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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1130 Methods in Mathematical Logic Proceedings of the 6th Latin American Symposium on Mathematical Logic held in Caracas, Venezuela August 1-6, 1983
Edited by C.A. Di Prisco
Springer-Verlag Berlin Heidelberg New York Tokyo
Editor Carlos Augusto Di Prisco Instituto Venezolano de Investigaciones Cientfficas Departamento de Matem&ticas Apartado 1827, Caracas 1010-A, Venezuela
ISBN 3-540-15236-9 Springer-Vertag Berlin Heidelberg New York Tokyo ISBN 0-38?-15236-9 Springer-Verlag New York Heidelberg Berlin Tokyo
This work is subject to copyright.All rights are reserved,whetherthe whole or part of the material is concerned,specificallythose of translation,reprinting, re-use of illustrations,broadcasting, reproduction by photocopyingmachineor similar means,and storage in data banks. Under § 54 of the GermanCopyright Law where copies are madefor other than private use, a fee is payableto "VerwertungsgesellschaftWort", Munich. © by Springer-VerlagBerlin Heidelberg1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE The Vl Latin American Symposium on Mathematical Logic was held in Caracas, Venezuela from the i st to the 6 th of August 1983. The meeting was sponsored by
Asociacidn Venezolana para el Avance de la Ciencia
(AsoVAC), Consejo Nacional de Investigaciones Cientificas y Tecnoldgicas (CONICIT), Fundacidn Polar, IBM de Venezuela, Instituto Venezolano de Investigaciones Cientificas (IVIC), Universidad Central de Venezuela, The National Science Foundation of the United States of America, The British Council, The French Government, Division of Logic, Methodology and Philosophy of Science of the International Union for History and Philosophy of Science, Organization of American States. The Program Committee was formed by Xavier Caicedo (Universidad de los Andes, Bogota, Colombia), Rolando Chuaqui (Pontificia Universidad Catdlica de Chile, Santiago, Chile), Newton C. A. da Costa (Universidade de Sao Paulo, Brasil) and Carlos Augusto Di Prisco (Instituto Venezolano de Investigaciones Cientlficas y Universidad Central de Venezuela). IVlC's Centro de Estudios Avanzados sponsored a seminar consisting in five short courses of ten hours each. The courses were: Non-Clsssical Logics by Newton C. A. da Costa, Some Aspects of the Theory of Large Cardinals by Wiktor Marek (Warsaw University), Mathematical Practice and Subsystems of Second Order Arithmetic by Stephen Simpson (Pennsylvania State University), The w-rule by E. G. K. Lopez-Escobar (University of Maryland), and Logic, Real Algebra and Real Geometry by M~ximo Dickmann (Universit~ de Paris VlI-CNRS). The courses were attended by participants fron Argentina, Chile, Mexico, Italy and Venezuela. The Ateneo de Caracas and the Consejo Nacional para el DesarroIio de la Energfa Nuclear hosted a panel discusssion on the Philosophy of Mathematics with the participation of Newton C. A. da Costa, Rolando Chuaqui, George Wilmers (Manchester University) and Vincenzo P. Lo Monaco, Pedro Luberes and Juan NuNo of the Institue of Philosophy of the Universidad Central de Venezuela. The local Organizing Committee included Arturo Rodriguez Lemoine, (Universidad Central de Venezuela), Pedro Lluberes who organized and chaired the panel discussion on Philosophy of Mathematics and Jorge Baralt (Universidad Simdn Bolivar. Professor Wiktor Marek provided invaluable help organizing the symposium.
IV The final version of these proceedings was prepared by Mrs. Magally Arvelo-0sorio.
My sincere appreciation for her skilled work.
Carlos Uzc~tegui proofread large portions of the book. This volume was edited with the collaboration of Xavier Caicedo, Rolando Chuaqui and Newton C. A. da Costa. Carlos Augusto Di Prisco
LIST OF PARTICIPANTS
Jos~ A. Amor Ayda Arruda Jorge Baralt
Wiktor Marek Maria V. Marshall Adrian R.D. Mathias
Susana Berestovoy
Gisela M~ndez
Lenore Blum
Irene Mikenberg
Everett Bull
Julia ~ l l e r Kenneth McAloon
Xavier Caicedo Ana Cavalli
Anil Nerode
Rolando Chuaqui Roberto Cignoli Peter Clote
Juan NuNo Jeffrey Paris
Manuel Corrada Luis Jaime Corredor
Ruben Preiss Alexander Prestel
Newton C.A. Da Costa
Cecylia Rauszer
MAximo Dickmann Carlos A. Di Prisco
William Reinhardt Arturo Rodriguez Lemoine
Itala M. D'Ottaviano
Rafael Rojas Barbachano
Sergio Fajardo
Gerald Sacks
James Henle Jorge Herrera
Maria G. Schwartze Antonio M. Sette
Jaime lhoda
Stephen Simpson
Ramdn Pino
Thomas Jech
Roger Soler
E.G.K. L6pez Escobar
Jacques Stern
Alain Louveau
Elias Tahhan
Maria Jimena Llopis Pedro Lluberes
Carlos Uzc~tegui Carlos Vasco
Menachem
George W i l m e r s
Magidor
Jerome I. Malitz Professor Michael Makkai was unable to attend the meeting. is, nevertheless, included in the proceedings.
His paper
TABLE OF CONTENTS Xavier Caicedo Failure of interpolation for quantifiers of monadic type. Rolando Chuaqui & Leopoldo Bertossi
13
Approximation to truth and theory of errors. 32
Peter Clote Partition Relations in Arithmetic.
69
Manuel Corrada On the axiomatizability of sets in a class theory.
76
Maximo A. Dickmann Applications of model theory to real algebraic geometry.
A survey.
Carlos A. Di Prisco
& Wiktor Marek
151
On the space (i) < Itala M.L. D'Ottaviano The model extension theorems for ~ 3 - theories Sergio Fajardo
157 174
Completeness theorems for the general theory of stochastic processes. James M. Henle, Adrian R.D. Mathias & W. Hugh Woodin
195
A barren extension. E.G.K. Lopez-Escobar
208
Proof functional connectives. Michael Makkai
222
Ultraproducts and categorical logic. Jerome I. Malitz
310
Problems in taxonomy, a floating log. Jeffrey Paris & A. Wilkie
317
Counting problems in bounded arithmetic. Ramon Pino & Jean-Pierre Ressayre Definable ultrafilters and elementary end extensions.
341
VII
Alexander
Prestel
351
On the axiomatization
of PRC-fields. 360
Cecylia P~uszer Formalizations
of certain intermediate
logics.
Part I. William N. Reinhardt
& Rolando
Chuaqui
385
Types in class set theories. Jacques
395
Stern
Generic
extensions which do not add random reals.
FAILURE OF INTERPOLATION FOR ~UANTIFIERS OF MONADIC TYPE Xavier Caicedo Universidad de los Andes - Departamento de Matem~ticas Apartado A~reo 4976 Botogt~ D.E.,
Colombia
ABSTRACT It is shown that no proper extension of first order logic by LindstrDmMostowski quantifiers of monadic type, that is quantifiers of the form QXl...Xn($1(x I) ..... ~n(Xn)), satisfies the many sorted Craig's interpolation lenmm or even the one sorted, if closed under relativizations. For example L or any of its admissible fragments can not be generated by any number of these quantifiers. This generalizes previous results of the same type shown under stronger hypothesis. In contrast, all monadic logics generated by cardinal quantifiers satisfy interpolation. §0.
INTRODUCTION.
In the context of abstract model theory few
"natural" logics seem to satisfy Craig's interpolation ienmm. No proper compact extension of first order logic satisfying this property is known; and besides L and its admissible fragments there are not too many uncompact examples. On the other hand, there are several general non-interpolation results, starting with Lindstr~m [9], who shows that if a logic extends L ~ w in infinite models, is generated by finitely nmny quantifiers, and satisfies the downward LDwenheim-Skolem Theorem, then it does not satisfy even Beth's definability In [7]
'
theorem.
Friedman shows that no logic between
L~
(Ch)
or L
(Q)
with
e I and the large logic L w(Ch,Q~l~ c Ord) satisfies Beth's theorem, where Ch is Chang's quantifier and Q~ is the quantifier "there are at least m ...". In [3], we observe that this generalizes to sublogics of L ( Q I Q E Mon) where Mon denotes the class of all quantifiers of monadic type. Makowsky and Shelah prove in [ii] that no logic of the form Lwm(Qi i e I) with Qi ~ Mon satisfies many sorted interpolation,
provid-
ed it satisfies Robinson's joint consistency lemma, the Feferman-Vaught
property for sums of structures, and also to be a strong limit cardinal•
II I ~ ~
where ~
is assumed
See also Makowsky [I0] and
Mundici [13]
for related results. In this paper we show that for quantifiers of monadic type, the above hypothesis,
as well as the assumption that the logic contains one of the
quantifiers L(Qili
Ch
or
Q~, ~ ~ i, are unnecessary.
No logic of the form
~I), Qi cMon, satisfies many sorted interpolation, and it does
not satisfy even single sorted interpolation in case it admits relativizations of sentences. After considering some simple applications we show an analogous result for extensions of
L .
Then, we observe that in contrast, the result
does not hold for monadic logics, since all extensions of first order monadic logic by cardinal quantifiers satisfy interpolation. §I•
PRELIMINARIES.
We assume that the reader is acquainted with the
basic notions of abstract model theory as presented, for example, in [i], [8] or by
[ii].
Universes of structures
A,B,C ..... respectively,
For a formula
~(x~)
IAI
~,~,
~,... will be denoted
denotes the cardinal number of
in any logic, ~
=
A•
{(a I ..... an) cAnl@~
~[a I ..... an]}. Elementary equivalence between two structures with respect to a logic L will be denoted "~mod L", elementary equivalence in
L
will be simply denoted
"~
~"
We will consider quantifiers in the sense of Mostowski [12] as generalized by Lindstrom [8].
A ~uantifier
Q
is a class of structures of some
finite type closed under isomorphism, to which it is associated a syntactical rule which allows to form, from formulae ~i~i ) . ~n(~n), . . the . new . formula~ . . ~° = Q~I' ~Xn(~l(Xl)~.... '~n(Xn ))'with the meaning: ~ ~ o ~> ( A , ~ ..... ~ ) e Q o Note that we use the same symbol for the quantifier itself and for its syntactical expression. Given a family of quantifiers Q l i~I, the smallest logic closed under the first order logical operations and the quantifiers Qi will be denoti ed L ( Q li c I) and we will say that it is ~enerated by the quantifiers Ql For cardinals K ~, LK~(Qili ~ I) is constructed allowing °
>
conjunctions of size less than
K;
the subscript
K = ~ will denote
closure under conjuctions of arbitrary sets of sentences.
Finally,
L~w(QIIi ~ I) represents the sublogic obtained by restricting to sentences of quantifier rank less than the ordinal y (cf.[3]). Note that the infinitary character of a logic is a relative matter, and definitively it is not a model theoretical property,
since under very
weak conditions any l ~ s
the form
L(Qili
lies (could be proper classes) of quantifiers.
c I) for adequate famiThis includes the in-
finitary logics construed above (for Y limit). This is also the ease of any logic satisfying the interpolation lemma. This paper studies the case of logics generated by quantifiers of monadic type or . Well known examples of these are the cardinal quantifiers Qe = {(A,B) I IBle~ } as well as Chang's Ch = {(A,B) I IBI = IAI} and Hartig's H = {(A,B,C) I IBI = ICi}. One could consider many other possibilities b where
such as
Qx I .... Xn(%l(X I) ..... ~n(Xn )) ~> P ( I ~ I ..... I~n~l) = 0 P(x I ..... Xn) is a given diofantine polynomial,
Qxyz(~(x),~(y),o(z))
or
~=~ I~~] ÷ (1~I) 2
in the sense of Erd6s-Rado partition calculus. Obviously these do not include the quantifiers of Henkin, Magidor and Malitz, and all sort of order quantifiers (cf.[ll]). Without misgivings about classes of classes, let Mot be the class of all quantifiers of monadic type (cf. [8]). We will abbreviate L~w(Mon) for L
(QIQ ¢ Mot) .
Given formulae ~l(X) .... ,~n(X), which may allow extra free variables, we introduce for each function 6:{0,1 .... ,n - I} + {0,I} the formula ~: where
%i 0
is
~i
and
%i I
=
A ~i+l(X) 6(i) i IB'I e ~, because if IA'I = IB'] we would have by (iii), (iv), IP6nV I = I P 6 1 for all 6e2 n, and so by (ii) ~ ' = ~', contradicting (v). Now, consider sentences with an additional predicate additional predicates as necessary to say :
vi) vii) viii)
"E
E
and as many
is an equivalence relation of the universe"
"Each equivalence class is equipotent to the universe" "The number of equivalence classes is equipotent to V".
Then, if @ is the conjunction of
(i)
to
(viii), the class
K I = {(A',E') 1 ( ~ ' , ~ ' , .... E'...) ~0 for some ~ ' , ~ ' , . . . } is PC in L and contains ~(IAI, IBI) with ~ a n d ~ a s above. Moreover, it is disjoint of the class K 2 = {(A',E')I(A',E')
m D(K,K), K ~Card}
which is trivially PC in L. Since we have D(IAI, mod L ( M o n ) by lemma 1.2, interpolation fails in L.
IBI) ~ D(IAI,IAI) D
REMARK. Clearly the conclusion of the lemma follows from the weaker hypothesis: Interpolation ( L , L ( M o n ) ) . 2.2
THEOREM.
If
L = L(Qlli
relativization and extends properly
e I)
with
Qi ~Mon is closed under
Lw~ then it does not satisfy inter-
polation. PROOF. If the logic has the form L = L ( Q i l i c I) with each Qi of monadic type, then for any monadic structures ~ f , ~ such that ~ ~, we have by the l e m a : ~ Qi if and only if ~ Qi This means that Qi does not distinguish infinite cardinals in the sense of Mostowski
[12]
and LindstrSm [8]. Applying Corollary 3.2 in LindstrSm's paper, we have that L must satisfy the downward LSwenheim-Skolem Theorem. By LindstrSm's corollary I in [9], there is a PC class K in L with the following properties:
i)
(A,B)~ K
implies
IBI < ~ .
ii) For all n c ~ there is (A,B) ~ K such that IBI = n + I. Let ~(P) be the sentence which characterizes K (with the help of additional predicates) when P is interpreted as B. Add new predicate symbols
(I)
~,
(3)
and (3).
As an application,
M~(Qo)
satisfies interpolation.
This logic is
also equivalent to
M w(Q I) in validities and so it is recursively axiom-
atizable~ REFERENCES [ I] Barwise,K.J.
Axioms for abstract model theory.
Ann.Math.Logic
7(1974),pp.221-265. E 2] Caicedo,X.
Maximality and interpolation in abstract logics. Thesis, University of Maryland (1978)
[ 33
Back-and-forth
systems for arbitrary quantifiers,
IN: "Mathematical Logic in Latin America", Proc. IV Latin American Symposium on Math.Logic, [ 4]
Holland (1980). On extensions of
L ~ ( Q I ) , Notre Dame J. of Formal
Logic 22 (1981),pp. [ 5] Fajardo,S.
North
85-93.
Compacidad y decidibilidad en idgicas mon~dicas cuantificadores
cardinales,
Rev.Colombiana
de Mat.
14(1980), pp. 173-196. [ 6] Flum, J.
Characterizing
[ 7] Friedman, H.
Beth's theorem in cardinality logics. Math. 14(1973),pp.205-212.
[ 8] Lindstrom,P.
First order predicate calculus with generalized quantifiers.
E 92
logics.
Preprint
(1982). Israel J.
Theoria 32(1966), pp. 187-195.
On extensions of elementary logic, (1969), pp. I-ii.
con
Theoria 35
12
[i0] Makowsky, J.A.
Characterizing monadic and equivalence quantifiers, Preprint (1978)
[ii] Makowsky,J.A. and Shelah,S.
The theorems of Beth and Craig in abstract model theory I, Trans.AMS 256(1979) pp°215-239~
[12] Mostowski,A.
On a generalization of quantifiers0 44(1957),pp.12-36.
[13] Mundici, D.
Quantifiers, an
owerview.
Fund.Math
Preprint (1981).
APPROXIMATION TO TRUTH AND THEORY OF ERRORS "I~(~
Rolando Chuaqui (2)
Leopoldo Bertossi (3)
Universidad
Cat61ica de Chile
Casilla II4-D Santiago,
i.
Chile
Introduction. The purpose of this paper is twofold°
On the one hand, we shall give
in Section 2, a model for the theory of errors based on the probability structures
of Chuaqui
1983 and 1984.
recent result in non-standard
The construction
analysis°
tion of the notion of "approximation
of this model uses
On the other hand, a formaliza-
to truth", which includes
bility of random errors in measurement,
will be presented
the possi-
in Section 3.
This last section is mainly the work of the first author, who takes full responsability
for it.
A formalization
of the notion of approximation
to truth was present-
ed in a paper inspired by ideas of N.da Costa(Mikenberg,
da Costa,
Chuaqui
of random errors
198+,MDC,
for short), but there,
the possibility
in measurement was not taken into account.
As an introduction
ideas in the present paper, we shall briefly sunmmrize
and
to our
the formalization
of MDC. A relational physical
structure ~ = < A , R i > icl is thought of as a theoretical
structure about the objects
in A.
We think of this total struc-
ture (ioe. for each n-tuples of elements of A, it is determined whether it belongs
to the n-ary
for the objects
in A,
relation R i or not) as what the theory gives us A natural way to formalize
to consider scientific
theories
Suppes 1957, Chapter 12.
tuples
this point of view, predicates,
is
as in
What is actually known about the objects in
A,
= where each R i' is only a 1 icI' defined relation over elements of A. That is, only for some
is in the Rartial partially
as set-theoretical
structure
~'
it is determined whether they belong R i or not.
the tuples,
it is undetermined.
for (or compatible with) ~',
A theoretical
if ~ i s
with
an extension
R' where the latter is defined. i' in ~' if it is true in all extensions
theoretical
structure
adequate
i.e. what is true in ~'
structure of ~ ' ,
For the rest of ~is
adequate
i.e.
We define a sentence
Ricoincides
to be true
with the same universe A.
A
for ~' is one of these e x t e n s i o n s , ~ ,
is also true in ~,
and nothing that is false
14
in ~' is true in ~.
Thus, there may be several theoretical structures
which are adequate for ~', theories.
each one determined
(possibly) by different
When our knowledge about the objects in
obtain another partial structure
~",
A
increases we
which is an extension of
were compatible with ~2~ , but not with~j",
~'.
If
then it must be changed and
the corresponding theory also. Another possibility for the rejection of the theory is the following. Suppose that
~f'
is what we know for the objects
theory T, say classical mechanics, universe
~ r domain ) A, that is compatible with
we take a d i f f e r e ~ d o m a i n
rejected,
and that a
~f'
Assume that if
T
is incompatible w i t h ~ ' .
is what we
determines, Then
T
For instance,
at slow velocities and ly true for
T
if we take
A
for B,
should be
but we would say that it is still approximately true
objects in A.
for the
as the medium sized objects
as classical mechanics,
then
T
is approximate-
A.
