j AoPHP,
to w h e t h e r we can remove e x p o n e n t i a t i o n
in this
proof. SECTION
i.
In this section we mention some simple connections
between these problems. mod 4 since given (i = 0 ^ n E A
Clearly
M A~A0,
0<i<3,1Ann
(2) n ( A n A ( 2 ) () 2 )) 
^ ~ @z < n ( x < z A z ^ [email protected]
~A))v(i
if we can count mod 2 then we can count I = i mod 4
v (i = i ^  ] x < n ( x ~ A ^ x c A
= 2^@x
(2)"  n (A hA(2)) ( 2") " " " (2) n ( A n A ( 2 ) ) (2)
+ yAzeA))v
(i = 3 A ~ x < y < w < n ( x , y , w e A ^ x ~ A
(2) a ( A o A ( 2 ) ) (2) A  ~ z < n ( x < z ^ z + y^z
In general we can count mod k Pl .... 'Pj
<~=>
4 WAzcA))
just if we can count mod
are all the prime divisors
of
k.
However
Pl .... 'Pj
where
the following
p r o b l e m is open° PROBLEM.
Let
p, q be distinct
(standard)
primes.
mod p in M can we count mod q? In this connection we do have the following result.
If we can count
820
THEOREM p
then in
PROOF.
I. If M
in
M
we can uniformly
By the hypothesis we mean that given
cB
<~
n,m,p c M, In = 0 A m
for all primes
p
prime
A ~ A0
there is a
(in M) and
m
v[n>0AniEAAm>0A
ml, p> E
v[n>0Anl~A^m=0^
pl, p> e B]
"~ B
~
B eA
then
= 0]
v[n>0Ani
Then we can
p
we can count.
such that whenever
i.e.
count mod
~'A^
B]
m, p> e B ]
IA nn I = m mod p".
'define'
IA n n I = j
~=~ j is minimal
such that for all primes
I An~ To show that such a
j
p_
= j mod po
exists we show that by induction on n, for fixed
m,
Vn ~j _
p<m,
IAnnl
to show that as defined above
= j mod p. IA n n I = j satisfies
the required counting conditions. A close connection b e t w e e n Woods
[II].
Woods'
counting and the
AoPHP
result will be discussed
was discovered by
in section 4 (Theorem 20)
but for the moment we content ourselves w i t h the following special case. T H E O R E M 2. the
Woods
[ii].
If in
M
we can count then
M
satisfies
AoPHP.
Woods'
proof
(of Theorem 20) is quite long.
Perhaps
to prove the theorem is to show by induction on F : a
I~> aI
(i.e.
F maps a iI
into
IF n (yx(al))l
= Y
IF n (a×y) I < Y
y
the simplest way that if
al)
then
for
y
for
y<_ aI
F c A~
and
w h i c h gives the r e q u i r e d c o n t r a d i c t i o n a = IFn (ax(al))l Notice
that this only requires
Of course this result a set has a reasonable
< al.
a single counting,
is not at all surprising,
of
F n (Y0 x yl)
it simply says that if
definition of its size then there cannot be another
321
such definition
of its size i.e.
Further results
connecting
example,
assuming
ii onto al)
its size is unique.
theseproblems
counting mod 2, if
can be proved similarly.
F : a ~>>(al)
(i.e. F maps a
then by induction on y IFn (y×(al))l
= y rood 2
IF n (axy) l = Y mod 2 which gives a contradiction T H E O R E M 3. mod k.
Let
Then for
as above.
0
M F~A0,
~x,
for
y~a
for
y ~ ai
Similarly
and suppose F : X
Returning for a mo~mnt to T h e o r e m 1
that in
l~>>xp,
M
we can count
for all
0
b o u n d e d l y many primes.
~,
we w o u l d of course have had a much
easier proof of this result if we could have shown that due to Woods
For
In fact by Theorems
M
had un
i, 2 and the following result
this is the case although to prove it we seem to need
T h e o r e m I. T H E O R E M 4. [Woods,
[ii]0] If
M
has the
AoPHP
then
M
has arbi
trarily large primes. SECTION
2.
our knowledge)
0
]in this section we shall give the best results
concerning
the problems
(to
in section 0.
Of course the obvious way in arithmetic
of stating that
IA n nl = m
is
simply ~f,
f codes a
Unfortunately f
Ii
map from
m
onto
Ann.
this will give a best upper bound for
cannot in general be b o u n d e d by a term in LA.
suitable basic
conditions
on
A
we can do this.
idea (a trick due to Nepomnyascii,
IA n n I
= m<=~ [email protected] c (k+l)(n+l), increasing,
f
However
see [6]) by disecting
f ~ (k+l)(m+l)
f so that
such that g, f are
f(0) = 0, f(k) = m, g(0) = 0, g(k)=n [g(i), g(i+l)).
this device gives
T H E O R E M 5.