This picture, however, structure
A
B (which may include A), then ~ '
know about the objects in B, and that the theory a total s t r u c t u r e ~ t h a t
in
determines a total structure ~f, with
~,
~f' exactly.
given by T,
is not completely accurate.
The theoretical
in general does not extend the partial structure
We usually say that ~ coincides with ~', except for possible
errors in measurement.
It is this last factor that we want to formalize
in the present paper. In the measurement of a certain quantity, we usually assume that there is a theoretical value, actual value,
given in the total structure ~f,
obtained,i.e,
of errors in the procedure of measurement. for this error.
Sometimes,
procedure itself.
Second,
but that the
occurring in ~', may differ from it because There are three main sources
there is a systematic error derived from the there may be an error produced by the limit
of precision of the measurement method.
Lastly there may be random errors.
We shall disregard the first two types of error and consider just random errors, which,
in general are the most important.
The account presented
in Section 2 could be modified so as to take into consideration the other types of errors. Thus, if we include random errors in measurement, structure ~ m i g h t
anyway, be considered compatible with it. sentence true
That is, there might be a
in ~', but false, strictly speaking,
fact being enough ground to reject
~(and
this situation, we do the following. with~another
then the theoretical
not be an extension of the partial structure 4', but,
hence T).
in'f, without this In order to formalize
In the first place, we associate
structure that we call an error-structure ~
.
Now, a
15
sentence
~,
instead of being true or false in ~ ,
P ~ e (~)' which depends on ~ e. has low probability ~e'
where~is
compatible
If there is no sentence
according
theoretical
~'
that
according to
structure of ~f,
then @~is
with~'. is thought of as "degree of partial
for a justification
for this view).
error, we cannot get to truth; as possible
to low probability, try to approximate
Because of the possibility
~ true in
198 +, there is a discussion need sequences
falsehood.
~' but with to zero~
~e' have to be complicated somewhat
~fe~ that formalizes
of the experiments
and define probabilities
of Chuaqui
P~e"
falsehood with a Thus, our structures we construct
from
an unlimited number of repetitions for sentences
These probabilities
according to
are defined using the
198+'
This presentation
is offered,
not as a program for practical
but only as a way to illuminate
and evidence,
In Chuaqui
In order to do this, we
for this purpose:
~e' the structure
mentation,
i.e. given any e >0,
of how to approximate
decreasing
but just
In fact, we should
P~e(~ ) r]
m,n
for elements of ~.
for each
x,y,z...
The atomic formulas are:
jcJ
, for each
rc~, iEl.
]_
This language will be a two-sorted quantifiers elements
for both types of variables. of
A
the other sort. Let
L
for the variables Just as for
~fe' ~e
-language with finitely many IThe assignments
assigns probabilities
B i e(a o ..... ,an_ 1 ) be the product algebra of
w-times,
and
~
e (ao,..o,an_l)
s,ncw, adscribe
x,y,z .... and elements of ~ for the
B
to formulas.
(a ° .... an_ 1 )
its product measure.
Then,
~w Bie
= ~
~w and
~i e is its corresponding product measure. Bj~e 8~
and let ~
~j~ e are defined analogously.
and
= ~ ,
be its corresponding product measure.
An element ~e
If
:
For
then
8%
is a system
.
~k =
< ~ k (t):tew >
where
~k(t)c Bk~fe.
If
icl, then
27
~i = < ~i(ao ......an-l't) : ao .....an-~A'tcm ~ where
~i(ao .....an_l,t)~Bi (aO .....an_l)
for each t ~ o h, now, assigns to each formula ~ and assignment language an element of B~ , as follows:
= i I, if (i)
s
of the new
s(x) = s(y)
h(x=y;s) ~, otherwise~ ~, if
s(n) = s(m)
h(n = m;s) = ~, otherwise° (ii)
h(Rj x o ..... Xn_l,m;s )
= ~
lj, if ~j(s(m)) and
~k(t) =
(iii)
=
where
R ~ ( S ( X o ) ..... S(Xn_l))
~j, otherwise,
ik, for all
keluJ,
t~m, with
h([fi(x o ...... Xn_l,m) er];s)
= ~
~i(S(Xo) ...... S(Xn_l),s(m)) and
~k(ao ..... an_it )
=
(v), ~vii), and
(viii)
k~j
or
t ~ s(m).
where =
ik(a ° ..... an_l) ,
(a o ...... an_l,t) (iv),
= I
[fi(S(Xo) ..... S(Xn_l)) e r] for all
k ~ i
or
~ (s(x O) ......S(Xn_l),s(m)) are the same as before.
We need two more clauses: (ix)
h(~n~;s)
=
v
h(~,s~ )o
tom
(x)
h(¥n~;s)
=
^
n). h(~,s t
t~
Just as before,
the probability P~
Now, the repetition),
(~,s)
~e
= ~
in ~
is given by:
(h(#;S))o
A,t- partial m-structures Bare of the form
(or partial
structures
with
28
~
=
'
J
i~l, j~J'
--
where
f~
n.
is a partial operation defined on
1Axm into
~
and
R.~is ]
n.
a partial relation on
]Axe.
(Here,
nA is the set of
These functions may be partially defined on
A,m
n-tuples of A).
or both;
eogo
f~(a o .....,an_l,t) may be defined only for some ao, .... an_l ~A In general, if we assume that ~ for each
ao, ....,an_leA
and
t~m.
represents our actual knowledge, then,
there will only be finitely many
tc~ with
f ~ ( a o ..... an_i t ) defined. A complete extension ~ of ~ will have these functions defined everywhere in A and ~ , and extend those of ~ o Observe that in ~,_ or in any of its extensions fi~(ao,...,an_l,V), for
~
t,vem with
, we may have
f~ (ao,o.o,an_l,t)~
t ~v°
Satisfaction for ~ is defined just as for the partial structures without repetitions ~'. In the language that we have introduced there is a formula ~ and an assignment s such that i=T¢[s]
iff
Iti(a° ..... an_l) I < a
where ti(ao,ooo,an_l) is Student's t for the measurement f~(a o .... ,an_l,V) with
vE~
that are defined in
~
p
and
a e~.
That is M-
fi~f(ao ..... an_l)
ti(a o ..... an_ l) = SM where
M
is the average of the sequence
and
SMiS its sample standard deviation° As we mentioned in Section 2, there is an
a ~,
such that P ~ ( ¢ , s )
29
is low for all
u-E-structures
~e
, associated with ~o
Thus, the
following definition makes sense° is incompatible with ~ iff
We say that the partial ~-structure there is a formula ~ and an assignment (i)
~
(ii)
P~(~,s)
~e
~T~[S]
s
such that,
, is low, for every
~-E-structure
associated w i t h ~ .
(iii)
P ~ ( ~ , s ) is high, for a certain alternative to ~/, ~ , ~e and a certain m-E- structure ~ associated w i t h ~ o e If a certain
~-E-structure
reasons, over all other
~e
is preferred, because of theoretical
u-E-structures associated with@/,
might relativize the definition of compatibility to this changing
(ii)
to
(ii)'
: P~
then we
~,
by
(~,s) is low.
However, the definition given (with (ii) instead of (ii)') is preferable, because it is independent of inessential theoretical features, such as standard deviations. It can be shown, by arguments similar to those presented in Chuaqui 198+, that the statistical tests for hypothesis are a special case of these definitions for the situation of this paper.
In particular, we
can explain, in this fashion the approximation to falsehood by a sequence of probabilities decreasing to zero° Two possible extensions of the models discussed here may be mentioned. In the first place, than the normal one.
fi R
(ao, .... an_ 1 ) may have a different distribution
This may happen with some methods of measurement.
The second possible extension is to non-deterministic theories.
In
this case, the theoretical structure Sfitself may have random variables, i.e. fi@/(ao ..... an_ I) may itself be a random variable. This is a possible line of inquiring that we have not yet pursued.
30
REFERENCES Anderson, RoMo
[1976]
A non-standard representation for Brownian Motion and Ito integration, Israel J.Matho volo 25, pp~15-46o
Chuaqui, Ro
[1977]
A semantical definition of probability,
in
Non-Classical Logics, Model Theory, and Computability, Arruda,da Costa, and Chuaqui (editors), North Holland PublicoCoo Amsterdam pp~ 135-167o [1983]
Factual and cognitive probability,
to appear in
the Proceedings of the V Latin American Logic Symposium, Caicedo (editor), Marcel Dekker Inc., New York° [1984]
Models for probability,
to appear in the
Proceedings of the First Chilean Symposium on Analysis, Geometry, and Probability, Chuaqui (editor), Marcel Dekker, Inco, New York°
Loeb, P°Ao
[1985]
How to decide between statistical methods. To appear in Mathematical Logic and Formal Systems. (volume in honor of N.C.A.da Costa), de Alcantara (editor), Marcel Dekker Inc., New York.
[1975]
Conversion from non-standard to standard measure spaces and applications in probability theory, Trans~AmoMathoSoc~volo 113-12 2
211, ppo
Mikenberg, Io, NoCoAoda Costa and RoChuaqui
[198+]
Pragmatic To appear
Scott, Do and P. Krauss
[1966]
Assigning probabilities to logical formulas, in Aspects of Inductive Logic, Hintikka and
truth and approximation to truth.
Suppes (editors), North-Holland Pubblic. Co., Amsterdam, ppo 219-264o
31
Stoll, Ao
Suppes,
A non-standard
[1982]
Po
[1957]
construction
of L~vy Brownian
motion with applications
to invariance
principles,
(Mathematik),
Diplomarbeit
Universitat,
Freiburg,
BRDo
Introduction
to Logic,
D. Van Nostrand Coo,
Inco, Princeton°
(i)
This paper was partially and Technological American
Development
Universidad
de Investigaci6n
of
(DIUC) of the
Cat61ica de Chile°
The paper was partially written when the first author was at the Institute
for Mathematical
Stanford University, Memorial (3)
Program of the Organization
States and the Direcci6n
Pontificia (2)
supported by a grant of the Scientific
Studies
in the Social Sciences
Foundation Fellowship°
The authors would like to thank NoCoAoda Costa for many useful comments o
at
financed in part by a John Simon Guggenheim
PARTITION RELATIONS IN ARITH~L~flC P. Clote I U.E.R.
UNIVERSITE PARIS Vll de Math~mmtiques et Informatique
Tour 45-55
5~me ~tage
75230 Paris
§0.
Cedex 05, France
Introduction. Recall
ble scheme, we give Peano
- 2 Place Jussieu
the folklore
is provable
(infinite)
equivalents
Part of the original
a certain arguments
in certain
The results
presented
herein
, as a defina-
arithmetic.
In this paper,
for certain
subsystems
of
for this work was
to allow one to formalize
subsyqtems
contribute
of Peano arithmetic
~neorem
motivation
amount of machinery
binatorial
of fragments
that Ramsey's
in first order Peano
combinatorial
arithmetic.
to produce
result
com-
of arithmetic.
to the proof theoretic
- specifically
that of
study
Z n induction
(I E n) and E n collection or bounding principle (BEn). To see combinatorial significance of these subsystems, recall the well-known result
of J. Paris
[20]
recursive (or even Kalmar positive integer m gn,m(X)
that a recursive elementary)
= least
y
such that
function
in the function
Ix,y]
.... >
f
is primitive gn,m for some
(n+2)~ +I
if and only if
IE n
~
V x ~y "f(x)
= y"
if and only if BEn+ I ~
i
Vx~y "f(v) = y"
These results were obtained and presented while giving a course Models of Arithmetic in the fall semester of 1982-83 at the Universit~ Paric VII. AutHor's present address: Department of Computer Science, Boston College, Chesnut Hill, MA 02167 USA.
in
33 This result ]inks essentially
the growth rate of a fast grewing recur-
sive function with the proof-theoretic vably recursive
this paper was to find infinite of subsystems
the combinatorial of the recursion
wizardry
sets of
Ramsey Theorem-type
of [20].
The main result,
to be essentially
[6].
into
(possibly non-standard)
M
set.
~ B~n+ I
This characterization
M
satisfies
initial segment
I
BEn+ I.
iff
M
>
of the collection
of
~n induction.
of
M
is
M satisfies
if and only if
satisfies
a certain infinitary An
definable
non-standard)
En formulas,
subset of
(Theore~ 4).
En induction
~n
M
I is a model
is partitioned
and J. Paris
[9]
formulation :
a model
: when any
into (possibly
corollary
go to
(Corollary
for Boolean combinations
even prefixed by bounded quantifiers.
give an alternative
equivalent
then at least two elements
induction
an
induction if and only if
This yield the useful
implies
[9] :
A n pigeon hole principle
bou~dedly many pieces,
the same piece
sub-
there
scheme then answers a question
Mills and J. Paris in
n-Ramsey
M
~-element
(M) n . <M
Along the way, we give an easy combinatorial A model
En+ I
Schematically
about initial segments posed by G of
formalization
boundedly many pieces,
AI
6) that
Theorem ii, turns
a proof-theoretic
Our result is that a model
is an unbounded homogeneous
unbounded
characterizations
if and only if for any recursive partition of
M
in
theoretic analysis of Ramsey's Theorem done by C.G.
Jr. in
collection
Our original motivation
of Peano arithmetic with the intent to simplify some of
out (in one direction) Jockuscb,
notion of a function being pro-
in a system of arithmetic.
In Theorem
of 8
we
and proof of a result due to G. Mills M
satisfies
E2n
collection
if and
only if the filter product of the Fr~chet filter with itself n-times F x
F x
F x
.
.
.
x
F
n- times is Ao-Complete.
34
The techniques
used are model
ultraproducts)
and more especially
formalized theorem,
versions
Lemma
answer
!.
(involving
recursion
end extensions
theoretic
Forthcoming
some of the questions
low-basis
work by the author has further
theorem and other recursion
and
(involving
of the limit lemma and the Jockusch-Soare
14).
ed the low-basis
theoretic
theoretic
exploit-
techniques
to
posed in this article.
Preliminaries. A formula
in the language
said to be bounded
of Peano arithmetic
if it is built up from atomic
Boolean operations
and by bounded
Ao = F~o = ~o
=
{~:
quantification
is a bounded
{+,.,0,I,i
i > (M)<M.
M AI
argue in the same fashion,
and T h e o r e m 2 to have that M
a formula of the form ¥ y < s @ w h e r e is
Hn)iS equivalent
~ BE n and hence
@ is E n ( r e s p e c t i v e l y
in M to a En(respectively
Before going on, several that in the proof that for M ~ BEn+ I implies
but using the induction
First is
n e I of a proper n + 1-elementary
end extension K of M w h i c h is itself a model of P
definable
3)
~n)formula.
things are w o r t h noting.
the existence
Kirby and Paris constructed
([ll]Lemma
~ y < s ~where
a
"E
n
ultrafilter"
U on
subsets of M by defining an u-sequence
+ IZ o, Zn(M)
the En
X o ~ X I ~ X 2 ~ ooo such
that
(1)
each X i is an u n b o u n d e d
(2)
given the i th En definable partial
A n definable
subset of M,
function fi with bounded
38
range in an u - l i s t i n g w i t h infinite repetitions of all such functions
(recall that E
parameters
definable means definable w i t h n - it is at this point that c o u n t a b i l i t y of the
model M is used),
either Xi+ I is disjoint from the domain
of fi or fi is constant on Xi+l. The " u l t r a f i l t e r " U is then defined to be i e ~(X = Xi)} -
U is a "E °
U = {X~Zn(M)
u l t r a f i l t e r " in the sense that for a In n
d e f i n a b l e subset X of M, either X c U or there is a disjoint Zn set Y w i t h K
=
Y c Uo
K
definable
D e f i n i n g the r e s t r i c t e d u l t r a p r o d u c t
{f : f is a Zn partial function w i t h dom(f)~ U
then by u s i n g the fact that holds
:
: if ~ is a
M
~
and rng(f) c M } / U ,
IZn, the following form of Bog' T h e o r e m
Zn formula w i t h m free v a r i a b l e s then
~ ~([fl ] ..... [fm ])
iff {a E M : a is in domain of fl, .... fm and
One then easily verifies
M
~ ~(fl(a) ..... fm(a))}~U°
that K is a proper n + 1-elementary
end e x t e n s i o n of M w h i c h satisfies P- + IE . Each instance o
~(0,~) of a
En or
the form
&
v x(~(x,~)
~ ~(x + l,~)) ÷ v x ~(x,~)
~n i n d u c t i o n a x i o m w i t h parameters r e s t r i c t e d to M is of
En v Z n + I V ~ n + l
(resp. En v En+ I v ~n) and so true in
S i m i l a r l y every instance of a En_ I or
K.
N n - i i n d u c t i o n a x i o m w i t h para-
meters in K is true in K° Thus it is a natural q u e s t i o n w h e t h e r given a m o d e l of for n ~I,
BEn+ I
one can construct a proper n + 1 - e l e m e n t a r y end e x t e n s i o n
K satisfying
P
P R O P O S I T I O N 3.
+ IEn°
A n e g a t i v e answer follows from
If K is a proper n + 1-elementary end e x t e n s i o n of M
s a t i s f y i n g IE n w i t h n e 0, then M is a m o d e l of B E n + 2. PROOF. for n ~ i
(The case for n = 0 is part of T h e o r e m B in [ii]°
The p r o o f
arose in a c o n v e r s a t i o n w i t h Z° A d a m o w i c z and A~ Wilkie) o We
show that M
~ B~n+lO
39
Suppose
that M
~ V i V X<eo~(i,t,x).
K > V i < a ~ t < d V x < e o ~ ( i , t , x ).
Since
K
~ IEn,
let d o be the least d satisfying
every d ~ K - M satisfies M QUESTIONS. Kaufmann
this formula,
~
q d V i (2) 1<M o
an n-elemen-
that inthe
3)°
proof of Theorem 4, we actually
iff
M ~ I~lo
iff
M ~ BE2o
show that
One can also show that M A Recall
1 > (M)<M o
that the primitive
are provably
above schemes° Wrathall's Bennett's
recursive
functions
Also recall
Theorem Theorem
[18]o [I]o
the results
The linear time hierarchy Rudimentary
or not,
Go Wilmers
the linear time hierarchy
raised the question w h e t h e r 1 ~ (2)<M
than M
En
En+ 1
fiers b e g i n n i n g w i t h bounded existential
M
of bounded quanti-
quantifierso
showed that the answer is "no",
> (2)IM implies E
theory:
= Rudimentary°
w h e r e an E n formula is one with n alternating blocks
Ao Wilkie
in the
= DefoNo
1 > (2)<M is strictly weaker
M
those which
and hence
in complexity
Since it is still an open question whether collapses
are exactly
total in P- + IZ 1 (theorem due to Minc)
the exact level where
that M is a model of IE 1 depending on the logical
n
complexity function
of the
A ° formula expressing
(in the proof of Bennett's
be E 1 is an open question.
the graph of the exponential
Theorem)°
Whether
We present Ao Wilkie's
the graph can
proof.
47
PROOF°
Let n o be such that ~(x,y,z)
is an E n
formula r e p r e s e n t i n g O
x y = Zo If M ~ Va,b~c(a b =c) for some a,b ~ M then let X = {
: M ~ ay = z
&
y~b}o
If X is bounded then by I~o, X has a m a x i m u m element - say o then e X t h u s is u n b o u n d e d contradicts
contradicting
in M, then let our assumption
maximality
F : X÷{0,ooo,b} that M E
of o
by F()
> (2) 1 <M °
= y.