Let
ke~,
0 <e < 1
and
A = {m Im
if we put
We can now refine this
and Vi~ A n Iterating
of around n m so
A
n
~ 2 l°g(n)~ 
for
all
n ~.
AcA~.
For
e A}, Then
n~
let
322
B k = {lj 
By induction
For
k = i
on
k.
(log(n)T)k+l)}
Clearly we may assume
n
~A~.~
is large.
we have
m = min(IAn c [i, J)l,
log(n)
2+1) IC~
<=> ~f(f:m
IF>> A n n[i,
v no such
f
exists
Then with the given condition
j) ^m
and
m =
f)
log(n)2+l.
on the size of elements
coding
of A n and the usual
I~ f _< (21og(n) e) log(n)2
Now assume
the result
for k.
Then I~ (k+l) (l°g(n) T ) + I) ~=>
m = min(IAn n [i, j)l,
i~ <=> {@h
: (log(n)
such that
[
I~ +i) ~ [i,j] @g
m< (log(n)
~
:(log(n)
2 +i) + (m+l),
(k+l) , g, h increasing
I~ h(0) = i, h(log(n) T )
I~ = j, g(0)
I~ and Vs < log(n) ~ , g(s+l)
h(s+l),
n, g(s+l)
i~  g(s) _< (log(n) T )  g(s)>
or {no such h, g exist and This immediately THEOREM 6.
= 0, g(log(n)
~ B k}
kcl',1,
0 < ~ < I,
A n = {mlc A ^ m < n } Then there is a function
F ~A~
A E A~
_c 2 IOg(n)e
and
I~ (k+l) m = (log(n) ~ ) +I}.D
and suppose
for all
such taht for all
F(n) = min(IAnl , log(n)k).
k
or
gives:Let
~ ) = m
n ~
that
ne~.
323
Having got this amount of counting we can drop the condition A n ! 2 l°g(n)
Precisely:
T H E O R E M 6'.
Let
ke~,
AeA~
and suppose
that
A n = {mle A ^ m < n } . IN Then there is a function F ¢ n 0
such that for all n elg,
F(n) = min(IAnl, This theorem follows
from T h e o r e m
6
log(n)k).
D
and the following t h e o r e m w h i c h
will be proved in a forthcoming paper by the authors: T H E O R E M 6".
Let
there is a function
A
be as in Theorem 6'
n
F ¢ A~
and let
such that for all
E > 0.
Then
n ~
i+~ F(n)
: An
We remark here that Theorems k, ~, ~
still standard)
It> IAnl
5 and 6 hold w i t h
A~
Furthermore
M
p r o v i d e d we interpret
~F, F : log(n) k suitable
I] in place of
"]Anl
~> A n
1,1
(but
_> log(n) k'' by
for a
function F etc. combining
of the previous
this limited amount of counting with the methods
section we can show
THEOREM 7 •
For
k~l~
and
M ~ ~ ~x,
F E A~ F : log(x) k
Ib> log(x) k  I.
As an aside we mention here that this can be combined with Woods'
proof
of Theorem 4 to give T H E O R E M 8.
For
k clg,
It is not clear at present
M ~ Vx 3 prime p, p > log(x) k. that T h e o r e m 6' can be similarly
D generalized.
However we can show the following w e a k theorem: T H E O R E M 9. < B Then
Let
A ¢ AM
= ~(i + log(~) k) A (~) e A M . U
O
< B
and suppose (kelq).
that for all
a,B eA,
324
PROOF.
Let
N c M.
Since
(I + pl)p ~ 2
for
(I + log(n)k) ~ e n Hence we can define
F : A nn
>
p > 0,
for some
log(n) k+2
by
F(x) = least B, (I + log(n)k) B ~ and count
Ann
SECTION
by counting
x < (i + log(n)k) B+I
Ann.
D
3. In this section we give an illuminating
of counting mod 2. machine based relations
F"
~ ~ log(n)k+2o
This characterization
description
corresponding
by Bel'tyukov to describe
in
to the function
[2].