If
But
X
But this
(This part of the argu-
n o
ment is due to Go Wilmers)o M
>
Hence
I
(2)< M implies
that
M N Exp,
En o
w h e r e Exp i s ([3],po31),
the
statement
¥ x ¥ y ~] z d 0 ( x , y , z ) .
P - + 1E O + Exp p- N a t i j a s e v i c '
1'I s a t i s f i e s
the
least
÷
M ~ ~ x0(x,u)
M~
element
principle
By D i m i t r a e o p o u l o s
Theorem. for
We now show t h a t
Z1 f o r m u l a s .
Suppose that
÷
& 0(a,u) where
V ~ (0(x,~)
(2) IMO
D
En o Notice also that the infinitary w i t h the finitary A ° pigeonhole
A ° pigeonhole
scheme
scheme has little to do
(due to Ao Macintyre) o Ao Macintyre's
question P- + IZo
"finitary A o pigeonhole
scheme"
48
is still an open question, V ~ (V x(V i ~ x
where
~ o' y < x @ ( l ,"y , u +)
the latter scheme is ÷ ~ i # i' ~ x ~ y < x(0 (i,y ,u) ÷ & +
e (i' ,y,u) ) ) where @ is a Ao formula. A. Woods
by an ingenious
argument,
[17] has shown that ~-- "finitary A ° p i g e o n h o l e " + V
P- + I E o 3.
In this connection,
x ~ y(x -< y
i
M --~--> (M)nM_ m PROOF°
homogeneous
it turns out that Y will be definable
the above scheme is written w i t h
THEOREM
is an unbounded
an i < a with F"[Y] n = {i})o
iff
M
BZn+m~
This part is easy and uses of a partition By induction
that n = k + i and . . k+l M --~--> (•) <M
m
while
the limit lemma to step down
stepping up the complexity
on n : if n = i, this is Theorem
2°
of the Suppose
56
We show that M
> (M)kM
.
km+l Using
BZm+I,
apply the limit lemma to obtain a
Am
approximation
G : [M] k+l ÷ a w i t h lim G(Xlm ..... Xk,S) = F(x I ..... Xk). S
If
Y
is homogeneous
homogeneous
for
(÷ )
for
G
and u n b o u n d e d
in M, then
Y
is also
F. This direction
is more difficult and proceeds by a series
of lemmas. First, v ~(v
Ik
let
÷ x(¢ (x,u)
be the scheme
n
¢(x,u)) ÷
where ¢ is LEMMA 12. PROOF.
En For
and ~ is
+
((¢(0
Suppose that
and trivial
a e X e An+I(M).
the set
for
n = 0.
for non-empty
For
for the characteristic
function of
~ g(x,s)
n ~ i, we show
An+ I definable
By the limit lemma,
M ~ V x ~ a ~ s V t ~ s g(x,s)
sufficiently
+ 1,u)))+
~n"
X s = {x~ a : M Then
+ ¢(x
V x¢(x ,u))) +
that the least element principle holds
A n approximation
& v x(¢(x,u)
n a 0, P- + IE ° + BEn+ 1 D IAn+ 1.
This is w e l l - k n o w n
sets.
, u÷ )
X;
let
g
be a
let
= 0}.
= g(x,t),
so by BEn+ 1 , let
large to bound all these values of s.
Now apply
s o be IZ n to
X s to obtain a least element - this is also the least element
of X. REMARK. In fact,
Independently
and much earlier,
in [4] it is stated that
over the base theory
BEn+ I
H. Friedman noticed and
IAn+ I
this lemma.
are equivalent
P- + 1E o + V x @ y(2 x = y).
Before giving details of the proof of T h e o r e m ii, we give a sketch of the idea behind the proof that M ~ BE 3
implies
2 M -----> (M)<M AI
57
First, tree
in the usual manner,
associate
T ! M < M with the p a r t i t i o n
a
Al
definable
F :[M] 2 ÷ a
branch B of the tree is "pseudo-homogeneous" bi,bj,b k ~ B
and
b i
iff
(M,f s) ~ ~(a) where
M ~ Sat o ( r- ~, -l
<s,a>)
~ is bounded.
FACT.
If
M ~ P- + 1E o + BE n for
function
f
is piecewise
PROOF.
By hypothesis,
given.
By
Bln let
b
n e 2,
then any total
En
definable
coded in M.
the graph of
f
is
be sufficiently
M > V i i
and T is an unbounded
M, then there is a An+ 2 definable
(This is the arithmetized
fini-
1-generic
version of the "low basis"
theorem) . PROOF.
We w o r k in M.
numbers
of the levels:
k(0)
First define a function k bounding
= 0 = sequence number
k(a+l)
= least x such that
for V y>-x Vs- ~ t &
Sato((i)o,t,~(i)l))))
0 if V m ~tcT(lh(t) e m & ~ t Vi ~ a(g(2i + I)=0 + ~
&
@y < lh(t) Sato((i)o,t,y(i)l)))
g(2a+3) =J @y (S)lh(s)_ 1 such that
- l(F((s)i,x ) = F((s)i,(s)i+l))}-
in M. ~2 d e f i n i t i o n ,
by
IE 2 the set
Then
Y = {x:x > (t) lh(t)_l in M.
the fact immediate
then since X has a
element
x V y>z(s'~
are at most
are u n b o u n d e d l y
CLAIM.
H(=O
= m÷~
a-many
V i < lh(s)
is u n b o u n d e d
and u n b o u n d e d
II 2 n o t i n g
To see that
a maximum
= (t)i+l).
show
can be done u s i n g
original
= F ( ( t ) i , ( t ) i + I) ÷ y
= F((t)j,(t) k , ) =
tree.
M ~ V m V s ~T(lh(s) This
j 0 ~ ycS(d(x,~)<e)
distance).
Let
K
0
be a real closed
A function
F : S ÷ Km
field,
m,n e i, and
is called semi-algebraic
S a (s.a.)
if its graph Gr(F) is a s.a.
subset of
K n+m.
Note that the domain set,
= {<x, F(x)> [x~S}
S = Dom(F)
since it is the projection THEOREM 4.4.
Let
(a) The set of K-valued
s.a.
of s.a.
(c) The image of a s.a. PROOF.
As above;
operations
write
are actually
functions
trivial
proofs
if desired 4.5.
EXAMPLE.
elementary dS : Kn÷
analysis, K
on a s.a.
and use
facts without
definitions
0
quantifier
elimination
since some of these
elimination).
elimination
of the relevant
(*).
surprising,
to quantifier
=
subset of K n
is s.a.
by giving non-trivial
this is done,
sometimes).
As an example
going beyond the standard
consider,
for a fixed s.a. set
the (Euclidean)
distance
from
is facts
One can avoid the proofs,
examples
of
S c K n, the function
defined by:
ds(x)
F.
is s.a.
down the standard
using quantifier
(in fact,
a s.a.
defined operations.
set under a s.a. function
(this is hardly
equivalent
defined
pointwise
(which are first-order),
a headache
is necessarily
K n of the graph of
functions
The proof of these elementary usually
over
function
K ~ RCF.
is a ring under the standard, (b) The composition
of a s.a.
x to S;
89
dS
is a continuous s.a. function.
General properties of these functions,
yielding non-trivial information, will be studied in §9. B.
D
THE REAL NULLSTELLENSATZ. A fundamental feature of algebraic geometry over an algebraical-
ly closed field, k, is the one-one correspondence between a variety
V
k-points of
and (proper) maximal ideals of the coordinate ring
k[V]
established by Hilbert's nullstellensatz. This correspondence fails in the real case : the polynomial X 2 + y2 + I does not have real points, while its coordinate ring is not reduced to
0
and hence does have
(proper) maximal ideals. In order to develop algebraic geometry over
IR, it is of the utmost im-
portance to restore a correspondence of this kind, that is, to determine which type of ideals correspond to points.
This amounts to proving a
"real" version of the nullstellensatz, which we will presently do. The key notion comes from the observation that the ideal ~ (a) of real polynomials vanishing at a point a el~n has the following property: n p2i
i=l
C
I(~)
implies
PI'
.
.
'Pnc . . ~(a)
Generalizing this observation we introduce the following notion: DEFINITION ing
k
and
4.6.
Let
be an ordered field,
A a ring contain-
I c A an ideal.
(a) We say that
I
n 2 i~l Piai el,
is real over k
with
aicA
iff
and
Pick, Pi>0,
implies a I .... ,ancl.
We will also need the following weaker notion: (b)
I
with
is called semi-real over k Pick,
(c) If 4.7.
Pi>0,
I = {0}
we call
REMARKS.
teristic # 2, extending that
and
(a)
iff
-I
2
is not of the form
~piai/l i
aicA. A If
real
(respectively, semi-real) over k.
I is a prime ideal and
A/I is of charac-
then I is real over k iff A/I has~oan ordering of k iff -i is not of the form ~PiX~. with i
Pick, Pi>0, and
xi
in the field of fractions of (b)
I real over k implies
(c)
Let
D
A/I. I radical.
be an ordered extension of the
90
ordered field over
k
,
and
vanishing at
S (cf. (d)
ideal of A, then
I
S ! Fn.
Then the ideal
I k(S) of polynomials
§2.1) is real over k.
If
A
is a noetherian ring and
is real over
k
iff
I
is a proper
I is radical and is the inter-
section of finitely many prime ideals real over k. (e)
The example
I = (X 2 + y2),
A = k[X,Y], shows
that semi-real is a notion strictly weaker than real. THEOREM 4,8.
(Real nullstellensatz).
its real closure, and I PROOF.
Let
be an ordered field,
I ! k[Xl,''-,Xn] an ideal•
is real over
k
iff
I =
D
Then
I k(V~(1)).
The implication from right to left is just the remark 4.7 (C).
For the converse we need only prove the inclusion
Ik(V~(I)) ! I, since
the other inclusion is trivial. Let
I
be generated by
The condition
QI .... 'Qr ~ k[Xl ..... Xn]"
Q E Ik(V~(I))
r (*) ~ ~ V~[iAiQi(v)= = 0
is expressed by
~
:
Q(v-) = 0 ]• l
Using remark ideals
4.7(d) we can find a representation
Pj are prime and real over k.
I = ~ P. j=l J
Then the condition
Qel
where the is equi-
valent to the conjunction of the conditions (**) k[X I ..... Xn]/p j for
j = I ..... £ .
>
Since each of the rings
ing extending that of Lj
Q(Xl/p ,j .... Xn/e.)j = 0
k
k[X I ..... Xn]/p j has an order-
(by 4.7(a)), we may consider the real closure
of its field of fractions with one of these orderings.
we have
k =__ Lj ;
Hence the formula
by Proposition (*) holds in
we get
Lj as well.
Applying (**) to QI ..... Qr
r Lj ~
Now, using
In particular,
3.10 this inclusion is elementary.
(*)
with
A Qi(Xl/p i=l J, ~ =
Q ( which implies that
QcPj.
X
) = 0. n/pj
x J .... , n/pj>
/p , .... J
x/
we conclude that
p ) = 0, J
Since this holds for
j = I ..... £,
we
91
get
Qcl, as contended.
D
Every ideal is contained in a smallest, possibly improper, real ideal; namely: ~{J
I J
is real over
k
and
I ! J ! A}. R
DEFINITION 4.9.
We shall call real radical of
est ideal of A, real over
k
and containing
I, v ~ ,
the small-
I.
The real radical admits a purely algebraic characterization: PROPOSITION 4. i0. a ~ R Iv~-- iff
there are
n,mEN[ ,
m > 0,
Pl .... 'PnCk+
and
b I ..... bneA
such that a 2m
2 ~ Pibi e I. i=l
+
This result is proved in Krivine [51], Dickmann [72;
Dubois-Efroymson [47]
and
Ch.lll].
We have the following consequences of the real nullstellensatz: COROLLARY 4.11. closure and
Let
I ~ k[X I ..... X n]
be an ordered field, an ideal.
k
its real
Then:
R
Ik(V~(1)) =
~
. R
PROOF.
By the nullstellensatz
R
Ik(V~ (IRE-)) =
V~,
and by the
R
preceeding Proposition COROLLARY
4.12.
V~ (I) = V~ (I~Y--). With the notation of the preceeding Corollary we
have: R
I E I~
iff
V~ (I) = ~.
D
We restate this corollary in more geometric language: COROLLARY 4.12 his.
(Weak real nullstellensatz).
variety over an ordered field of
and let
[
Let
V
be a
be the real closure
Then the following are equivalent:
(i)
V
(ii)
The ideal
has a
k-point,
i.e.
V(k) #
~.
l(V) is semi-real over k.
D
R
(Note that an ideal COROLLARY 4.13. let
V
I
is semi-real iff
Let
be a variety over
k.
and
The map
#-~
k
is proper).
be as a Corollary 4.11, and a
~-~
I(~)/i(V(k)),for
a~V(~),
92
establishes a one-one correspondence between of
k-points of
V
and ideals
k[V] which are maximal among ideals real over k.
PROOF. (i)
We need to show: I(~) contains
real over k; (ii)
l(V(k))
and is maximal among ideals of
Every ideal of this type is of the form
PROOF of (i). that
I(a)
m l(V(k))
I(a) is real over k.
then
V~(1) ~ {a},
PROOF of (ii). By 4.8,
i.e.
Let
4.11
M =I(V(k))
k[X I ..... Xn]
and
and
then
because
a~V(k).
Finally, if
V~(1) = ~ ;
I(a) for some Remark
l(a) ~ I with
hence
i~I
a~V(k).
4.7(c) shows I
real over k,
by Corollary
4.12.
M = l(V(k)) be maximal among ideals real over k. 4.12
we get
aEV(k).
Also
V~(M)_# ~
Let
~V~(M).
Since
M c l(a), and maximality implies the
equality.
0
Exercise.
Prove that in a ring of the form
k[V] every ideal which is
maximal among real ideals, is maximal. Historical rehabilitation.
The authorship of the real nullstellensatz is
usually attributed to Dubois proved
[46]
and
Risler [57].
it long before, as well as Proposition 4.10;
C.
However, Krivine see his paper [51].
THE SIMPLE POINT CRITERION. The weak real nullstellensatz
terms the algebraic condition
4.12
bis expresses in geometric
"I(V) is a semi-real ideal".
We are in-
terested in finding a geometric expression for the closely realted notion "I(V) is a real ideal". It turns out that this condition has a very interesting geometric content: it says that ).
V
has a non-singular
When the variety polynomial
V
is a hypersurface
F(X I .... ,Xn) -
(k = the real closure of
~-x.~F (i = i,. .. ,n)
-i.e. it is given by a single
we know from elementary geometry that a point
aeV(k) is called non-sinsular tion
k-point
This is what we will prove below.
(or simple) if at least one of the deriva-
does not vanish at
a.
The correct definitives
i
in the general case is as follows: mials
PI,...,PI~k[XI ..... Xn] ,
assume that
V
is given by polyno-
and consider the Jacobian matrix,
93
/
~PI .....
~P£ \
~X 1
~X 1
~PI .....
~P£
J(PI ..... P£ )=
This is an
nx£
matrix with entries in
I_< s _<min {£,n}, J(PI'''''P£); s
let
we have
culated modulo
r
Let
For a fixed
sxs
r
s,
minors of
denote the largest
is just the rank of
J(PI ..... P£) cal-
I(V).
DEFINITION 4.14. by
M~l ~k[Xl'''''Xn]"
Ms i ~ I(V);
such that some
k[X I ..... Xn].
M si (i = l,...,t i) denote the
P1 ..... P£
A point
a~V(k)
of a variety
is called non-singular
space to the variety at
a
a
over
k
defined
(or simple.) iff the rank of
J(PI ..... P£)(a), the Jacobian matrix at It is easily seen that a point
V
a,
equals r.
is non-singular
has dimension
n-r
D
iff
the tangent
(cf. Shafarevich [14;
pp. 74,77]). Our main result is: THEOREM 4.15. and
I
Let
be an ordered field,
a prime ideal of
has a simple PROOF.
k-point
Let
k[Xl,...,Xn].
iff
the ideal
A = k[Xl,...,Xn]/i and
its real closure,
Then the variety defined by I
K
k
I
is real over k. be the field of fractions of A.
(a) We prove first the implication from right to left, which is a simple consequence of the second transfer principle. By assumption
K
has an order ~ extending the order of k;
the real closure of rate
I.
With
r
.
Assume that the polynomials
denoting the rank of
J(PI .... 'P£)
in
let
L
PI,...,P£
denote gene-
A, as above,
we have: £
X1
K ~ A
Pj(
j=l In particular, (*) holds in
X
/I ....
tr
n/I) = 0 A '
X1
V Mr. (
X
/I''"
i=l i
n/I) # O. "'
the statement
£ tr r @v I ..... @vn ( A P-(v I ..... Vn) = 0 ^ V Mi(v I ..... Vn) # 0), j=l 3 i=l L.
Since the parameters of
morphism) we have
k c L, then
(*)
lie in
(*) also holds in k.
k
and (up to k-isoThis means, precise-
94
ly, that the variety defined by
I
has a simple
k-point.
(b) The proof of the other implication requires some high-powered commutative algebra, we only sketch it. Let ~ V ~ (I) be a simple point and M = I(a-)/I its maximal ideal in A. A (deep) result from c o ~ u t a t i v e algebra tells us that, under the present assumptions, (t)
the following holds:
The localized
AM
of
A
at the ideal
M
is a regular (local)ring.
(For the definition of a regular local ring and the proof of Atiyah-Macdonald
(t), see
[I; Ch. ii].)
Next observe that the evaluation map at
a,
ev~ : A ÷ k, defined by
ev~ (Q/I) = Q(a), is a surjective ring homomorphism with kernel
M; hence
A/M = k , A/M ÷ AM/MA M
On the other hahd, the map
is an isomorphism (easy verification).
AM/MAM
-
A/M
which sends
X/M
into
x /MA m
Hence
_- ~ .
It follows that the residue field of
AM
is orderable.
Now, another basic
result from real commutative algebra tells us: (tt)
If
then
B
B
is a regular local ring whose residue field is orderable,
is real.
For a proof, see
Lam
[54; Prop. 2.7].
From (tt) we conclude that
AM
is real;
this implies that
A
itself
is real, as one can easily verify.
D
The general case, when the variety is not necessarily irreducible, easily derived from the preceeding theorem.
is
We will not prove this, but
in order to state the result we need to know that a proper ideal in a Noetherian ring is contained in only finitely many minimal prime ideals (that is, minimal among the prime ideals containing it); Macdonald
[i; Thm. 7.13
and
Ch. 4].
THEOREM 4.16.
Let
V
its real closure,
and
Jl ..... Jm
k[X 1 ..... X n]
cf. Atiyah-
be a variety over an ordered field ,
containing the ideal
the minimal prime ideals of I(V).
Then, the following are equi-
valent: (i)
The ideal
I(V)
is real over k.
(ii)
I(V) is a radical ideal and each variety
V~(Ji) , i = I ..... m,
gS has a simple point. For details, see D.
0
Lam
[54;
Prop. 2.9 and Thm. 6.10].