C2A ~
is a natural
of the classes
sf,(the f)
characterization
variation
Grzegorczyk
and the class
We shall show that these machines
and hence highlight
what
it means
for
of a
class of
A~
given
can be used A~
to be
closed under counting mod 2. We first describe Such a machine
Bel'tyukov's
M
consists
Stack Resister
Initially
the input goes The program
where each L (i) (ii)
x 0 .... ,xm and the other registers
M
is a sequence
of instructions
t i := ti+l & Vj
tj
q+l'
zI + z2 = z3
goto
Li, else goto Lj,
(iv)
if
zI . z2 = z3
goto
Li, else goto Lj,
z, Zl, z2, z 3 are chosen i
and
at most one of
M M
from
0, x, t, r.
L!,L2...L p contains
eventually
halts with
tially with Bel'tyukov's ly]. f, g : ~ ÷
if there is a SRM (i) (ii)
M halts @i, j el~
M
t k = 0.
and
M accepts
[Our definition
in order to make
~
In this program
the instruction
halts just if he is told to go to Lp+ I.
For functions
are
LI,L2...Lp
:= 0 & goto Lq+ I,
if
each
r.
has one of the forms
(iii)
where
if
q
t0,t I ..... t k and a work register
into
for
(SRMSs).
of a finite number of input registers
Xl,X 2 ..... Xm, stack registers
all zero.
Machines
A c ~]m
for
ti:=ti+l
an input
disagrees
x
inessen
the later work run more smooth
we say that
A c SRM Space
(f,g)
with the following properties:
on all inputs
~
and accepts
such that for any input
iff
x, during
the computation
325
of
M
on
x r ~ fi(max(~))
and
to,t I ..... t k ~ gJ(max(x)). Let
~(x) = i, etc ., P2(X)
0(x) = O, THEOREM
I0.
= x 2 + 2 for all
(a) follows
Lin. Space
= ~
Then
(a) [Bel'tyukov,
[2].]
SRM Space
(p2,P2)
(b) [Bel'tyukov,
[2] .]
SRM Space
(@, p2 ) = A~,
(c)
SRM Space
(~, p2 ) =
C~
(~
Sm~
Space
(2 , p2 ) =
C6A~ ,
(e)
SRM
Space
(B , p2 ) =
C~
immediately
= Lin.Space,
,
.
[Actually Bel'tyukov shows that for reasonable functions Result
x~.
f,
since by the result
SRM Space(f,O = s of Ritchie
[9],
.] P2*
PROOF.
Result
(a) is proved
in
line a proof of (c) borrowing The proofs We first
of
(d) and
show that
suitable machines it only remains tion,
(e)
[2] and
heavily
are along similar
C2A ~ ~ SRM Space to decide
if
to show that
SRM
Space
quantification
this last property
suppose
that
M
is a suitable machine
We may assume tk = I.
We amend
M
M' looks
and
Xl.X 2 = x 3
p2 ) is closed under nega
and counting mod 2.
A _c ~m+l,
so
A ~ SRM Space
(~ ,p2 ) and
(with the above notation)
that at the end of any computation
of
To show
to accept
M, t k = 0
Ao
or
Letc B
Suppose
( ~,
[2].
Clearly we can find
x I = 0, x I + x 2 = x 3,
bounded
We outin
lines°
(~ , p2 ) .
conjunction,
that
(b) is stated there.
on ideas of Bel'styukov
~=~ C2x < y, <x, ~> c A.
to form a suitable
like
M
but has further
that on input
r = 0
if
machine
z = y
then apply
If
z ~ y
(so
z
stack registers
at some stage
C 2 x < z,
If
M' to accept
M'
t_l,
B.
Phisically
tk+ I, tk+ 2, tk+ 3.
has tk+ 2 = z,
tk+ 3 = 0
<x,u> c A, else r = i. tk+ 3 := tk+ 3 + 1 (etc)
in fact)
set
if
r # 0 and halt.
tk+ 2 := tk+ 2 + l(etc).
Then if
326
r # 0
set u>
tk+ I := tk+ I + i (etc) ioe.
with
of this computation if
t k = tk+ I
Clearly class
we have
for
(i.eo by setting
M' is
Bo
It remains
to show that
SRM
and program
r = 0
that
If at the end
t k + tk+l,
on all inputs.
Let
M
p~ (max(x))), Ii
and
r = i
r = t_l)
program and the accepted (~, p2 ) is closed under are similar.