HILBERT'S 1 7 ~ h p r o b l e m . In its original formulation this is the problem of knowing
whether a rational function with rational coefficients, such that
R(~) e 0
at every
x c~ n
vanish, is a sum of squares in
~(X I ..... Xn).
A positive answer was given by Artin [19] breaking work of Artin-Schreier [23]
R ~ ( X 1 ..... Xn),
for which the denominator does not
in
[42], [43].
1927, based on the pathIn the 1950's Robinson [22],
gave an alternative (and generalized) proof using the second trans-
fer principle
3.9.
It should be emphasized that the ideas underlying
Robinson's technique apply as well to other situations where the methods of Artin-Schreier are inapplicable; real closed rings orderings
for example, to polynomials over
(see Dickmann [83]), or over fields with higher level
(see Becker - Jacob [81]), or even over p-adically closed
fields (see Prestel-Roquette Robinson's proof
[86]).
"trivializes"
In a certain sense, one may say that
that part of the work of Artin-Schreier
which deals properly with Hilbert's
17 ~h
problem.
Robinson's proof was the first truly mathematical application of quantifier elimination.
The first chapter of Delzell's thesis [20]
extensive historial account of Hilbert's
contains an
17 ~h problem.
Since the subject has been widely treated in the existing literature, we confine ourselves to the essential points. THEOREM 4.17. and
Let
R~k(X I ..... X n)
that Then
R(x-) e 0
for every
x c~ n
R with PROOF.
Pick,
Pi >0
Assume
=
and
R
Thm. IX.2])
extending that of
(*)
L
its real closure, Assume
2 ~ PiRi l Riek(X I ..... Xn).
(cf. Dickmann
The standard
[72; Prop. I, 1.5]
implies at once that
k(X I ..... X n)
k,
Note also that if
for which
P, Q ~ k [ X I ..... Xn], then Let us denote by ment
~
for which its denominator does not vanish.
does not have this form.
criterion of positivity [13;
be an ordered field,
a rational function with coefficients in k.
Q # 0
R o[x]}
is either empty or connected. (ii)
If A
# ~ then A-- (=closure of A ) = { x ~ n I ~ >$[x]}. G ' G Thus, Thom's lemma says that any finite family of one-variable polynomials closed under derivation is separating. The announced generalization is this: THEOREM 5.6.
(Separation theorem). Every finite family of polynomials,
PI .... ,P£ c ~ [ X I ..... Xn] , can be completed into a separating family PI .... 'P£' P£+I ..... Pz+s" Furthermore (a)
The number
function of
£, n
s and
of additional polynomials is a primiteve recursive d = maximum of the total degrees of the poly-
nomials PI .... 'P£" (b) There is a primitive recursive function f(n,Z,d) giving an upper bound on the total degrees of the new polynomials Pz+j(j=l,...,s). (c)
The coefficients of the polynomials
mial functions over
~
of the coefficients of
P£+I ..... Pz+s are polynoPI ..... P£"
D
The proof is not easy. A word of caution: adjoining only the partial derivatives of PI ..... P£ family;
counterexample:
does not produce, in general, a separating the polynomial
(X sin ~ - Y cos ~)(X sin ~ - Y cos B) + i, with 0 < ~ < B < ~ (Houdebine[76]). All
known proofs proceed by induction on the number
n
of variables,
using several of the following ingredients: Quantifier elimination; The continuity of the functions of order of
~,
x
which give, in the increasing
the real part of the (complex) roots of a polynomial
equation of the form pn(~ )
yn + ... + po(~ ) = 0,
Po,...,Pn c ~ [ X I .... ,Xn], at values ~ n such that (cf. Gillman-Jerison [85; Thm. 13.3(a)]);
Pn(X) # 0
The following result, known as the "cylindrical decomposition" theorem:
100
THEOREM 5.7. P
Given a s.a. set
U ~
~n
c ~ [ X 1 ..... X n] there is a partition of
A 1 .... ,Am, such that for each
and a polynomial
U
into finitely many s.a. sets
i = 1 ..... m, either one of the following
properties holds: (i) P(x,y) has constant sign (>0, 0}. £ m
and (finite) unions commute with closure, we obtain:
k
which gives the desired representation Coste/Coste-Roy
can be extended
a change of notation we may then assume
[62] K=~
of C.
derived the statement
D for arbitrary
K > RCF
by use of the first transfer principle
(a) and (b) of the separation
: using points
theorem 5.6 they show that the statement
of Theorem 5.9 is first-order. Instead of working
through this transfer argument we outline below a
direct model-theoretic the separation
theorem.
proof due to van den Dries
[79]
This method has the advantage
which bypasses of solving a
number of related problems which cannot be solved by the preceeding technique. Van den Dries remarked
that (the dual of) Theorem 5.9 just says that any
formula defining a closed set tifier-free
equivalent modulo of K).
C ! C n is equivalent
to a positive
L-formula in any real closed field extending the theory
The model-theoretic
derived from the compactness
T = RCF + criterion theorem.
the quantifier-free
quan-
K (i.e., diagram
to be used in this case is easily
103
5.10 L
MODEL-THEORETIC
and
(i)
~(v I .... ,vn)
There is a positive,
that (2)
CRITERION.
and
Let
L-formula.
T
be a theory w i t h language
Then the following are equivalent:
quantifier-free
L-formula
~(Vl,...,Vn)
such
T k ~ 4. Given
phism
9~ , ~
f: ¢
for any
an
~ T,
÷ ~,
L-substructure
¢ c
~
and an L-homomor-
implies
j~ > ~[f(~)]
then
E ~ cn.
D (general case).
PROOF OF THEOREM 5.9. model-theoretic
criterion.
We check condition
(2) of the
Given the situation
F UI A
(*)
with
f =
L
UI
UI
K
K
F, L, K N RCF,
A an ordered subring of
phism of unitary ordered rings
such that
F
and
f ~ K = id,
f
a homomor-
we have to show
that F > ~[~3 where
implies
a ~ A n.
Before proceeding w i t h the proof, additional (i)
assumptions
Replacing,
remark that we can make the following
on the given situation
if necessary,
field, we can assume that
A
the field
is a convex
is just Lang's h o m o m o r p h i s m extension (2)
We can assume that
Otherwise, (3)
replace
By changing
f[K] = f[A] subfield of
. A
A
K, if necessary,
containing
(a)
R is algebraically
(b)
R is real closed.
by a larger real closed
valuation ring of
is a local ring with maximal
K.
F.
This
at
ideal Ker(f) o
Ker(f).
we may further assume that
This is seen as follows.
fiR] = f[A].
A
A
L
(*):
theorem 4.21.
by its localization
and
Since
L > ~[f(~)],
By Zorn,
let
We have to show that
R R
be a maximal is real closed
closed in A (by maximality).
is a convex subring of
F ~ RCF,
the intermediate
value pro-
104
perty holds for polynomials sign between in
A[X],
a
and
P~A[X].
b (a,bER,
Q has a root
ceA,
Now, if
Q~R[X],
Q # 0, changes
a < b), then, viewed as a polynomial a < c < b.
Hence our claim follows from
(a). (c)
fiR]
(d)
fiR] =
Otherwise,
f[A].
let
dental over tible in
is a real closed field.
xeA
be such that
f[R].
A,
f(x)~f[R].
By (c), f(x) is trascen-
It follows at once, using (2), that Q(x) is inver-
whenever
Q~R[X],
Q # 0;
hence
R(x) c A, contradict-
ing the maximality of R. Now we can complete the proof of Theorem 5.9. F p ~[a]. Since
Using (3) choose
f rK
bcK n
aEAn be such that
such that f(b i) = f(a i) for i=l ..... n.
is injective it suffices to show that K > ~[~].
By assumption the first-order statement If
Let
b E(~ #)K, then there is
" ~
is open" holds in K.
EEK, C >0, such that
n
(**)
Vv I ..... v n ( i ~ l ( V i - b i ) 2 < s
holds in K;
by transfer it also holds in F.
....>. ~ ~(v I ..... Vn))
Since A is convex in F, it follows that the maximal ideal Ker(f) of A is convex in A (cf. Cherlin-Dickmann [82; Lemma 4]). As Ker(f) nK = {0} then
we conclude that y < ~ for all y ~Ker(f). Since ai-bieKer(f), n i~I= (a i - bi )2 < e. Condition (**) implies, then, that
F ~ ~ ~[a], contradicting the assumption of the theorem. COMMENTS.
Applying the same model-theoretic principle to algebraically
closed fields, van den Dries
[79]
ness of projective varieties,
gets a simple proof of the complete-
a basic result in classical algebraic
geometry (for the geometric meaning of this result, see Shafarevich [14; Ch. I, §5]).
The author has used it in
[73]
to prove the follow-
ing result, a refinement of Theorem 5.9, which answers a question of BrScker
[59; p. 261]:
PROPOSITION 5.11. closure, and of polynomials
U
Let
be an ordered field,
an open s.a. subset of Pijek[Xl ..... Xn] U = ~ ij
K n.
K its real
Then there is a finite set
such that
{~cKn I P i j ( x ) > 0 } .
D
105
REMARKS.
(a)
This section contains only a few of the most basic
results on the topology of s.a. sets; interested
in pursuing
Prop. 3.5
and (b)
much more is known.
The reader
this line of enquiry may consult Coste
§IV,
§V],
Hardt
[75],
Mather
[68;
[77].
As a part of his research on Tarski's problem
(cf.3.11
(iv)) van den Dries has embarked on the project of showing that the major topological
theorems of real algebraic geometry are consequences
of the structure of the parametrically
definable
has conjectured that similar results hold for
subsets of
<JR, +,-
~
.
He
, ;
cf. [ 8 7 ] . §6°
THE REAL SPECTRUM. A.
INTRODUCTION. One of the milestones
introduction
of the prime spectrum of a commutative
When this construction
is applied to the coordinate
V over an algebraically the set subset-
of classical algebraic
V(k) of
closed field
k-points
in the disguised
obtain a manageable
of
V
An obstacle
and generalization [14],
Hartshorne
to the development
information
and M.-F.
Thus, we
[3]
about its geometry.
theory of schemes,
of classical or
algebraic
Mumford
of algebraic
real numbers was the lack, until recently, of the prime spectrum.
ideals.
space not too far removed from the ori-
point of view leads to Grothendieck's Shafarevich
ring of a variety Spec (k[Vj) contains
form of the set of maximal
topological
ring by O. Zariski.
with its Zariski topology as a dense
ginal variety which gives valuable reformulation
k, the space
geometry was the
This
a far-reaching geometry.
Cf.
[i0].
geometry over the field of of an appropriate
analogue
The discovery of such an instrument by M. Coste
Coste-Roy meant,
therefore,
a fundamental
step forward in the
study of real varieties. To be sure, notions for quite some time; -
for example:
the space of orderings Prestel
-
akin to the real spectrum of a ring have been known
[55]
of a field with the Harrison
none of these ancestors
tric information
see
and Lam [53];
the subspace of real prime ideals of Spec(A);
However,
topology;
see Dubois
suitably reflected
[45].
the wealth of geome-
contained in the rings arising in real algebraic
geo-
metry. The crucial question
in setting up a theory of the real spectrum is to
decide which objects
are to be its elements.
Through a sequence of
106
attempts initially motivated by topos-theoretic considerations, Coste/ Coste-Roy
[69], [61], [62] realized that the appropriate choice of
objects was a hybrid made of pairs of the form (*)
<prime ideals
P
of
A;
total orderings on the quotient ring
A/p> In order to give a motivation for this choice, recall that the members of the prime spectrum, i.e. the prime ideals of a given ring
A, descri-
be its homomorphisms onto integral domains (exactly the subrings of algebraically closed fields) modulo the natural equivalence relation corresponding to cormmutative diagrams
j
R1
R2 Substructures of real closed fields (in the natural language L
for
RCF) are ordered domains, and a moment's reflection shows that the pair (*) classify the corresponding class of epimorphisms. Let us now show how the presentation of the objects (*) can be simplified in order to facilitate formal work with them. an axiomatic characterization of the set
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
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
,( X-r)
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 = X-r.
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
(r-g,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 X-r,
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 half-lines
{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.
Xe-l.
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, 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
(a-s,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)2-Q(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 and similarly for
such that
F(x,/x3 + x 2) e 0
on (-I,-i+~)
- ~ x 3 + x 2.
Thus, at each non-singular 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 half-branches: ~,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 half-branches 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 half-branches" 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/Coste-Roy [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
quasi-compact,
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 quasi-compact.
(3)
The irreducible
closed subsets of SpecR(A)
are the closure of a
unique point. In particular: (4)
SpecR(A)
(5)
Let
is a
To-space.
~, 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 c-b
B ~ {~}
iff
~ c ~
to
is in
B
and e, and
~ , it is easiest to check
implies
~ % {B}
or
Y j B , get
beB-y
and
~ c B , y,
B % {e} cey-B.
Since one of b-c
we obtain either b=(b-c)+c ey or
a contradiction.
REMARK.
correspondence
of such point
is To:
B i y
c = (c-b)+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) = f-l[~] 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 quasi-compact.
(ii) X has a base of open quasi-compact 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
L-sentence 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] ~
A-algebra (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 = g-I [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 geometrico-topological 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 Elman-Lam-Wadsworth
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 non-empty
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 sign-changing
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:
.
= k(B),
! k(B)-
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 first-order.
k(B) Let:
and
k(B)
F
extending
to : £ ~ A Pi(a) >0. i=l
B we
117
~(x,y,a O ..... an_ I ):
x0' i=l
Below we prove that these conditions are equivalent
> @x
£ 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/Coste-Roy 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
and any variety
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.,
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(X-xo)k/P
in some interval (Xo,Xo+~) , a N ~ 0.
function p defined on expansion.
0 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) = f-l[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 0, it follows
v(U)u+--~0
we have
C
is compact, the statement of Theorem
fact first-order, 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 uk-N/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
K-valued s.a. functions on a s.a.
K a real closed field, which are continuous in the euclidean
The following result is proved by Carral-Coste then, by transfer, for any real closed field PROPOSITION 9.8. and
gEcK(s-z(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 S-U) 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 well-behaved 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 z-ideal (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 non-negative 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 Gillman-Jerison
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 obta-ned
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]
Bochnak-Efroym-
to the subject, while Bochnak-Efroymson 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-
r-fold continuously
the following is important:
realvalued f
Semi-algebricity
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 Cherlin-Dickmann
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 Bochnak-Efroymson
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 non-empty
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 non-zero 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 non-zero 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. Bochnak-Efroymson 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 Artin-Mazur
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
(Artin-Mazur).
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 counter-example.
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(~n-u)
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
rE gi2 cP, i=l
=
but
Similarly h 2 ~ P, contradicting REMARK.
f ~C 2 , ~ 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 _, manuscript
See also : [24].
(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 40506-0027
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
K-complete normal non-trivial
the sets of the form
p = {Pc(%)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 Non-Logical
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 M-valued
~ is the Borel measure
the same space M.
(a)
(i.e. complete
X:~ × [0,I] ÷ M
P x B where
(c)
space
A function
Lad-terms
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
Lad-Formulas
(i)
For each Lad-term
(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 Lad-formu-
it
to use these quan-
helps
the reader
to
have a better understanding. Remark
3°9
suggests
other possible
extension
of the definition
of
Lad-formula. 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 Ft-measurable. (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])-
o-algebra 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
o-algebra 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
o-algebra
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
G-algebra 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],
o-algebra
to P.
there are other ways of defining
o-algebras.
[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
o-algebra
if it is measurable with respect to 0.
o-algebra
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-
o-algebras
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)
o-algebra 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 Lad-term 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 × Bn-measurable,
Borel o-algebra on [0,I] n, n the Optional o-algebra. (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 n-tuple
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 P-almost 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 two-sorted
(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- 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- to
Let
Yt
of
since
(6), F 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]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 well-founded
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 ~ 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 non-trivial
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
C-proof
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 "pa-proofs
of
A
~
is to be formally
for every Curry algebra
C
and
because
quired uniformity
a sentence
A
of the constructive
C-proof
(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 proof-assignments.
is the first-order
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 first-order
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
C-proof
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[X-y]]-
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 first-order 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 first-order language
for all Curry algebras
C
and all
iff
L~ such that proof-assignments ~ pa:t C e pa(A).
In view of the remark (*), the condition "t C ~pa(A)" can be replaced by: (**)
The first-order 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[((K-x)-y) ~ x],
~x~y~zE(((S-x) .y) .z) ~ ((x-z)- (y.z)) ] and
THMcA
be the set of logical consequences of
CA.
G@del'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 proof-assignments
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
.
in symbols:
consequence
pa:
,b r Epa(B r)
We shall say that the term
is a semantical
of sentences,
txl ..... Xr
C-proof
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 = {IP >
VALS is a recursively
,
VALS
Define
~A[X] of section 2.2
A}.
one can then show that: enumerable
set.
FOR DERIVABILITY.
.... be an enumeration
recursive
function.
Then let
AXMS
of
be the
set of sequents:
{, .... } 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
{ IA
(M),(D),(~ ~ ) * and ( ~ ) * ,
where the last two rules are: (~ 3)*
F., (... (((.B~B)
~B) . . . ~.B..)..... ~... A
r, B ~ A, (~),
F~
(...(((A~A)~A).o.~A.~
.......
F~A, respectively. After
verifying
sequences
one
3.3.1
that easily
the
above
obtains
THEOREM.
the
A
rules
following
For any sequent r ~
3.4
mentioned
iff
preserve
completeness
F ~
which is closer to Gentzen's sentential Gentzen's
calculus.
FOR DERIVABILITY.
the conditional of
Sequent calculus
schema for the rule.
Unfortunately
although
it may appear as simply sentential
The modification
is in the applicability
calculus of
(for the introduction
of the rule, not in the
Some persons may find the restriction of interest
since it is a global restriction of the inference,
§3.3,
for the intuitionistic
for the intuitionistic
and conjunction.
~ in the succedent)
given in
We now present another axiomatization
In fact, at first sight,
axiomatization
theorem:
A.
The set of axioms of the axiomatization is of no practical use.
con-
:
A SOUND AND NATURAL AXIOMATIZATION
recursive,
semantical
involving all the nodes above the node
and not just those immediately
above.
a price has to be paid for such naturalness
and although
215
we can show that the a x i o m a t i z a t i o n ed in showing
In order to further
emphasize
write
in the form
the sequents 3.4.1
the relation
to Gentzen's
3.4.2
~A.
CUT RULE OF INFERENCE: F,A~
B r
3.4.3
F ~ A
~
B
STRUCTURAL RULES OF INFERENCE r~A
(M3N)
5=>A provided
every
A r~ ~ A
r 3.4.4
, sentence
r~ (REP)
occuring
in
r
also
(= ~)
r, A ~ B
3.4.5
r~
r
~ A
r~
B
(g~)
the rule
(--> g)
speaking
(~
F
and
B look the same.
PRE-DERIVATIONS.
~
C
C
in obtaining
F,B ~
C
~
C
r,B g A
is not universally
A pre-derivation
N, while $N;
$ ~ N
and
~
~
C
applicable. of
in other words
sentential
consists of a finite tree
defined on T.
SN
is the
is the name of the rule schema a pre-derivation
is to all intent and purposes
of the intuitionistic
C
the latter statement.
and two functions
sequent occuring at the node Z
B
~
g) may be applied when the (sub) derivations
We now proceed to make precise
T (of "nodes")
~
F,A g B
Loosely
r ~
r,B
~ A g B
As already remarked
language
F,B
r ,An
A r
followed
5.
RULES FOR STRONG CONJUNCTION.
g)
3.4.7
in
RULES OF INFERENCE FOR THE CONDITIONAL.
=)
~ A
occurs
A
r~A=B
(~
systems we will
"r ~ A".
A X I O M A SCHEMA: A
(~
is sound, we have not yet succeed-
it is complete.
calculus
a derivation
in the
(with analysis)
of the conditional
and usual
conj unc t ion. Given a p r e - d e r i v a t i o n structure of
P.
P
=
The reduced
,
then
logical structure of
is called the logical P
is the pair
216
where
unless
~N
R
is the function defined on
is either
(MON),
(REP),
(~)
T
such that
or
(~)
=
in w h i c h case
A
RN
=
0.
Then two pre-derivations equivalent,
in symbols:
PI = <TI'$1'~I> PI ~ P2'
and
P2 =
iff their reduced logical
are
structures
are isomorphic. 3.4.8
DERIVATIONS.
A pre-derivation
w h e n the following condition At each node
N
of
tions immediately
T,
T
above
at which N
~
=
(~),
D
=
is a derivation and
then
D
is a derivation of the sequent
in case that there is a derivation F ~
is a derivation
the two sub-pre-deriva-
are equivalent.
If
If F is empty and
is met:
A, then
~
is the root of the tree
$~.
We write
of the sequent
A
"F ~ A" just
F ~ A.
is a (formally)
derivable
sentence
of Z . 3.4.9
AN EXAMPLE OF A DERIVATION.
First consider the following
two derivations: A = B DI
(A=B)
--> A = B
~ (A=C)
D2
(A~B)
~ A
~ A = B
(A=B)
A = C
A
~ (A~C),A
~ A ~ C
g (A~C)
~
B
A,A = B
~
B
C
~
C
A,A ~ C
~
C
~
A
B
~ A
~ A ~ C
(A=B)
B
g (A=C),A
~
C.
A moments reflexion
shows that they are equivalent.
is a derivation of
(A=B) ~ (A=C)
DI (A=B) ~
Thus the following
= (A~B~C): D2
(A=C),A .~ B (A=B)
(A=B)
~ (A=C),A
(A=B) ~ (A=C)
~ (A=C),A
~
C
B ~ C A=B
~ C
(A~B) ~ (A=C) = (A=B~C) 3.4.10 derivation n
SOUNDNESS THEOREM.
An induction on the length of the
gives us that to each derivation
we can associate
a term
tD x0,-.-,Xn_ I
D
and each natural number
such that
:
217
(i)
if
D
is a derivation of the sequent
then
(2)
if
tD Xo,--.,Xn_ I DI,D 2
validate
are equivalent
DI =
S
derivations,
then
t
x0,..-,Xn_ I term
tD
3.4. Ii
x0,-..,Xn_ I.
can then be used to show that:
THEOREM.:.... If
F ~
A,
then
F
~
A.
A NORMAL FORM THEOREM FOR DERIVABILITY.
of the natural a x i o m a t i z a t i o n
given in
normal
t h e o r e m for it.
form (cut-elimination)
the proof of n o r m a l i z a t i o n (the structural
(A~B)
§3.4
and
An interesting aspect of
D
in w h i c h
t-cut-free
the usual rule of repetition).
there are no cut-formulae
equivalent
N
of
T
w i t h cut-formula
(A~B),
derivation with cut-formula
that around the node
D
F, (A~B)
have
[D 2]
is
A
or
B.
and a D
~ C
F ~
--> (A~B)
C
case is w h e n the last rule of inference applied in (~ 3)
[(~g)
respec.]
In such a situation we w o u l d
(for example): 2
Di F,A
~
_F.., (A~B)
D2 C
F~A ~
C
. F....-~
F
F
~
B
(A~B)
-~ C
Then we transform the above derivation
to an
Thus assume
is as follows: D2
F
D1
D =
one can transform
either
N, the derivation D1
The critical
of the
derivation.
The first thing we show is how given a derivation cut-node
Another advantage
is that we can prove a
is the essential use of the rules of repetition
rule (FDN)also includes
Let us call a derivation form
~ A,
D2
t
The
S = B 0 .... ,Bn_ I
into the derivation:
218
Di F,A
-~C
F,A
~
F~A
C
F r
~
r ~
~A
A
C
Iterating the above p r o c e d u r e one can then t r a n s f o r m a given d e r i v a t i o n into a d e r i v a t i o n w h i c h is
~-cut-free.
Then traditional methods of r e d u c i n g a cut-formula of the form (A=B) can be m o d i f i e d so as to apply to our axiomatization.
I n t e r t w i n i n g the two
reductions one can then obtain the n o r m a l i z a t i o n theorem. §4.
R E L A T I O N B E T W E E N THE CONCEPTS INTRODUCED. 4.1
THE C L A S S I C A L CASE.
of i n t u i t i v e l y valid sentences, provable sentences respectively°
Let
IVALc,
SVALc,
THM C
be the sets
s e t - t h e o r e t i c a l l y valid sentences and
(in some natural axiomatization)
of classical logic
Then one usually argues as follows:
(.i)
IVAL C
!
SVAL C,
because if a sentence is valid in all possible structures then it certainly is valid in all s e t - t h e o r e t i c a l structures. (.2)
THM C
~
IVALc,
b e c a u s e the axioms and rules w e r e chose so as to be correct• Combining
(.i)
and
(.3)
THM C
(.2) !
Then GSdel's completeness
one then obtains IVAL C
!
SVAL C •
theorem, using a rather w e a k set-theory,
gives
the m a t h e m a t i c a l result: SVAL C Combining the latter w i t h (.4)
THM C
=
(.3) IVAL C
i
THM C •
then gives us: =
SVAL C,
We remark once again that in order to derive assumptions 4.2
(.4) certain existential
in set-theory were required. THE CASE F O R THE (CONSTRUCTIVE)
instead of 4o1.1 (.i)
LANGUAGE
~.
First of all,
we have FVAL~
~
IVAL~,
b e c a u s e if we have a term (of c o m b i n a t o r y logic) w h i c h validates a sen-
219
tence
A, then we do have an intuitive construction that proves A.
The converse is by no means obvious. The soundness theorem, combined with (.I) then gives us (.2)
THM~
!
FVAL~
2
IVAn.
Now a completeness theorem for the axiomatization of know that F V A ~
§3.4 (we already
is recursively axiomatizable) would give us only the
mathematical result that TH~
=
FVAL~.
Unfortunately the above result does not produce the analog of the classical 4.1.4.
In order to obtain such a result (even in the presence of
a mathematical completeness theorem) we need first to justify that (.3)
IVAL~
2
FVAL~
The latter could be immediately obtained if one could show that any intuitive construction could be represented by a term of combinatory logic; in other words, 4.2.3 Church's thesis.
would be a consequence of (some form of)
If Church's thesis is to be involved, then it is probably advisable to consider number theory in some more detail. §5.
NUMBER THEORY AND STRONG CONJUNCTION. 5.5
A CONCRETE MODEL FOR THE SENTENTIAL LANGUAGE ~.
Using
the techniques of Troelstra [1979] one can show that there are primitive recursive terms each sentence
where
PRA
E, p, %, A
of
~
primitive recursive predicates a primitive recursive predicate
P, ~ PA
(.i)
PRA
~
PB~C(~) ~ P(~,FPB(X)~Pc(~(~,x))]),
(.2)
PRA
~
PBEC(~) ~ P(n,FPB(%n)^Pc(Pn)^%ngpnT),
(.3)
PRA
~
n ~ ~
(.4)
PRA
~
PA(~) ^
(.5)
if
and for
such that:
A (n-mm=m~n) ^ (n~m^m~r=n~r) n ~ m = PA(m) ,
A c FVAL~, then for some n,
PRA ~ PA(n),
is primitive recursive arithmetic and F ] gives the numeral
corresponding to the G~del number.
Let Then
5.2 A FORMAL SYSTEM OF NUMBER THEORY FOR STRONG CONJUNCTION. HA be intuitionistic number theory formulated in a sequent calculus ~ HA(E) is the extension of
HA
obtained by
220
(I)
Enlarging
(2)
Add the inference
(3)
Add the (binary)
(4)
Define pre-derivations ly to
(5)
schemas corresponding
(& ~)
and ( ~ & ) ,
and equivalent pre-derivations
analogous-
§3.4.7. in
HA(&)
FORMAL REALIZABILITY
complexity
is a formula
xrA
lizability
A.
of
to
&.
rule of repetition,
Define derivations
5.2.1 logical
the class of formulae so as to include
FOR
of the formula of
HA
analogously HA(&).
A
of
§3.4.8.
Using an induction on the
HA(&)
one can show that there
which formally expresses
The only addition required
to
the recursive rea-
to Troelstra
[1973]
is the
is an almost-negative
formula
clause: xrCA&B) For each formula of ~ .
of
(.i) Now let
then give us that:
HA(&)
[1973]
(.2)
~ (.i)
xrA
[1973] )•
~ A, then
HA
ECT 0 be the schema of the
Troelstra
Combining
if
xrA ^ xrB
HA(~),
(see page 193 of Troelstra
The usual techniques
over
A
=
~
~x[xrA]
"extended Church's
~
~x[xrB]
and
HA + ECT 0
(.2)
iff
HA + ECT 0
(with respect
connectlve
)•
of
§3°4
HA)
However
that the completeness
will involve some form of Church's
thesis.
that the results obtained in this
connectives
do not find that strong conjunction
are viable concepts,
candidates would be given by: A ~A~B(C)
~-> B
=
(A~B)
we
is, per se, of great interest.
much more interest would be some kind of strong equivalence;
(2)
HA
for & to be considered as
the contention
Although we feel confident
(I)
of
over
CONCLUSION.
note show that proof-functional
possible
HA
In any case, the above conservative
extension result further supports of the axiomatization
In
of
is conservative
is conservative
(which in our opinion is a minimum requirement an "intuitionistic
HA(&)
to the formulae of ~(&)
B
~ B.
we then obtain that
more interest would be to show that
§6.
thesis".
, page 196 it is shown that for formulae
~ (B~A),
~ ~(c,F~A(X ) ~ ~B(X)7).
two
Of
221
But of course, it was strong conjunction that led us to the concept of a proof-functional connective. It perhaps should be remarked that accepting proof-functional connectives, such as strong conjunction, requires rejecting the assumption that a construction proves a unique sentence and thus forces us to distinguish between a construction as an object and a construction as a method. REFERENCES Pottinger, G.
[1980]
A Type Assignment for Strongly Normalizable %-Terms, in To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Edited by J.P. Seldin and J.R. Hindley, Academic Press, N.Y.
Troelstra,A.S.
[1979]
The interplay between logic and Mathematics: Intuitionism.
Report 79-01 of the Mathematisch
Instituut, University of Amsterdam. Troelstra,A.S.
[1973]
Mathematical Investigation of Intuitionistic Arithmetic and Analysis.
Lecture Notes in Mathe-
matics, vol. 344, Springer-Verlag Publishing Co.
ULTRAPRODUCTS
AND CATEGORICNL
LOGIC
M. Makkai* McGill University Department of Mathematics and Statistics 805 Sherbroo~e ~t. West Montreal H3A 2K6 Quebec, Canada Introduction In categorical
logic, there is a natural way of considering
idea of a logical operation construed
as an operation
a given category,
in general.
A logical operation
acting on finite diagrams
yielding
finite diagrams
the
can be
of a given type in
in the same category
as
values. In symbolic certain standard symbolic
a standard
development category,
in categorical
most frequently
The next step is to consider
disembodied
the original abstract
become formulas
in symbolic
In symbolic life situation
the
and morphisms,
including sentence
only has to codify in precise
In an attempt
universe,
just as sets and relations
in choosing
from the real the rules of
and inference.
in natural
languages;
In one
terms what is already given intuitively
form.
be hampered
or the "true",
nature of
rather than helped by our linguis-
This background may have been shaped in an arbitrary
manner by an evolutionary mathematical
in arbitrary
of the diagrams become
formation,
to clarify the general,
logic, we may, however,
in
logic.
this we are uniquely helped by experience
tic background.
logic starts by
'same' operations
to the symbolic one consists
complete
the
Set, the category of
logic the main step of the transition
formal manipulation,
in a practically
to considering
acting on finite diagrams
sets and functions
objects
having
governed by formal rules of mani-
certain selected operations,
categories;
expressions
and one proceeds
uninterpreted,
The parallel
considering Set.
interpretations,
expressions
pulation. sets, and
logic, one starts with symbolic
process.
logic is essentially
If one believes,
as I do, that
a part of the makeup of the physical
rather than being an addition
life, then a theory of logic independent
to it contributed of concrete
by intelligent
aspects
of language
is desirable. A particular
difficulty
arising in symbolic
encountered when trying to coach in symbolic * The author's
research
logic is the one
terms a general
concept of
is supported by NSERC Canada and FCAC Quebec.
223
logical operation.
In abstract model theory,
at such a general concept,
one considers
operations on strings of variables,
in an attempt at arriving
generalized quantifiers,
and strings of formulas.
It is
strikingly clear that the formal aspects of the concept are too closely modelled on previous abstract integrity,
linguistic patterns;
the concept does not have an
a good chance of being comprehensive.
In categorical
logic, the main step of the transition to the
abstract situation is to decide what constitutes, arbitrary category, standard category, natural one.
in a more or less
the same operation as the one we start with in the say Set.
One considers
The general shape of the procedure those properties,
as many as one can find,
or maybe, only some of them, of the given concrete operation combinations
of various given concrete operations)
in the language of categories "theories",
(or
that can be expressed
(i.e. in terms of composition of arrows),
and one simply imposes these properties categories;
is a
as requirements
on the kind of
that serve to embody the abstract essence of
the logic under consideration.
The crucial point is that the "theory"
is a structure of the same kind as the standard structure we start with: they are both categories.
In the symbolic context,
a structure radically different
the formulas
form
from the entities making up the standard
interpretation. Although the choice of the properties may not be straightforward, and a priori
it is not clear if the proposed procedure
in capturing at least as much as symbolic
can be successful
logic does, the proposal at
least has a clear enough outline to constitute a program for an investigation of logic in a general manner. This is not the place to give a historical categorical
logic.
that categorical
Nevertheless,
introduction to
we may make the blanket statement
logic has been successfully developped along the lines
indicated above to cover at least first order, and also, higher order logic, in both the so-called classical and intuitionistic versions. particularly attractive character.
E.g., the definable concepts
order theory on the one hand,
(roughly,
but those two structures construction,
are connected,
functor, mapping the theory-category
a category.
Net only that
the so-called evaluation
into the category of functors from
This, or other similar constructions,
ultimately, be explained without so gotten are clumsy,
of a first
as a result of a general cat-
by a crucial functor,
the model-category to Set.
formulas)
and the models of the theory on the other
hand, both form the same kind of structure: egorical
A
feature of the resulting theory is its unified
could,
category theory, but the explanations
and they lack the coherence of a good theory.
224
In first order logic egorical
(our sole interest
logic has revealed
in this paper),
cat-
that, in order to have a good general
of logic, one should consider
theory
certain kinds of 'definable concepts'
associated with any given first order theory that do not fall under the usual definable
sets or
relations
of symbolic
able sets arise either as a disjoint set of equivalence definable
set.
has become [CHL].
important
Those new defin-
sum of definable sets, or as the
classes of a definable
Recently,
logic.
equivalence
relation on a
the use of the more general
in stability
(These developments
theory;
c.f.
definable
sets
ESh] and especially
have been independent,
so far, of categorical
logic proper). The central
concept of first order categorical
the notion of first order theory, notion that underlies, stability abstract logic.
implicitly,
theory mentioned algebraic
above.
geometry
view of logic,
has its central
since it enjoys completeness
that the concept without
the generalized
definable
The concept originally
(sheaf theory), without
The notion of pretopos
logic, replacing
is the notion of pretopos.
cannot be made to comprise
role,
sets of
arose in
any connection
each indicating
logical operations
causing the notion of theory to lose essential properties.
The subject matter of this paper is one such completeness It is based on a very general operation on diagrams, operations) complement.
~in fact, too general)
of pretopos
operations
functors
is shown to be not too general).
follows a familiar pattern:
logical concepts
in Set
on Set are
'algebraic'
the definition The result,
characterizations
a stronger
Gaifman's
form of Gaifman's
theorem.
again in essence,
theorem
The proof of the theorem of the arguments.
The comparison with
result is made harde~ although only in a superficial
the fact that Gaifman's
terminology
Part 3 of the paper contains The exposition
(not using categories)
the details of the comparison.
to categorical
in the categorical
w a ~ by
is different.
of the theorem is used as an opportunity logic.
to give
The methods used here differ from
those in [MR] inasmuch as I have tried to use, wherever inherent
of
consist of a proof of Gaifman's
theorem followed by additional
an introduction
of
The main result of the present paper is, in essence,
present paper will, original
of
via ultraproducts.
The main result is closely related to an unpublished of Haim Gaifman.
('logical'
augmented with Boolean
(Thus, after all, in the given context,
a logical operation
property.
concept of
and it says that the operations
that commute with the ultraproduct
exactly the composites
course,
to
from the point of
properties, further
It is the
formulations.
I could, methods
All unexplained
category
225
theoretical terminology can be found in ECWM]. I owe special thanks to Andrew Pitts,
from whom I have learned
much on conservative and quotient morphism.
I also thank Michael Barr
who pointed out an embarassing error in an earlier version of the paper. Part 1. I.i
Logical Operations
Operations o ~ Diagrams
Logical operations are seen, on the simplest on sets:
the operation,
level, as operations
applied to one or more sets
arity of the operation),
yields another set.
(depending on the
On the second level, the
operation on sets is abstracted, with retaining its essential properties, and one ends up with an algebraic operation, the customary sense: ations union, algebras,
or a logical operation in
an operation acting on formulas.
intersection,
The set oper-
etc. giving rise to abstract Boolean
and giving rise to the logical connectives,
are the obvious
illustration of this state of affairs. In fact, not only the connectives, be so construed.
but also the quantifiers
To do so, one should, however,
idea of an operation on sets.
can
slightly refine the
Instead of talking about operations
acting on ordered finite tuples of sets, one has to consider ones operating on finite systems of a slightly more elaborate kind.
It is
a basic insight of category theory that, by talking about systems of sets and functions,
each mapping one of the given sets into another one,
complex ideas concerning sets can be elegantly expressed.
A system of
the kind we need to have is usually called a diagram of sets and
functions.
In this paper, we are interested mainly in finite diagrams.
For a while,
a diagram will always mean a diagram of sets and functions.
A finite diagram is a system of finitely many sets and functions given shape'.
'of a
An example would be the diagram f > B
A
h
~C
g consisting f is
A,
of three and
f
sets
maps
and t h r e e S
into
functions,
B, e t c .
The
such that 'shape'
the
of this
domain of diagram
is this:
÷ -
such a thing is called a graph.
;
(I)
226
Before we make these concepts present
the existential
a finite diagram,
quantifier
general
and precise,
as an operation
let us re-
that yields,
from
more sets and functions making up, together with the
given one, a larger finite diagram. One starts with a relation over a set
X,
its second-place
can also write of
X.