(I, p2 ) ! C 2 A 0" M
Let
if
SRM Space
Let
LI,L 2 ..... Lp.
tk+ 2.
quantification
Space
M on the input
:= t_l + i, (etc)
[2].
to, t I ..... t k (bounded by
and run by
out as a suitable
and bounded
is due to Bel'tyukov
space and halting
everywhere
t_l
The proofs
conjunction
r = 0
t k = 0 put
this can be written
negation,
proof
x 0 replaced
and
The method
be an SRM running
of
in this
have stack registers input registers
x 0 ..... x m
be the instruction
amongst
LI,L 2 .... ,Lp(if any) which starts t i := t i + I & Vj < i, tj For convenience that
L I = I0
(accept) Then for
or
we make
i (reject)
appears
in
we can find
(i.e. having
&0
about
we have just applied x, r) ~ >
r
rather
~iq(~,
of
instruction
than
t k
x, r) e A 0
and
M Ii
the stacks
A0
0
functions
contain
~, ~, r
and
then
[the next instruction ed is
M, namely
the answer
graph and value bounded by a polynomial)
such that if in any computation
0iq(t,
assumptions
and that at the end of the computation
0 ~ i,q ~ k
Fiq(~ , ~, r)
some unimportant
:= 0.
of the form
Ip
to be execut
I ] or [q = k+l and we will halt without q another instruction of the form Ip].
applying Furthermore
if
9iq(~,
when we are just about eiq'S exist for more
follows
than,
of the form I i, So
M
say,
~, r) then to execute
x, r) is the value of
or halt if q = k+l. q from the fact that no computation of M p(m + k + 3)
otherwise
is equivalent
Fiq(~, I
moves without
it would
to a machine
contain K
whose
applying
That such can go on
an instruction
a loop. instructions
r
are:
327
(i)q (q
In state
:= 0 f o r
In state ~q = V i
L
j
L
if
if
~q(t,x,r)
and return ~k+l(t,x,r)
to
set
r:= Gq(~,x,r),
tq := tq + i,
L.
set r := Gk+l(~, x, r) and halt, where
(0iq(~, ~, r) A t i + 0 A A t. = 0) V (q = 0 A r = 0 ^ A tj = 0) j
and Gq(~,x, r) = z <=~ ~/ (0 iq(t,x, t. = 0 A Fiq(~,~, r) = z) ~ r) ^ t i ~ 0 A A i j
K'
~q, ' G'q c C2A 0.
=
0Am
equivalent
=
to
0).
K
with one less
Repeating this process until we
are left with just the final "halt" instruction gives the answer. We shall remove the loop corresponding and consider a computation of L.
K
It will continue to execute
arrive at a value knew
s O of t o
to
(i)0, producing
r0, rl, r 2 .... until we
such that
Since
rs
0
we reset
can only take values
t = tl,t2,...,t k
0, t, x, r and in state
~0(s0,
s o and rs0 in the remaining instructions for
Let
starting with
So, rs0 beforehand we could bypass
we apply (i)q
(i) 0.
t, ~,r).
(i) 0 by replacing t o and r by of
K. (Notice that whenever
t o to 0).
0, 1 we can define
r s = 0 ~=~ ~b < s [b is maximal such that
Now if we
rs
by
G0(b, t, x, 0) G0(b, t, x, I)
.x
and either or
(G0(b, t, x, 0) = 0 A C 2 x < s(b < X A G0(X , t, X, 0) = i))
(G0(b, t, x, 0) = I A
or [no such
b
~ C 2 x < s(b < x A G 0 ( x ,
t, x, 0) = I))]
exists and either
(r = 0 A C 2 x < s, G0(x , t, x, 0) = I) or(r = IA, C 2 x < s, G0(x , ~, x, 0) = I)]. Hence d J s o = least s
to
such that
~ C0(s , t, x, r s)
K' with instructions :
328
(i)~ (! ~q ~k)
In state
r := Gq(S0, t , ~x ,
L
if
~q(S0, ~, x, rs0)
set
rs0 ), tq := tq + I , t.J := 0 for 0 <j
and return to L o (ii)'
D
In state
L
if
~k+l(S0, t, ~, rs~ set
r := Gk+l(s0,t,~,rs0 )
and halto The theorem now follows, The fact that SRM Space (2, p2 ) = SRM Space (3, p2 ) is rather surprising and suggests that perhaps the same result holds for I and 2 , i.e. that if you can count mod 2 then you can count mod 3. Of course a natural conjecture at this stage is Conjecture.
SRM Space ~ ,
p2 ) = C(k+l)~A~
For other investigations along these lines see Hicks coming paper by HandleyParisWilkie.
[5] and a forth
By directly applying the last theorem we can obtain a further characterization of C 2 A ~ due to Gandy (by a different proof). THEOREM II. [Gandy, unpublished.] Let G be the class of functions formed from the characteristic functions of the graphs of +,~ by substitution and (bounded) primitive recursiono characteristic functions of the sets in C ~
Then
G
is precisely the 0
SECTION 4. In situations such as we have with these counting problems a currently popular reaction is to prove some oracle independence results, thus confirming (what we already knew~) that these problems are not entirely trivial. results.
In this section we give some such
The relativized versions of A~ are formed in the obvious way. Namely add to LA a new unary relation symbol X, define bounded formulae, A0(X),
of
LA(X)
there is some
as before and for
A, B ~
#(x, X) c&0(X ) such that
B = {~c~
Notice that by induction on the length of is m~ ~ such that if n > i and A n {~n
nm~
=
]~(~, A)} =
say that
iff
l~(n, A)}.
~(x,X) we can show that there
m D n n ~ then {~n
B eA~
I ~(~, D)}.
329
Our first theorem is very straightforward. THEOREM PROOF.
12.
@A _c IN
Suppose that
ly unspecified.
such that
An p
If
p
A0 A
is closed under Counting.
has so far been defined and
Ap is complete
has the form
m
~, X) ~ , ~, k>
~_
and
peA
~=> [{x
Otherwise put Whilst
F~(x0,
~(x0, ~, X)
p e' A.
¢ A0(X ) set
The result
= k.
follows by the above remark.
this tells us that any proof that
ing cannot relativize however
n0
is not closed under count
it really gives no new insights.
Our next result
seems more promising.
THEOREM
13 •(*) @A c IN
such that
A A0
is not closed under counting
rood 2. PROOF.
The proof is an easy application of a very beautiful
due to Ajtai
[i] characterizing
theorem
sets of the form
{A ! n I ~(n, A)} for
~(x, X)
the special Let
e A0(X) o [Actually Ajtai's result is much more general case we shall need. ]
First we introduce
be the language of arithmetic but with
relations
and for
domain n.
n¢ IN
let
n
It is straightforward
there is a formula
~(x)
e L (X)
be the obvious
+,. treated as
structure
to show that for any such that w h e n e v e r
than
some notation, 3ary
for L with
~(x,X)¢ A0(X) 'A ~ n, ne IN,
~(n, A)<>~ ~(A). This result,
for
result for
L (X)
For
let
f£ n2
subset of
n
L, is proved in
[7]°
Xf = {m Im < n ^ f ( m )
into
2}
and for
Ajtai's
• 
T h e o r e m w h i c h we need is
(*) of
n
[Ajtai,
eventually
[i].]
c= 0}.
fe ~n2 n
T H E O R E M 14.
An = ~ the
is a simple g e n e r a l i z a t i o n
n o t a t i o n will only be used w h e n
for all
Since we assume
~n 2
Let let
Let
There is a striking resamblance
e >0
maps a
f = {he n2I h = f}.(This
is implicit.)
~S c T c i n 2
= {f If
and
Then the version of
~(X)¢
L (X) .
Then
such that
between
this theorem and results
[12] which were unknown to the authors at the moment of writing this paper.
330
(i)
{f I f E T}
(ii)
f cT ~
form a partition of
I dom(f) l
= n  n
le
n2,
, le
(iii)
l{h en21~ 6(Xh) }& U f fcs
I < 2nn
Here A stands for symmetric difference. {h E n21~ 8(Xh) } simple set
U f. f ES
both sides of h En2
(iii)
The result says then that
can be approximated very closely by the
To see just how good this approximation
is divided
by 2n.
that
This shows that the probability
chosen at random is in one of {h en2I
~
0(Xh)}
~
but not in the other is at most
, ~ fcS
I/2nle
As a simple corollary to this we have. THEOREM 15. ly we can find
Let cn qe  2
g et2,
0(X) E L(X).
such that
Then for all
n
eventual
q ~ g, Idom(q) l ~ n vrn+2 t+l and
either _~ {h on21p 0(Xh)} _c {h E n21 p ~ e (Xh)}.
or
PROOF. Let
Let
n
be large and let
H = {f c r I f n g ~ ~}
Ifn{hcn21
T
etc be as in theorem with ~ = !~ .
and suppose that whenever
~ 8(Xh)}l,
If n{h En21
f EH,
Xh> ~ ~ O ( X h ) } l
>2t.
Then 2 nvr~ >
I{h En2]F0(Xh)}A U fl > 2t. IHl . fcs
But
U f ng feH
= U (fug) fcH
IHl
=
•
which gives a contradiction. find f E H such that
Hence, since n
2
g
so by counting elements, >Igl
=
2nt
So, without loss of generality we can
If n{h En21~ e(Xh)}l~ 2 t c n and f ug E 2, Idom(f u g) I < n  v~ + t. cn is large we can find q ~  2 such that q ~_ f u g,
Idom(q) l < n  ¢~ + t + 2 t<_ n  v~ + 2 t+l
and
331
n {h ~n21
~ 0(X h)} =
as required. PROOF OF THEOREM found
A nt
13.
and
We construct
~(x,X) c A0(X)
{nI~(n,A)} Fix
n
large.