Now,
five sets:
Rxy R
for
R.
variable
itself is a subset of R
i: R -+ XxY.
j: ~yRxy --~ X.
The fact that
'explained',
#2: X×Y --~ Y.
Thus,
first-place on
Y
XxY.
variable
(usually,
ranges
Y = X).
~yRxy,
We
a subset
So, we have the following
These sets are connected by maps.
being a subset of
inclusion map: will be
whose
We want to talk about
X, Y, X×Y, R, 3yRxy.
First of all,
Y
R
X×Y
means
the presence
of the
We will have another inclusion map: X×Y
is the Cartesian product
by two projection
the operation
gives, i
R
>
maps:
X
and
from the diagram XxY
X
one further set and one further
of
#i: X×Y -+ X,
Y
function,
extending
the given diagram
to i
ayRxy
> XxY
~ X
We have not mentioned yet how actually we only w a n t e d to make clear its operations
yields
Y
the operation
'shape', or arity.
from each of certain diagrams
7\ a diagram of the shape
is defined:
In fact, the
of the shape
227
/\ extending the first diagram in the obvious sense. The above example following way.
can be streamlined,
We start with a single
in the above situation, clearly,
the subset
of the function with any
f.
f
~ X
;
is the composite
3yRxy
of
of
i
and
~I"
is the same as the range
X
So, we consider
Then (or image)
the following operation:
f
R
in the
function
f
R
and generalized,
~ X
we associate f
R
with S the range of latter operation
f,
and
has the
j
~ X
the inclusion map of it into
7
It is the next step in category
theoretical
that not only the form, but also the content quantifier
language of diagrams,
~and many others)
theory.
For an indication
the idea of
added:
in the
the concept
of
Doing so, we are right in the middle of
look at the above simplified explicate
insight to realize
of the operation of the
can be expressed
with one new element
composition of functions. category
The
'shape'
° ~
existential
X.
of what would be going on,
operation.
'inclusion'.
let us
The first thing would be to
This is done by the notion of
monomorphism, whose definition I will not repeat here (c.f. ECWMI). To be sure,
a monomorphism
in Set is not necessarily
an inclusion
~the
228
converse
is true, of course) but at least,
category of sets) R
) U
inclusion,
it is the case that
any m o n o m o r p h i s m is i s o m o r p h i c
is a m o n o m o r p h i s m ,
and i s o m o r p h i s m s
R
then there are ~ R',
U
(in the
to an inclusion: R'
* U'
if
) U', an such that the
square
In fact,
~ommut~8.
U
~ U'
U'
the i d e n t i t y
R
+U
R'
+ U'
can be chosen to be equal to
U,
with
function.
S t a r t i n g w i t h the d i a g r a m f
R
+ X
we consider an e x t e n s i o n diagram f
R
/, ,
X
S with the following properties: q: R
~ S
j
is a mono;
R
commute
moreover,
there is
making
[since
j
following p r o p e r t y
is a mono, (q
is an
whenever
..............l. X
q
is n e c e s s a r i l y unique] and having the
'extremal epi'):
R
q
) S
q' S' commutes,
and
k
is a monomorphism,
then
k
must be an isomorphism.
229
(We have described phism
f).
the extremal
epi-mono
factorization
It turns out that the property
uniquely up to a unique isomorphism: S
(with the same isomorphism S
defines
j
whenever
.................... J
S'
described
of the mor-
~ X
............ ~ '
,~ X
then there is a unique
X) both answer the description, ~ S' making
"X
commute.
In fact, the inclusion
does answer the description; tion, we have described, 'range ' What is important ultimately,
of the range of
thus, by the above
f
into the set
'categorical'
up to a unique
isomorphism,
the concept of
about the above
'definition'
of range,
that of the existential
quantifier,
elements of the sets.
mentioning
that:
their elements about the
as objects
'category of formulas'
existential sets.
quantifier
In the foregoing terms of category
Therefore,
in a category.
discussion,
theory:
without
abstract,
it is natural
commute,
that
to try to talk
In fact, we will give in
the same definition
(or 'range')
We will have to continue for explanations,
representing
their being unspecified means in particular
are not given.
'formulas'
abstractly,
Now, it is clear that in logic we do
we talk about formulas
"unspecified", sets;
and
is that it makes sense
in a context where we talk about sets and functions precisely
X
defini-
of the operation
of
as we did above in the category of
I already mentioned category,
some technical
monomorphism,
to use such elementary
categorical
isomorphism. concepts;
I refer to [CWM].
A graph (see [CWM~), is like a category but with reference to composition and identity morphisms removed. The appropriate structure preserving maps between graphs are called diagrams. Since every category is a graph
of type
G
(has an underlying
in a category
C:
graph), we may speak of a diagram
a diagram
D: G
~ C.
230
Given a graph C
form the objects
transformations type
G
and a category
of a category
(see [CW~]).
(G,C)
E.g.,
(I) is a system of arrows
C,
f
in
between
C
G
in
are natural
two diagrams
of
as in the picture:
h
B
of type
whose morphisms
a morphism
~, B, y
A
the diagrams
, C
f'
A' satisfying the square
-+ B'
g,
three commutativity
A
conditions,
f
.
A' should be commutative;
one for each arrow
~ B'
and two more similar
conditions.
is an invertible
it is one in which every component
In particular,
natural
(in the example,
transformation;
~, 8 or ~)
We are ready to give a very general should be.
Upon reading
definition
of what
the definition,
it turns out that what one would
are not sufficient
Let
term C
in the important 'operation'
be a category,
C
of type
C
(K' c Ob(G',C))
(i)
K'
diagrams D~I c K'
is closed under in
(G',C),
(i = 1,2),
and
(G,G')
of first order
logic.
Thus
of G'.
An
~:
G a subgraph
is given by a class K' of diagrams satisfying
Di c K',
that restricts
analogy
context
(if then
Di = D!IG1 (restriction then there
K'
operations'
isomorphisms
is an isomorphism, to
the following Di
and
Di c K') to G), and
is a unique isomorphism
are isomorphic
and
(ii)
~: D 1
if + D2
~': Di -----+ D½
~'IG = ~.
algebraic
operations).
of
conditions:
D½
may be called the graph of the given operation to ordinary
How-
'fully defined
G and G' graphs,
type
in
instead.
call
is preferable.
operation in G'
an operation
the reader may
feel that we should have talked about partial operations
the shorter
is,
an isomorphism.
in a category ever,
in (I):
~ B
f,
an isomorphism of two diagrams individually,
h'
(in obvious
231
An operation
is finitary
(having finitely many objects
if
G'
The domain of the operation (G,G)
which are restrictions
It is easy to see that
K
(hence
G
too)
is a finite graph
and arrows). is the class
of ones in
K
of those diagrams
K': K = {D'IG:
D': G'
is closed under isomorphisms.
We are going to call the operation fully defined if We will soon see an important
example
When we have two categories, so that the two operations consider
in algebra.
of a non-full
with an operation
to be the
specified
In particular,
"operation-preserving"
functor of any diagram in the graph of the operation
Before we come to examples, satisfy a stronger
arbitrary morphisms,
in fact,
the modified
Let us call such operations important
operations
which
of the
done with ordinary operations (with re-
this is the case if the image under the
gory is in the graph of the operation
that,
to
we have the important notion of a functor
from one category to the other being
operations
in each
'same', or "realizations
as it is frequently
spect to the given operations):
meaning
K = (G,C).
operation.
are of the same type, we might want
the two operations
same operation-symbol",
in
~ C}.
let us mention
condition, not just
condition
in the domain cate-
in the codomain
namely
category.
that the most
important
(ii) above with ¢ and ~'
isomorphisms is stronger
strong operations.
(it is easy to see than the original
Nevertheless,
do not satisfy the stronger
one).
there are
condition.
Logical operations are operations in the category of sets, and, more generally,
operations
operations
in Set.
less, e.g.
our description
in other categories
Of course,
the designation
of pretoposes
The reader
(operations
as operations
should be like
(or look up, say in [CWM])
in categories. in a different
on models).
in the above sense,
With any given category, limit
operations
Finite
for us as logical operations;
and colimits will be important operations
neverthe-
operations.
is now asked to recall
of limit and colimit
will be important
is imprecise;
in Set. 1.2. Pretopos
concepts
'behave like'
in the next section will be
explicitly based on the idea that pretopos certain operations
that
Limits
also,
role,
the
limits and colimits as
and colimits
infinite
limits
'semantic' can be construed
in the obvious way.
we obtain one
(partial)
operation
of
for each graph serving as the type of the diagram the limit of
which we take; the operation,
this graph is the type of the diagrams denoted
G
above.
E.g.,
in the domain of
in any category,
the operation
232
of product of two objects is given by the class
K'
of product-
diagrams C
of type the graph
G':
the domain of the operation consists of those pairs of objects (diagrams of type G: • .)
which have a product in
colimit operations are, in fact, obviously,
C.
A, B
The limit and
strong operations.
A limit,
or colimit, operation of a particular kind is fully defined in a given category just in case that category "has all limits given kind", in the usual terminology.
(colimits) of the
In Set, the category of sets and functions, all small limits and colimits are fully defined, and they have meanings which are familiar from many contexts.
E.g., any diagram of the form B
Ax
Ad
with
A × B
~l() Set:
the
= a,
Cartesian ~2()
B
product: = b,
is
A × B = {: a product
aeA,
diagram.
beB},
and
The diagram
in
C
is a copreduct diagram if and only if both C
i, i'
is the disjoint union of range(i) and range(i').
are one-to-one, (Therefore,
and
in Set,
233
coproducts
are also called disjoint sums).
The limit of the empty diagram is called the terminal object; Set, any one-element
set serves as such.
diagram is the initial object; It is a well-known terminal object
fact
it is the empty set in Set. (see [CWM])
G
is now the empty graph],
and has equalizers of parallel pairs of morphisms,
then it has all finite limits. consequence,
that if a category has a
[this is a ~-ary operation:
has binary products
in
The colimit of the empty
for colimits.
An exactly dual statement holds, as a
The notion of pretopos will be based on
finite limits and finite colimits,
the latter suitably restricted.
A diagram q
A
, B
(I)
q' in Set will be called an equivalence relation if the map a l
~ 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 J-covering of A), with
is called
the following
(i)
A (Grothendieck)
limits.
J
required
conditions:
A' --=---+ A
is a 1-element
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 J-covers
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 J-continuous 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 pre-topology],
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 pre-topology, 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 J-continuous
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 J-coverings
¢(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 sub-poset 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 J-covering
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, ~ 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
Beak-condition ÷ 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 so-called 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 Beck-condition,
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
F-image 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 so-called 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 quotient-conservative
factorization.
The definition of a quotient-morphism 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
2-category
"objects" of the 2-category),
(c.f. [CWM]):
it (the
certain functors between them (the
"morphisms" of the 2-category), and certain natural transformations between those functors
(the "2-cells" of the 2-category).
is the 2-category of pretoposes, pretopos-morphisms, transformations between such.
E.g., Pretop
and all natural
Similarly, we may talk about Lex, the
2-category of categories with finite limits, functors preserving them ("Lex-morphisms") and all natural transformations between such. The 2-category of coherent categories is
Coh.
The 'standard' 2-category is Cat, the 2-category 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:
V@. !
Cat, and
Therefore, we have in mind three setfor
i = ~, i, 2,
with
@0 < @i < 82
256
inaccessible
cardinals.
Ob(Set)
= V@0 , Set e Ob(Cat)
c V@I,
Cat e V@
. Small categories are those in V@0. 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 2-categories. 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 2-category
+~(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 Lex-morphism
> T'
F: T
can be factorized
into Lex-morphisms: 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.
rfs-l~.
in diagrams,
E are indicated by double arrows.
To define
composition,
what we need to show is that,
in the
situation
the morphism Consider
gt-lfs -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: rgt-l~
o
rfs-ln
= r(gf,) (st,)-l~.
T" This definition equivalence
is legitimate;
classes
again be equivalent, in
if different
are chosen,
simply because
of the
right-hand-sides
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 rfs-l~ into F
representatives
the resulting
is well-defined,
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 rsf-l~,
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
rfs-l'.
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
rfs-ln
and
rgs-l~
simultaneously
259
t'
S
It is clear that, with rgt-l~ = 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
rfs-l~
rfs-l~
~
,
g (in T);
T',
es
=
since
= r(ft')u -I~,
rgs-ll 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 rfs-l~, rgs-l~ 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 quotient-conservative 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 Lex-morphism
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 Lex-morphism, uniqueness
in 2.3.1 of the quotient
in the quotient-conservative
satisfies
of this factorization
the condition holds
construction
the condition.
(2.3.1'.),
factor-
Now, by the
one easily deduces
for any quotient-morphism.
that
262
Conversely,
assume
quotient-conservative
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 quotient-conservative
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 quotient-conservative
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 image-mono
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 subobject-fullness
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 Lex-quotient,
then it is a
Coh-quotient. m 2.3.4. Corollar[
2.3.5.
on subobjects
Proof:
The
'if' part, 2.3.4.
A Coh-morphism
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 Lex-quotient,
i.e.,
be given.
Find
A'
and an isomorphism
the composite FA' m X
f', the subobject
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 Coh-morphism
category. iff it is a
Pretop-morphism.
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 Coh-morphism.
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 Coh-morphism,
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 quotient-field sums;
(Cohds for short),
satisfying
conditions
is a Coh-morphism
(4) in 2.4.2 holds.
(1),
Cohqe(T)
as follows.
A Cohds-cat-
preserving
for short,
is a coherent
We prove two propositions.
is also a Cohds-category,
Let us put ourselves
and
One is our
the other with Pretop has the addendum that
provided
T
is a Cohds-category.
imply the one to be proved.
To carry out the first construction,
let us make some preliminary
into a Cohds-category.
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 Cohqe-category 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 2-category
Cohds-morphism
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 one-to-one
correspondence
with
of
270
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
Cob-category
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 Coh-morphisms
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 Coh-morphism objects, statement that,
quotient-like if it is full on sub-
and every object in its codomain of the
'moreover'
for a Pretop-morphism,
a quotient.
is covered by it (see the
part of the last proposition). being quotient-like
But first, we prove
We will show
is equivalent
to being
274
Lemma 2.4.6.
A quotient-like
conservative
Pretop-morphism
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
: equivalence
f;
let
, FAxFA. B
~ AxA,
relation,
Y
be the subobject
By assumption, say, whose
and
F
F-image
+
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 subobject-full. • 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 Pretop-morphism
on subobjects,
is a quotient
and every object
if and only if it is full
in its codomain
is covered by it (it
is quotient-like). (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 subobject-full,
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 Coh-morphism 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
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 quotient-conservative factorization. Pff0position 2.4.8. In any one of the 2-categories Lex, Coh, Pretop, the following is true. If T is any object of the 2-category, N is any set of morphisms in
T,
then there is an arrow Q: T
~ T[~ -I ]
in the 2-category 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 2-category).
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 L-structures,
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 many-sorted 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 Coh-functor,
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 L-structure,
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 Lex-functor
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 kernel-pair, 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 J-continuous,
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 well-known diagrams this:
in
sense.
the Cob-operations)
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 Pretop-morphism
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 first-order
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
(2-categorical)
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 Coh-category
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 G-diagram
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 category-based
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
(Gentzen-sequents)
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
"2-catalgebraic
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 Coh-functor
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 2-category
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 Groupoid-enriched rather than Cat-enriched;
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
Gpd-category), Even
the reader can see here is intimately
defining Boolean pretoposes
I.i).
in the context of BPretop,
a quotient-morphism
C,
had to be defined
the fact we are considering
related to the fact that not all operations are strong
of
as 2-cells.
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 Coh-morphism (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 Coh-morphism 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 L-formulas
appropriate).
that there
satisfy @,
~(~)
÷ o(~))
then
Given a
in a countable
is consistent
T ~V~(¢(~)
(for all L-sentence
Let us return
(see [CK]).
a set
given a countable
~(~)"
in the opposite
is complete
if
is non-isolated,
"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 L-formulas 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 L-structures 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 well-known 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 counter-example
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~wenheim-Skolem
~
^ ¢(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*(N),
M, N e Mod T,
Mfm)
incl.
Then there is an L-formula
, 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 L-isomorphisms 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 L-isomorphisms 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 Keisler-Shelah
<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
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 one-sorted
are all those that are isomorphic 'replete'
Given two
in [CK] stated for the case
case has the same proof,
from the one-sorted 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 U-ultrapowers 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 Lex-quotient
we can still say that
precisely,
With
with
property
even if C
+ F(C)
generates
T
some analogs
a canonical
Lex-morphism,
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 Lex-morphism
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 Lex-morphism
"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
÷
Z-categorical
I remind the reader of a
adjunction.
in fact could be any category.
There
Let
S
= Set;
at
is an adjunction between
the 2-functors
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
2-functors 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 functor-category
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 Pretop-morphism. The adjunction-isomorphisms Below,
referring
certain restrictions appropriate (0 = @C,N
are defined as before.
to the last adjunction,
of
2-category,
~(F),
or
G(F),
and LH~, ~H l
for some appropriate
for
we will write F* for
for a morphism
F
@(H),
respectively
e-I(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 right-hand-side 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 left-hand-side 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 quasi-inverses of ~*, ~'*, and composing them with the inclusion, we obtain the upper square in the left-hand-side 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 right-hand-side, by the help of our adjunction. Second, Q and r form the quotient-conservative 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 well-defined,
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.
('pre-ultrafunctor'
in which ultraproducts
is a functor
fied isomorphisms.
as follows:
~
it is clear that
A u-functor
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 transition-structure 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). u-functor is one in which the [X,U]'s are all identities. The u-functors K1 ÷ K 2 form a category, u-transformations as morphisms. A u-transformation natural additional
transformation condition
between of
the
compatibility
functor-parts 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 1-cells and 2-cells: 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 Pretop-morphism; easy to see (using Los's theorem).
this is
301
If I: T 1 ~ T 2 is a Pretop-morphism, then I*: Mod T 2 --~ Mod T 1 is a strict u-functor. Similarly, for a diagram DO: G , T, (Do)*: Mod T
, (G, ~
is a strict u-functor.
Now, let us turn to the diagram under (2) and (3). (~')*
is a strict u-functor;
is one.
also, the inclusion of
As we said,
K'
in (G', @ i s °
It is easy to see that the quasi-inverse of the functor-part of
a u-functor whose functor-part
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 quasi-inverse with isomorphisms at head and tail that are u-transformations,
but we don't need this additional fact).
in (3) can be made into a u-functor, 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 quotient-conservative
factorization of
conservative, and hence
is conservative, by the uniqueness of the
qr
H.
Since
q = qK'
is
q-e 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 u-functor. 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 u-functor, 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 2-category
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'
+ 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
0A E
v[n>0An-l~A^m=0^ e B]
" ~ B
~
B eA
then
= 0]
v[n>0An-i
Then we can
p
we can count.
such that whenever
i.e.
count mod
~'A^ 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_ a-I
(i.e.
F maps a i-I
into
IF n (yx(a-l))l
= Y
IF n (a×y) I x-p,
M
we can count
for all
0 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 + p-l)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 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 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
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-
is odd.