~ {n I I A n n l
We arrange
enough
A n [t,n)
Without
Then
so that by earlier remarks
~ ~(Ann)
it will be
n ~ A (2)
~=> cn
q~
2
such that
q F t ~ t2,1dom(q) I
and q ! {h~ n2 I~ ~(X h)}
or
~ ~ {h ~ n21 <~, Xh> = ~ ~(Xh)}.
loss of generality h c n2
assume
such that
the former.
h ~ q
and
Then since
IXhl
is odd.
Idom(q) I
A n n = X h.
n c' A (2) whilst
In view of our present elements
~ ~(Ann).
inability
0
to count even sets with fairly
it would be nice if we could arrange
small.
However
similar
open problem
is closed under Unlike
n ~ A (2)
either
we can find
that
to satisfy
By Theorem 15 we can find Xq ~t = A n t
Suppose we have
to arrange
= 0 mod 2} = A (2)
<=~
A n In, n m~) = ~
to find
in stages.
that
~ @(n,A) We shall set
A
and we wish
Ajtai's
For example
is whether
A~
results
IAI
here to be very
does not seem to give this. or not we can find
counting mod 2 but not,
some oracle
any proof that
Theorem
say,
few
A
Another
such that
closed under
A
counting.
this one gives rather more than just that
is closed under counting mod 2 will not relativize.
an obvious
way to attempt
to prove
that whenever
0(x) e A 0
then
{x I e(x) }(2)cA 0 would be by induction
on the complexity
of
block on doing this in most natural ways. the theorem if
e.
But the Theorem puts a
For example with
A
as in
332
A I = {2n I nElq}, then we do have
A (2)
A 2 = {2hi n c A} u {2n + II n c' A} A~2) ~ A
'
but we do not have
(AInA2)(2)eAA
0"
In [I] Ajtai also uses his Theorem 14 to show that distinguishing between sets of odd or even size is difficult. T H E O R E M 18.[Ajtai, of Peano Arithmetic,
coded in
K
.
[i].]
a e K,
such that in
T H E O R E M 19.[Ajtai, of Peano Arithmetic,
2
Then
~P ! a ,
K
K
be a countable n o n  s t a n d a r d model
a nonstandard. K,
[I]
IPI
.]
is even,
Let
K
Then IQI
@P, Q c a, P, Q is odd and
be a countable n o n  s t a n d a r d model k and R c a coded in K.
a ~ K, a n o n  s t a n d a r d
P coded in K
is even and for all in
Let
Precisely:
such that for all
AI, .. ., A m _c a coded in
i
l{jl ~ P}J
K @R' , Q, AI, ' "''' A'm
coded
such that
i < a,
,Am> ~
I{jJ c Q}I
Let
F
+ ~x, F : x
lit does not matter if we allow quantifiers above
in
of the PHP.
be a new unary function I&0(F)
140 (F)
F
..,A~>
is odd.
We now turn to look at oracle versions PROBLEM.
Q, A'
symbol.
Then is
i~>xI consistent?
to appear in the bounds on the
since we can always replace
F
by the identity
"x".]
At present we know of no full solution to this p r o b l e m although there are several partial results. T H E O R E M 20.
[Woods,
Firstly Woods has shown:
[Ii].]
Let
Def(G,
F) be the a x i o m
G(x,y) + I Yx,y [ G ( 0
,
y) = 0 ^G(x+I,
Then IA0(G , F) + Def(G,
y)
=
G(x, y)
F) ~ ~ ~x, F : x
if
F(x) ~ y otherwise
].
~e>xl.
D
Notice that Theorem 2 is a special case of this. Our next result shows that we can give a positive for the fragment T H E O R E M 21. I ~I(F)
answer to our p r o b l e m
I ~l(f). I @I(F) + Ix, F : x
le> xi
is the induction schema for formulae of
is consistent, LA(F)
where
of the form
333
~x I ~x 2 ..... @Xn0, 6 quantifier
free.
PROOF.
K
Let
cK
be a countable
and ~ nonstandard.
argument.
Forcing conditions F(Xl)
with
~<~,
formula
nonstandard
We produce
~
<e
from
is defined
F II. and
= Y2 ..... F(Xn)
For
~ a forcing
~]I % ~=>
dependent
If ~ k(b)
n
k(x) = ~ye(x,y)~ ~ ~
or
and
~T Z o,
e I .... ,ek
for
for % not involving
@I(F)
TI~ %(b)
and
a and
IT  o[ _
appearing
oil ~ l(b)
and is compatibile
is nonstandard decides
0(b,a),
and a is finite. T IIO(b,a)
We can now pick a complete that
~
I~I(F)
that every nonempty have a forcing
F, etc.
Then there is a fixed
Fi(ek ) for
are the constants Then either
i< j
or
~I(F)
with ~.
Then
n clq,
b c M either To see this notice
i< j,
in 0.
some fixed J,
Suppose
@v ~_ ~,
This
of
there are
9116(b,a~ ~(~
is possible
r u ~II 9(b,2)
for ~(el)
since
so since T
and hence T I~ k(b). sequence
+ F : e+l
condition
a
(f(a) = b) e O,
a. Hence we can pick T =_ o such that T specifies ~(b), ~(al)
Fi(ek )
~(x)
6(x, Yl ..... ym ) we only need to know the values
such values.
some
~=~
M ~ k
F i (x) . Fi(yl . . .) . . .Fi(ym . . ) . Fi(el . . ) where
condition
only on 0 such that for any condition
that to decide
= Yn'
e.g.
~[~ F(a) = b
that
forcing
~ ~ M,
in the obvious way,
Now suppose
axioms,
by a simple
are finite sets of the form
= YI' F(x2)
and
LA(F)
model of Peano's
F : ~+I i~>~
of forcing
conditions
to ensure
i~>~. For it is enough
set has a least element.
o and a If ~(b)
to force
So suppose we
for k ~ ~I(F).
Then for
n
as above {a_
]~ forcing
is definable T 20.
in
K
condition
T ~ o,IT  o I _
and hence has a least element,
e say.
Let
T I~ %(e).
Then by the above remarks
o I~ , k(a) for all
a <e
so • forces
that e
334
is the least element satisfying
X(x)
in
.
D
Along similar lines we have T H E O R E M 22. (successor)
[Goad,
and let
[4].]
TO
T O + full induction
Let
be the for
L0
be the language with just < and
L 0 theory of ~.
L0(F) + ~x, F : x'
Then l~>x
is consistent.
D
Goad proves
this by a quantifier
be somewhat
simplified by w o r k i n g with models
elimination
argument.
a countable n o n  s t a n d a r d model of Peano Arithmetic,
let
"integers"
L0
J +
~j,
of
J
and
in that order.
standard.
Define
so
F : d + n since
K
be the structure
Then
F : K ÷ K
F(x)=
not,
let
{
K ~ TO .
dcZj
x+~
if
xcJ,
x + G  i
if
J<x
x  d
if
d~x
x
if
d + n ~ x,
is definable
from parameters.
for
Let ~j
can
be
be the
w i t h domain
and
n c J, n non
+ D,
Alsosatisfies in
J,
full induction for,if
J must be definable
in
But we can now construct an isomorphism a of
fixes these parameters
J
by
~> d + ~  i.
Let
His argument
as follows.
but does not fix
J,
whic~
hence giving the required
contradiction. SECTION
5.
In this short final section we indicate how our previous
results might be improved by making various IA 0.
It must be admitted however
course is
A~
if
we obtain,
for
f : ~ ÷ ~,
T H E O R E M 23.
about
A~
and
that in all cases except for Theorem 26
the general opinion is that the assumption Recall that we know that Lin.Space
assumptions
is false.
is closed under counting and so of
A~ = Lin.Space.
By directly amending this proof
Assume that A 0 Z Spaee(f(n))
and let
A e A 0 ' A c ~ 2
and such that for all n c~, A n = {m IcA} ! 2f(l°g(n)) Then { I k =
1Anl}~A~
D
3~
The assumption here concerns the amount of computational in
A~.
space available Assumptions about available time also yield more counting.
Precisely let
f :~ + ~
have
A~
graph,
f strictly increasing and
f(n) e n l°g(n) for all n. Define A~(f) as for &~ but with the new function f added to +, (allowing f to appear in the bounding terms in the quantifier). A~(f) =
A0
In [8] an equivalence between the assumption
and the computational
THEOREM 24. that for all
Assume
A~ = A~(f)
and let
A ~ is proved.
A~A~,
A !
~2
and such
n ~,
IAnl = l{m I m < n Then
time available in
&~A} I ~log(fJ(n)).
{Ik = IAnl}4 A~.
0
This theorem follows easily from Theorem Turning now to the status of the THEOREM 25.
we mention the following result.
Assume that
IA 0 + Vx, x l°g(x) exists Then
AoPHP
6.
~
MatijasevicRovinsonDavisPutnam
IA 0 + Vx, x l°g(x) exists
+ Con(IA 0) ~ AoPHP.