We now turn to look at oracle versions PROBLEM.
Q, A'
symbol.
Then is
i~-->x-I 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-->x-l.
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--> x-i
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 ~ non-standard.
argument.
Forcing conditions F(Xl)
with
~-
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[ _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 ~ T E a l ,
al
where
and hence
as
=
QYr < ak 'Yr"k _c Rg(F)
i ~ i ~4
yq(n)
if
Pi = { ~ O n k
M i ~ f~g
{ ~ O n
if
i
By
An .
Put
~n-Completeness
is bounded° m'cm
ioe.
is
A
n
and
An }
Recall
is
if
f,gcM i
then:
, and
A n to ensure
f~M i
it is
that will prove the theorem.
U,
where
meM.
M ! M I"
By definition of
Suppose that for every m ' e m i.e.
f-l(m')~ U.
onk\f-l(m')EU
B m, = onk\f-l(m'), n Bm'cU m'em
,
This contradiction
M k ~ f = m'.
function
~~eOn k . Then
for every
M k ~ fcm,
}cU
Then, by definiton of is
A is
f~D i.
~ = m Cm(~)
: M ~ f(~) = m' } ~U
f-l(m')
f
m is identified w i t h the constant
such that
and
Mk,{~On
'
M _ce Mko
If
Cm:On k ÷ M
AcU
~ f(~)~g6)}~U
f:On k ÷ M
Now, we set a series of lemmas LEMMA
and
: M ~ f(~) = g(~)}~U
and
enough to find a witness
f~D. if i
A~.
i~{l ..... k}.
Mi ~ f = g
Remark that,
for the fact
f is bounded over
but
for every
m'~m, because
for every m'em.
n Bm' = m'~m
shows that
and
f-l(m')EU
~U
because
for some
345
LEMMA
6.
PROOF.
M ~ MI
M i ~ Mi+ I
We have seen that
to take
feD i
e~i(A) A
A
for
and therefore
Let us define
Pi
Pi~Mi
i~{l .... ,k-l}
To see that
f~Di+ I. feD i.
Let
So,
f
M i ~ Mi+ I it suffices be an element of
f"(A[)
by
Pi (~) = ~i' ie{l,...,k}
Pi is bounded over
(onk)~
ie{l .... ,k-l} there is
over
A~ .
That is
Thus
{xeon
Pi+l ~Mi
x ~i(A)
x ~i(onk).
A c On k and for
such that
Pi+l
is unbounded
because the elements of U are unbounded.
M i ~ Mi+ I.
We show that k
That
It is clear
for each
Moreover it is easy to see that for every unbounded every
Di, then
is bounded for every
f c Di+ I.
: Onk ÷ M
because
for
f" (A~) is bounded for every y c~i+l(A).
is a witness for
that
M ! MIo
and show that
there is a witness
is
and
Pl ~ Mo
=
: aex I}
clear that
Let e be an ordinal in M.
Note that
has bounded complement, thus {xcOn
pl~onMl.
So,
M1 ~e{Pl
for every
k
: ~eXl}~ 0.
~eon
And it is
and therefore
Pl ~ M. LEMMA PROOF°
7. Let
tion of geM i.
f
Mi+l, Then
(AnB)~
M i ce Mi+ 1 and
g
for
ie{l,...,k-l}
be such that g e M i and Mi+ I ~ feg. By definik ~ : M ~ f(e) eg(~)}eU. Let B be a witness for
A = {~eOn
AnB~U
for every
and
AnB witness
x c~i(AnB)
and
f
f~M i because
g is bounded over
is bounded by
g.
LEMMA 8. ~o~ lemma for (ZnU~ n) formulas holds for each Mi,ie{l ..... k} That is for every E or H formula n n M i ~ ~[~]
{ ~ O n
k
:
~ M ~ ~[f(~)]} ~ U.
(*)
Before the proof of lemma 8 we have to remark that the hypothesis of theorem 4 holds; a U'
An-based on
On.
M ~n-COllection
if
this is so because if there exists
Hn-ultrafilter U on
On k
then there exists a
By induction, since
M ~ V = L,
~n-ultrafilter
M has A -Skolem functions. n
Then (theorem ic) there exists M' such that
M ~ e M' n+l "
M ~ V = L, M is resolvable and therefore (theorem Ib)
But
'
as
M ~ ~ -collection. n
346
PROOF OF LEMMA 8.
By induction
the lemma is true by definition Note that if (*) holds holds
for the boolean
on the complexity
for the formulas combinations
The proof of (*) for Ao-formulas En-formulas
assuming
(*) for
of (*) for
A -formula.
(*) for
Hn-l"
Then there exists
Then
{~cOn k
~ ! En, then
(*)
of F.
is essentially
the same as we give for
So, we will
omit the proof
for
Suppose
that
g~M i such that
Hn-I formulas
: M ~(g(~),
~(~))}cU,
M i ~ 3×~(x,~)
M i >~(g,f).
because
where
~ is
But, by the above
(*) holds
for En_l-formulas.
and therefore
: M ~x~(x,~(~))¢U.
Conversely,
suppose
such that
A
is
A
n g: On k ÷ M by
Define
-~ k B = {~On
: M >~×~(x,f(~))}~U.
and
because
A~U,
g(~) = y
E~EA ^ ~(y,f(a))
g is well
define because
about M, see [P]).
As
A
also
Clearly
tive hypothesis, Therefore is
An .
Let
C
{~EOn k
If
i Ad + Hn-Collection So
this proof),
is
so M i ~3×~(x,~). If
for
fact
Vz< L y -~ ~(z,~(~))
the remark before
: M >~(g(~),
^ y = 03.
(this is a standard
An
there exists e such that
because
( ))3 v [ ~ A
geM i.
be a witness x c~i(D )
are
(c.f.
M i >~(g,~),
it remains
~
There exists A ! B
A -based. n
A VZ<ey ~ ~(Z,
and
M > Hn-COllection
An .
U is
M >An+l-foundation
An, since
So if
of some class
of formulas
En_l-formulas.
In_l-formulas.
(*) holds
{a~On k
If ~ is atomic
O
Assume
remark,
of ~.
of M i,
see Thus
for some B.
gEM i.
347
LEMMA
9.
M ~n Mi and
Mi ~n Mi+l for every
i {I ..... k-l}.
PROOF.
Let ~ be a In-formula. Then, by Lemma 8 -~ k M >~(~) ~> {a~On : M ~-~(a)} = onkc U ~> M i }=~(a) i.e.
M 4n Mi" _l
Let
~
be such that
fcM i. By Lemma 8 (~ is In). k ~ ~ {~eOn : M ~%(f(e))}(U Mi+ I ~+(~).
M i >~(~) Joe.
Mi ~ n Mi+l"
LEMMA I0. PROOF.
M ~n+l Mk
Let ~ be a
In+l-formula.
M ~n Mk (Lemma 9) and To show that
M ~n+l ~
by
V×~y~(x,y,z)
M ~[~],
then
it suffices to show that M k.
So, let ~ b
where ~ is ~n-l"
f(x)=y P(g(~), fog(~),~)}EU M k ~ ~(g,fog,~) for any
LEMMA PROOF.
, and
g~M k.
as
fog is
So, for any
In+l-formulas.
Hn+l-formulas satisfi-
M ~Nn-Collection°
therefore
M k ~-~[~] because
In-elementary extensions preserve
ed in M are satisfied in form
If
f
~ of the
Define f:M ÷ M is
A n because
geMk,
An, we have, by Lemma 8,
geMk, M k > @y P(g,y,~), and
Mk > V ×qy P(x,y,~). ii.
If
M !+l
Mi
for every
M i > ~×~(x,f)
So, by lemma 9,
ie{l .... ,k-l}.
where ~ is
Mk ~(h,f)
and hence
Nn'
then
M i > ~(h,f) for some h o
M k > @ ~(x,f).
So by lemma I0,
M ~ @ x%(x,f). The converse is a consequence of lemma 9. LEMMA PROOF.
12.
Mi ! + l
Let ~ be a
Mi ~n Mk (Lemma 9) extensions°
Mk
for
every
In+ I formula. and
If
ie{l ..... k} M i >~[~]
then M k ~ [ f ]
because
In+l-formulas are preserved by In-elementary
848
Conversely,
assume
M k P~[~].
~xV yP(x,y,v), where P is Note that
M k ~ "V
M ~ "V~L~exists"
and
Since ~ is En+l, ~ is of the form
In_ I. L ~ exists" because
"V~Leexists"
M k > 9xVy(eek ~(x,y,~) because M k >VycLPk~(h,y,~).
Since
is ~2"
n >- 1 '
M ~n +I M~,
So, M k > LPk exists.
Mk > @xVy~(x,yT).
Take
h<M k such that
Vy~LPk $(h,y,~) is ~n' by the Lemma 8,
{~eOn k : M > Vy~L k P(h(~),y,f(~))}e[~.
It follows that
B = {~¢On k : M ~ ~×VycL Let
A
be such that
~(x,y,~(~))}~U. ~k A ~_ B, A is An and
AcU.
Define
g-On k + M by:
_A
_A
g(~) = x (~¢A ^ VycL k~(x,y,~(~))
_~
A VZ~(g,y,~)
An
These three
~ ~ {~¢On k : M > V y c L ~(g(~),y,f(~))}¢U. _~ k M k > VycLPk~(g,y,f). Now , remark that M i £ LPk
M i ~ V = L, M k ~ V = L (V = L is
Assume
are
is An.
M ~n+l Mi' M ~n+l Mk and
so
M > An+l-foundation.
for any
and
~2' M ~ V = L and
onMi --~ Pk
(Lemmas 6 and 7).
ycM i _c LPk, M k >~(g,y,~) but
YeMi, that is M i > Vy~(g,y,~)
M i ~n Mk'
and therefore
M i ~ ~xVy~(g,y,f). It only remains to prove the assumption M k > @xVy~(x,y,~).
Take
g e M i.
We have
h' such that M k ~ Vy ~(h',y,~). Then, by
Lemma 8, E = {~¢On C I _c E,
k
~ ~ : M ~ Vy P(h'(~),y,f(~))}¢U..
C I is
An
and
CICU.
Let
Let
C I be such that
C 2 be a witness for
~¢M i
Put
349
D = AnCInC 2. witness Take
D is
for
As
f"(D~x) ! L6
D
Recall is a set
nition of D,
L
for
that D is a
~ c M i, there exists
M > En-separation
Put
" L B. b = m~ " '~D - -xJn
M ~ VZcb ~xVy ~(x,y,Z).
on
T
We are going to prove
is a witness that
•
M ~ ~TV Zeb @xcLTVY ed by
DeU.
g ~ M i-
xc~i(D).
~'(D~..)nLD~~
A n and
~(x,y,Z)•
(see [P])
B such
so
It is easy to see, by defi-
But now, by
Hn-COllection,
This last fact implies
that
g
is bound-
D- . x
Theorem 4 now follows
from lemmas
We don't know if the converse
5,6,7,10
and 12.
of Theorem 4 is true.
has carried us to the equivalence
of two concepts
But this question
that we give immediate-
ly. DEFINITION ke2.
14.
Let U be a
U is said to be
for every X such that f:X + M such that On k-i x f-l(a)eU
or
{(al ..... ak )~O~k
:
DEFINITION said to be
15.
i) ii) iii)
An, then there exists
~k-i < rkof Let
n
conditions
if for any
X ~ On i and
and
k iff
be such that there exist
ic{l ..... k}
Mi ~n Mi+l
for every
ic{l ..... k-l}
1 M i ~ Dn-foundation
for every
and
ic{l ..... k-l},
On k-i × XcU
and for every
a~M such that
ne 1
and
M I , M 2 .... ,Mk
k e2
~iEonMi
such that
@xeaO onM
0
we conclude (KS,P)
Since such
~
K s is d e n s e that
Hv
in
and
qij (~) > O) .
algebraically
true,
forget
finite
.
, there
(Pi(X)
are
trivially
(L,Q)
of
extending
(L,Q)
Since
qij (~) > o~
j
qij (v) A 3 (K~,P)
by
> O
(6), w e
.
find
elements
%
of
Ks
358
(~'P)
As
Pi(ap)
= 0 , this
By c o n t i n u i t y
~
~ 3
qij (ap) > O
implies
(K~,P)
we can find some
(K~,P)
~
~ ~(ap)
~p 6 K *, ~p # O, s a t i s f y i n g 2
Vv(Jv-apJ
.
_< ~p
~
qij(v)
> 0).
] Expressing
s,r
this f o r m u l a
6 K[X],...,Xn,Z] (K*,P)
~
in q u a n t i f i e r - f r e e
form we find polynomials
such t h a t
(S(ap,~p)
= O ^ A
r
(ap,~p)
> O)
v
and, w h e n e v e r
(7)
(K~,P ') a l s o s a t i s f i e s
(K~,P ') ~
T h e set of
number
ap
=~
(finite) Aj
of c l o p e n
Yp
cover
sets
of
Yp
X ~ . Since
already
covers
X ~ such t h a t e a c h
X ~.
X
and some
i
such that
replaced
by
the B l o c k A p p r o x i m a t i o n
la - a
(7) h o l d s P'
for a l l
a
P' £ X
a finite
X ~ = X I 0...U X 1
and
in some ~
with
% 0
Yp in
%,~p,i
a ,~ ,i
to find e l e m e n t s
for all
(7) h o l d s
Let
is c o n t a i n e d
w e thus can f i n d s o m e e l e m e n t s
Since
the c l o p e n s e t
X ~ is c o m p a c t ,
X
order
then
qij (v) > O)
(7) thus c o n t a i n s
For each
N o w we a p p l y
conjunction,
H(rv(ap,~p)).
the sets
be a p a r t i t i o n
~P2 ~
i ~
P' £ X * s a t i s f y i n g
Yp
Clearly,
Vv(Iv-
this
6 X
a
of
Ip, S 2
for a l l . Thus
(K~,P)
~
K
X~
A j
in
(K~,P ') #
(a)
P' 6 X
, we o b t a i n
f i n a l l y we g e t
V i
to this s i t u a t i o n
such that
for all
P' £ X
of
qij ([) > 0
A j
qij
. K~
359
for all
P 6 X ~. Since for every
course of the proof,
(K*,P) b
Moreover,
Pi
i
w h i c h was a c t u a l l y used in the
was the zero polynomial,
vi (Pi (~) = O ^
given a n o n - z e r o p o l y n o m i a l
assume that xl,..°,x n
g2
is
among the
qij
> 0
g 6 K[X I ..... X n] for each
holds for all orderings
assumption, we also
find
d e p e n d s o n the choice of s a t u r a t e d n e s s of
g(a) ~ 0
$ O
K , clearly of
L . By this
g 6 K[X]. U s i n g once m o r e the
V (Pi (x~) = 0 ^ A 3
for all n o n - z e r o
are a l g e b r a i c a l l y i n d e p e n d e n t over
since
. Here the choice of the elements
i g(x~)
Q
we may
i . Indeed,
K ~, we can even find elements
(K~, P) b
for all
qij (~) > O).
are a l g e b r a i c a l l y i n d e p e n d e n t over
(L,Q) ~ g2(~)
and
¢
we even get
q
~Kt +-
x~ in K ~ s a t i s f y i n g
(x~) > O) ij
g 6 K[X]. Thus the elements K
and satisfy
x~
(K~,P) ~ ~(x~)
P 6 X ~. This finishes the proof of the theorem.
References
[i]
ERSHOV,Yu.L.: T o t a l l y real field extensions. 25,No.2, 477-480 (1982)
[2]
ERSHOV,Yu.L.: Two theorems on r e g u l a r l y r - c l o s e d fields. J. reine angew. Math. (to appear)
[3]
HEINEMANN,B., PRESTEL,A.: Fields r e g u l a r l y closed with r e s p e c t to finitely m a n y v a l u a t i o n s and o r d e r i n g s (to appear)
[4]
PRESTEL,A.: Matem~tica,
Soviet Math.Dokl.
L e c t u r e s on f o r m a l l y real fields. M o n o g r a f i a s de Vol.22.IMPA, Rio de J a n e i r o 1975
[5]
PRESTEL,A.:
[6]
PRESTEL,A., ZIEGLER,M.: Model t h e o r e t i c m e t h o d s in the theory of t o p o l o g i c a l fields. J.reine a n g e w . M a t h . 2 9 9 / 3 O O , 3 1 8 - 3 4 1 (1978)
[7]
SCHMIDT,W.M.: Equations over finite fields. An e l e m e n t a r y approach. Lecture Notes in Math., Vol. 536, B e r l i n - H e i d e l b e r g New York: S p r i n g e r 1976
P s e u d o real closed fields. theory. L e c t u r e Notes in M a t h . V o l . 8 7 2 , S p r i n g e r 1981
In: Set theory and m o d e l B e r l i n - H e i d e l b e r g - N e w York:
FORMALIZATIONS OF CERTAIN INTERMEDIATE LOGICS Part I Cecylia Rauszer Uniwersytet Warszawaski - Instytyt ~ t e m a t y k i Palac Kultury i Nauki 00-901 Warszawa, Polska
It was Godel who
first observed the existence of a continuum of
logics between the intuitionistic predicate logic LI and the classical predicate logic LK. logics".
These logics were named by Umezawa "intermediate
There are many interesting results connected with intermediate logics. One of them asserts that only seven propositional intermediate logics have the interpolation property. Maksimova [7].
This result has been proved by
She showed that the interpolation property is equivalent
to the amalgamation property and then she proved the existence of seven classes of Heyting variaties with the amalgamation property. For predicate intermediate logics the problem of the interpolation property was first examined by Gabbay [2].
He used the theorem that Craig's
interpolation lemma is equivalent to a weaker version of Robinson's consistency theorem to prove Craig's interpolation lemma for the intermediate logics: LI, LM, LMH, where (LM)
LI + ( ~ u ~
(LMH)
LI
+
Vx ~
~ a) ~
a(x)
=
~
~
Vxa(x).
Eight years later in [3] Gabbay extended his model theoretic methods to the so-called logic of constant domains logic (LD)
LI + V x(a(x)u~) ~
where
x
LD, i.e. for the intermediate
(Vxa(x)uB),
does not appear as a free object variable in 8 and he also prov-
ed Craig's interpolation lemma for In 1981 L6pez Escobar
[4]
proof-theoretic methods, used by Umezawa
[i03
LD.
proved Craig's interpolation lemma for LDusing The same formalization for
LD
was earlier
to show some syntactical properties of LD.
It turned out that both papers [33
and [43
Kripke model for an LD theory constructed in domains and the Gentzen type formalization of
contain gaps. [33 [4]
A certain
has no constant is not complete.
361
It should be emphasized papers.
that both authors know about the errors in their
L6pez-Escobar wrote
two papers
the Gentzen formalization
of
Let me cite the following
two sentences
by J.Barwise,
[5],
that every reasonable
[6] partially
and the interpolation
connected with
property for
"Alan Anderson often argued
formal system has both a Hilbert-style notion of proof.
it is certainly
notion of
While this may overstate
true that a Gentzen-style
approach,
emphasis
on rules, rather than on axioms,
inherent
in any given logic in a way not done by a Hilbert-style
Gentzen-style
formalization
of an intermediate
definition was given by Ldpez-Escobar sense of his definition formalization for LD.
logic
L
formalization
we should know
of L.
respect
to
BS
i.e.