Thm. 0
A proof of a rather more general result will appear in [i0]. The assumption Con(IA 0) can be weakened but our proof still needs some consistency. It would indeed be surprising however if there really was a formal connection between the consistency of
IA 0 and the
AoPHP,
For our next result we show that the problem of section 4 on the relativized AoPHP viz.
has an affirmative answer assuming the CookReckhov conjecture
Vk @n
such that every proof in the propositional A
V
i~n
j
uses more than
nk
THEOREM
26.
~
V
i<e~n
V
j
(Pij ^ P
ej
calculus of
)
symbols. Assuming the CookReckhov Conjecture, IA0(F) + @x,
F : x
le>xI
is consistent. PROOF.
Let
K
be a countable proper elementary extension of
~.
Pick
336
nonstandard
c, b e K A i~c
uses at least For
8(x)
symbol, calculus
V j
cb
+
in
V i<e~c
K, every proof of
V j
(Pij ^ Pej )
symbols.
a formula
and
such that,
~ ~c
from
L(R),
where
R
we define a formula
is a new binary relation
8(a)* of the propositional
as follows. e
8
R(a I, a 2)
Pal,a 2
aI + a2 = a3
s
aI
t
new propositional
al,a2,a 3
1
variables
a2 = a3
al,a2,a 3
aI = a2
e
el A e 2
8*
al,a 2 ^ e*
1
me I
1
~e* 1
~x8 (x)
V
@( a ) *
asc
Vxe(x)
A
e(a)* etc.
asc
Not let
TO
consist of
A i
V ^ A j
A ~ (PijA Pej)... j
t
together w i t h {e*
le is an atomic or n e g a t i o n of atomic sentence of L and e+l p e}.
Then there is still no proof of an inconsistency from T O using less b than c symbols. Now we can form increasing sets of sentences TO ! T I ! T2 ~ (i) less than (ii) n ¢I~
"
such that each
T n is coded in
There is no proof of an inconsistency
K
and
(in K) from T n using
cb/2n symbols. For every formula
such that
@(a)* ¢ T n
@(x) or
of
L(R)
~ 0( a )* c T n.
and
a< c
there is an
337
(iii) e(d)*^
Whenever
V e(a)* ~ T n a~c
then
@n < m ~
and
d~ c
such that
A ~ 0(a)*~ T . a
Define
R c (c+l) x (c+l) by R(al,a 2)
~=>
Pal,a 2 e U
Tn .
nc~
Then it is easy to see that
for
a ~ c,
the formula
where
c ~
some
i ~}.
TO,
+ {6 I e * e
F : c+l
, F> > IA0(F) denotes
d
prove a rather THEOREM
such that
of
a rather
surprising
special
27. Assume
of the
K
a ~,
a.
that
A0
F c A~,
Then for each
{b < a ~ M ~ ~xl
& q ~r,
0
if
Pi =
0>
I
if
Pi =
I>
a
if
Pi =
2>
b
otherwise
as
YYl < ak
y~k c Rg(F)
~ T E a l ,
al
where
and hence
as
=
QYr < ak 'Yr"k _c Rg(F)
i ~ i ~4
yq(n)
if
Pi =& q ~r,
0
if
Pi =
0>
i
if
Pi =
I>
F(a)
if
Pi =
2>
F(b)
otherwise
F(a v, a, b,)
in F are restricted
depends
r.
It follows
where
to
that
a v.
such that
a2' ~3' a4 < a
for
quantifiers on
n
T c ~4.
[such that for all
F(ai)
I ~i~4,
if
this can be written
" a2 + a3 = ~4 ]
for
yq(n)
and finite
So using
[such that for all
k, T e ~ .
n
r is A 0,
Furthermore
and all
the formula
{b < a ~ I M ~ ~ F(a ~, a, b, b)}
F only
is not
of the form given in the theorem. COROLLARY F:
a2
28.
~>a. Then
PROOF.
that
av
M some
v >~,
{b c M IN ~ ~x I
V~2
g, h polynomials
the hypotheses written
of
F c A Mo
and
M ~AoH.
By M ~ A0H we mean that there is no fixed M A 0 set (without parameters) is of the form
every
with
Suppose
D
...
over
Theorem
in the above
Qx r
form then,
r ~
g(~l ..... x r,b)
= h(~ I ..... x r,b)}
To show this it is enough hold since if any with the notation
A~
such that
to show that
set can be
of Theorem
27, the set
339
{a ~
a
b>[b < a ~ & M ~ e(a ~, b
this set could be written but fixed,
value of
be the parameter, c