S
consists of two parts
[8].
speaking,
~LI
consists
: the first part called
logic
LI
of the system for is complete with
iff ~ has a normal derivation
in
BS, i.e.
The second part of the defined system
is complete,
As an application
logics
LM
and
LMH
the appropriate
formal system
and cut-free. of
L, where
The cut-free
for the
logic considered.
For the intermediate
lemma for
is different
of a certain set of logical rules which are specific
intermediate
LS
S
in [5] and in [6].
a modification
Intuitionistic
there is a cut-free proof for ~. S
A possible
there is no cut-free and complete Gentzen-style
basic system (BS) is roughly given by Sch~tte
system".
in [5] and he proved that in the
from the ones that were considered by Ldpez-Escobar
LI
with its
of a cut-free
In the present paper the basic concept of a formal system
Every formal system
the
lays bare the laws of thought
But if we want to examine the problem of the existence what is to be a cut-free Gentzen-style
LD.
from the paper Stationary Logic
Mo Kaufmann and M. Makkai:
proof and a Gentzen-style case a bit,
LD
calculus
LS
we show, among others,
L = LM, BS
introduced
a formal system for Dummett's
the Craig interpolation
LMH.
logic
in this paper can be extended LC.
to
This is done in a separate
paper. §i.
BASIC SYSTEM. We define a formal language with connectives
u (disjunction), fier),
n (conjuction),
~(implication),
i (falsum), ~ (existensial
quanti-
V (general quantifier).
Assume we are given countably variables,
sentential
infinite sets of free and bound object
variables
and predicate
symbols of each number of
362
el.
arguments
By an atomic
formula we mean every sentential
is an n-ary predicate variables The set
then FORM
I)
(n e i)
and
is defined
if necessary)
finite
(Bu(A ~
s))
as usual.
(possibly
and
F~
(an ~
(Bu(B I ~
p
formula.
F = ~I ..... an' & = ~l'''''~m ' ~,B ~ FORM, r~
falsum and if
a I ..... a n are free object
p(a I ..... a n ) is an atomic of formulas
(with indeees, If
symbol
variable,
Denote by F,A
empty)
sequences
.... of formulas.
then for brevity we write
for Sl~
(~2----> (... ~
(B 2 ~ ( . . . - -
(Sn~S)
...)))))...)))
and ~i ~
(s2 ~
If F is empty, A empty, A
then
('''(Sn ~ F ~
(Bu(A ~
then F ----> (B u(A ~
are empty,
Denote by
then
s)...)))
s))
~)) is the formula
F ------>(Bu (A ~
~))
F ~
Bu(A ---->~), if (Bus)
and if F and
is the formula of the form (Bus).
rules
F ~ (Bu(~(~ r~ (Su(s ~
(str~)
respectively.
is the formula
R ~ the set of inference
(str~)
,
listed below:
y))) y))
r ~
(Suy) ...
r ~
(Su(s~
y))
(str~) r ~
(~u(A ~
F ~
(ur B)
r ~
(Bu(A~ (Bu(A~
r ~
(nl B)
(~----~ ( ~ y ) ) ) ) Yi))
(Su(~ i ~
y))
F ----->(Bu((~ins 2) ~ (ul s)
r ~
(~u(~ I ~ y ) )
i = 1,2
(YlUY2)))
r ~
i = 1,2 y)) (~u(~ 2 ---->~))
F ----> (Bu((elUS 2) ---->y)) (nr B)
F ~
(Bu(A ~ F ~
yl ))
(Bu(A ~
£ ~
(Bu(A ---->y2))
(YlnY2)))
3~3 r~
(~ iB)
.(.S..u...(A---> ~ ) )
r ~
(Su(A ~ r
( I r B)
r ~
r ~ r ~ ~, B, ¥,
r B c RB (sir 2)
Yi c FORM r
i.e.
and
are any sequence of formulas.
is rule of inference
such that the formula
let us take for example as
r ----> r ~ (~ R
r, A
rB
the rule
B
(str~)
is the following rule of inference:
(str2) Denote by
y))
(Bu(~(a) ~ 7 ) ) .... (Su(Vx~(x)~))
then
does not appear, then
(Su(A~
r ~ . . ( S u ( . A ~ ~(a))) ....... r ~ (Su(: ~ -Tx~(x)))
(V i B)
Let
y)
(Su(A=> i))
....................
( ~r s)
where
~
........... S u . ( r , ~ ((~n)~y)))
-¥)
the set of all rules
such that
r
rB c RB
i.e
~
°
R = {r : r B e RB}. By the basic
system
from
and the following
R u RB
(ax I)
(BS)
we mean the set of all rules of inference formulas
(i ----->~)
(ax 2) ri--~ 6
(CUt)
F ~ (@ i)
~(a) ~
and rules:
y
~, 6, y, ~(a) e FORM
y (Vr)
r ~
~(a)
r
Vx~(x)
----->
and a does not occur as a free object variable
in any formula in the conclusion of the rules The formula
e)
6---->y
~x~(x) where
(~
(~ i) and
(V r).
~ in the cut rule will be called the cut formula and denot-
ed cfl. Let us m e n t i o n
that the rules
(jr B) and
some rules that will be introduced
later.
(Jr) are added only to simplify
364
It is not difficult to verify that: i.i
(ax I)
and
(ax 2) can be restricted to
(I ~
a) and (a ~
a),
where a is an atomic formula. 1.2
0
For each rule of inference in
nistically valid,
BS, if all premises are intuitio-
so is the conclusion.
A formula ~ is said to be provable in the abbreviated as
BS
(henceforth
"BS ~ e") iff ~ is obtained from the axioms by means of
the rules of inference. Notice that the subsystem of the cut rule,
(7 i) and
BS consistin~ of (axl),
(ax2), the set
(Vr) is the formal system for intuitionistic
predicate calculus described by Schutte in [8] and [9]. this system by 1.3
IS.
For any formula iff
(Cut-elimination for
a cut-free derivation in PROOF.
Let us denote
Then in the standard way we obtain
BS ~ ~ 1.4
R,
BS).
IS ~ a. For any formula
0 ~, BS ~ ~ iff
e has
BS.
It follows immediately from the Haptsatz theorem for
IS [8] and
1.3.
[]
A derivation in which the cut rule does not appear is called normal. COROLLARY. i.
For any formula ~ , the following conditions are equivalent
~cLl
2.
e has a normal derivation in
3.
BS ~
4.
~ has a normal derivation in
We say that a formal system holds for
S
iff
S
IS BS.
0
is s e p a r a b l e or the separation theorem
the provable formulas of S have derivations using
only the rules of inference containing those logical connectives appearing in the formula. It is not difficult to show that: 1.5
The formal system
BS
without the cut rule is separable.
1.6
The separation theorem holds for the system
IS without the
cut rule. 1.7 In a normal derivation of e (in BS) occur.
0
0 only subformulas of 0
365
Let
g~FORM.
F~
~, where
Recall that we can consider F = ~I .... '~n
and
will be called the antecedents Y = YI ~
Y2
and
Y2
called antecedent
of ~ and y the succedent of ~.
of ~ and
Y2
all antecedents
of ~.
occurring
A pair of E and
, where
Craig's ~cL.
, then YI
If is also
letters w h i c h occur in
[~,~ *] is said to ~*
is obtained
from
in ~.
be the set of all free object variables,
and predicate
y~
the end part of $.
Let Z be the sequence of antecedents
Let
In the sequel ~i .... 'an
is not of the form y~ ~
be a p a r t i t i o n of ~ if ~ is a subsequence by omitting
$ as a formula of the form
~i,YcFORM.
~¢FORM.
sentential variables
In the same w a y we define
F is a sequence of formulas.
Interpolation
Lemma.
Let
L
be an intermediate
logic and let
[~,~*] be any p a r t i t i o n of ~.
Let
A formula y such that I.
c n
2.
~ ~
y~L
and
y ~*~L,
is called an interpolant an interpolant a formula
for
[~,~*].
Sometimes we say that [~,~*] has
if there exists a y such that I and 2 hold.
~ has an interpolant
We say that
if every partition of ~ has an interpolant.
It is well known that 1.8
(Craig's
interpolation
lemma for LI).
If
~cLl,
then ~ has an
interpolant. 1.9
D
(Craig's
tion in PROOF.
BS.
interpolation
lemma for BS).
Let ~ have a normal deriva-
Then ~ has an interpolant.
Let us assume
that
BS ~ ~.
By the corollary
~LI.
According
to the 1.8 ~ has an interpolant. §2.
LOGIC
LM.
The intermediate following
logic
LM
is obtained by adding to
(~ ~ u ~ ~ e), ~ e
is ~ ~
i •
It is well known that the logic
LM
with the directed set of "worlds", for
LI the
schema
(M) where
D
LM, then
is characterized by Kripke i.e.
if
is a Kripke
structures structure
366
A t,sVu(t
~u
&
s ~u).
The next two lemmas show some syntactical 2.1
The following
formulas
(i)
(((~->
~)
~(~'~
(2)
( ( ~
~ ~) u (
properties
are equivalent
B)) ~
to
of LM.
(M)
---> ( ~ u ~ ) ~ ,
~)),
~
(3) (4) ( ( ~ (5)
~)
((~
~
~
(~
(6) ( ( ~ ~ ~ S ) (7) ( ( ~
~
(8) ( ~ where
~ u~y))
n)
2.2
If
~
u (~ ~
~ S)
~
n
( ~ u ~
then
(A
The main difficulty
(g ~=> ~), ~ n represent
±), respectively. (~aom
(~ ---->m B), ~ ~ ( ~
~, BcFORM,
~¥))),
~
B)),
and as usual
A,B c {(~nB),
~ B) u ( ~
(~au~ ~)),
(n----->~)), (q ~
(B => ~ ~),
((~----->
~)),
( ~ m u ~ ~) +
~, B, y~FORM,
((~
(~u~)),
B),
(~ ~ ~
~ B),
(m ~ B ~
~ B), ~ ~ (B ~-> m ~), ~ ~ (-.~
in constructing
the formal
system
MS
for
LM
MS
and some new rules characteristic
to the positive
should contain all inferences for nonpositive
tautologies
of
the following set of rules:
(mlB) r ----> ( S u ( A ~ (m2 B )
Some of the m e n t i o n e d Prucnal.
(~u~)))
r ----> ( e v ( ~ - - - - > r ~
*
is
rules inference.
Thus the system
Let us consider
B)} where D
It is known that the positive part of LM is equivalent LI.
u~
~>B)~LM.
the choice of suitable
part of
~ ~),
~)) (Su((~
equivalences
r ~ ~ ~
(eu(6~
y))
~) ----> y ) )
were indicated to me by T.
LI LM.
367
r
(m3B)
(Bu(-Tc~¥))
~
r
~
r
(Bu((c~
(Bu(-76~y))
~ ~)
~¥))
(m4B)
2.3
If the premises
of the rule
(miB) , i = 1,2,3,4 belong to
LM, so
does the conclusion. PROOF.
Notice that
immediately ly valid. PROOF of (69 ( ~
~ 6) ~=> ( ~ u ~
(mlB)
for
(m2B).
y)eLM,
(~
and
(m3B).
It is sufficient
then
((~ ~ ~ 6
y) E L M and
6) ELM.
The rule
Thus the lemma follows
(m4B) is i n t u i t i o n i s t i c a l -
to observe that if ( ~ y )
) ~y
) ELM.
E L M and
To prove this let
(6~y)ELM.
Notice that (((-l~y)
n (-~
----->6) n ( ~ y ) )
~
((-i - ~
y) n (-~a----> y))) ELI.
Hence (((-~ -1~u-~ ~)---->y)-->y)--~((-~ - ~ y )
n (-~)
which together with our assumptions We call system ed from
MS
LM
y))ELI
((~8)
as propositional
r
where
from
(m i) is obtain-
rB(cf. § i).
iff
(Cut elimination
for
D MS).
Every theorem of
has a
inductions,
one
the othter on the rank of the derivation.
Recall that the grade of a formula of logical connectives
in ~.
~, g(~), is the number of occurrences
The left rank
RI
of the derivation
defined as the largest length of any thread of formulas with the left hand side premise of the rule formula occurs
LM
MS.
To prove the theorem we carry out two complete
on the grade,
Now
MS ~ ~.
The standard proof is omitted.
THEOREM 2.4.
D
logic.
For any formula ~,~ELM
normal proof in
gously.
----->y)ELM.
part of the basic
(miB) , (mi), i = 1,2,3,4
(miS) in the same way as the rule
THEOREM 2.3.
PROOF.
that
the system based on the propositional
BS, and the rules
we will treat
PROOF.
proves
n (~
in the succedents.
The rank
R
is
R I + R r.
~ is
in ~ that ends
(cut) and in which the cut
The right rank
Rr
is defined analo-
368
According to
1.4
the only new cases arise when one (or both) premise
(or premises) of the rule cut is (are) of the conclusion of the rule (mi6), or (m i) i = 1,2,3,4• I.
Assume that
R = 2, i.e. R I = R r = i, and the theorem is proved
for the grade n. Case of (m I ). Suppose that the following derivation ~.
'{(ml)
r~ ]'~
g(Bu(A~
(Bu(A~(~ ~-~ 6))) (~,u(S~ (--,o~u--,6)))
(m~um
S~¥
6))) = n+l.
(A~
(6 u ( A ~
Consider
(~u-~6))
(~u--,~)))
~
~¥
(ul)
y
(cut) F ~ Then ~ is reduced to
y
~i as follows [
(ml) ....(A---->( 9 ~ ~6))~ (A~ ( ~ ~ 6))
(A~ (~= ~ 6))~ (A~(~u-,6)) (A~ ( ~ u ~ 6))= y : 1
6~
(cutl) y
(A~
(~
~6))
~
(ul)
r----> (6u(A~
(~
-,6)))
(Bu(A~
(~
~ 6))) ~ (cut 2)
Notice that
g(A~
r a y ( m ~ u m 6)) < n and
g(6u(A~
(~
-~ 6))) = n.
Hence
by the induction hypothesis (CUtl) and (cut 2) are eliminable from ~i' that proves that the proof ~, given above, can be transformed into a proof without the cut rule. The case when the right premise of the cut rule in ~ is the result of the rule (str2) is trivial• Case of
(m I) .
Suppose that
g(cf2) = g(-~ ~ u~ 6) = n + 1
the following derivation
(ul)
It is reduced to
and consider
369
r~
(~=
(m 3)
-, ~)
(~
~ 6)
~
~f
(cut) F ~ By the induction hypothesis is eliminable
on the grade
g(~
~ 6 ) = n, the cut rule
from the above proof•
Case of (m2).
Recall
Notice that according not be n •
y
So
that
R = 2 and theorem holds
for the grade n.
to the assumption on the rank the cut formula can
cfl = ( - ~
~) = q and let g(cfl)
= n + i.
Let the
derivation ~ run as follows
(m2)r ~ (-l-~ r ~ (~6)
]I
n) r ~ (~n) ~ n)
((~ ~
~) ~
n) ~ (cut)
Let
~' be a cut free
interesting rule
(~
MS-proof of
((~ ~
6) ~ q ) ~
cases are when the end-formula of
i) or of the rule
(m3).
y
The only
7' is the result of the
In the latter n must be of the form
nI CASE I.
z' runs as follows
(~ ((~ ~ Then ~ is transofrmed
I)
~)~n)~x
into
(cut I) F~
n
n~y (cut 2)
F ~
y
(cut I) and (cut 2) are eliminable by the induction hypothesis CASE 2.
7' runs as follows:
( n= ~ ~ ) i
on the grade•
370
•
,
(m 3) ((~6) Using the fact that the proof
~
~ nl)~
SI ~ (~ ~ ~
y (~63
~ (~ ~
6)))
~' is reduced to Sl-proof
(cuti) -I ~ e
~-~ (~---->~ (-i e ~
6)) ~n I ~
r ~
( ~
(~e~
~ n I)
~nl)~
(~6~
( ~
Y)
%)
y) (cut 2)
-I n I ~ y
%) r~ (6~ ~n I) (cut 3)
(6~ ~nl)~ (r~y) P ~¥
(curl), (cut2) and (cut3) are eliminable by the induction hypothesis the grade.
on
Now let the right premise of the cut rule be the result of (m 2) and let cfl = ( ~
8) , i.e. the derivation ~ run as follows:
f
•
°
(r) P~
(~ e~
(m 2) (-~ ~ ----->6) ~
~)
y
Fay The premise of the rule (r) can not contain ( ~ is i.
So the last step in
~'
must be one of the following cases:
CASE i.
(=l) F ~
( ~
6) as the left rank
~)
371
CASE 2.
8=61u
and
62
r~
(~e~
6 1)
(ur)
CASE 3o
~ = 6
i
n ~
or
~'
r~
(-~ ~ 6 2 )
r~
(-~ ~ 8 )
•( u
r)
and
3
!
I
CASE
4.
r~
( ~
6 = (~i ~
62) ~ 3
r~
(-~ ~ i F~
CASE 5.
*
61)
6 = -~ 61
(-i e ~
r~
(~ e ~ 2
)
and
62 ~
) ((~i ~
63
( ~ I)
62)~63)
and
r ~
(61 ~
-~-~ ~)
r ~
( ~
-~ ~I )
In the case when instead of (ur) we apply a rule of the form (r ~) we proceed in an analogous way as in the case (1)-(7). Note that the subformula (suB) of the formula ( ~ u ~ ) ~ y is decomposable only by the rule (ul).
372
CASE 6.
~ = -~ (6 1 ~
r ~
rl)
and
(-~a~
(-~
r ~
-i-~ 61 )
--~n )
(m 4) r ~
CASE 7.
(-~
-~ (61 ~
n))
and
6 = ~ 61 u ~ 62
r ~
( ~
(61~
-i 62))
r ~
(~c~
(-n 6 1 u-~ 6 2 )
(m I)
We check only the Case 4. is as follows:
Then the derivation
A
or
((61 ---->62) ~
~vv
( ~
((6 1 ~
62) ~
Thus ~ can be transformed
B
7"
of the right premise
C
or
63 ) ---->y (m 2) 63))3 Y
as follows:
6 1 ~ 61
6 2 ~ 63
(31) 6 1 ~ ((61~ 62)~ 63)
((61~ 62)~ 63)~ Y (cut i)
r~
( ~
6 l)
(-~e~ 61)---->y (cut2) Fay
By induction on the grade all cuts are elinfinable. Case of (m3).
Consider only the case when
cfl = (~ ~
~ 6) =~ n
and
373
let q be the form
~ ql"
Let
~'
such that the end formula of (~
~ 6) ~
More exactly,
y and
~'
~ nI ~
((~6)=~nl)=~
y
y
let the derivation
r --~ ( ~
be the proof of
is obtained by application of (m 3) to
~ n I)
r ~
~ ~) ~
~ n I)
~ runs as follows:
(~6~
~ q I)
-~ ( ~
~ 6)~y
-~ n I ~ y
(m 3) r ~
((~
(m 3 ) F ~ As before, we use that
SI ~ -, -~ e ~
Thus ~ can be transformed
y (-~ ~ 6 ~
-~ ( e ~
~))
into:
Sl - proof ~-,~(~-,6~
~ (c~
-,6)
-~ql~
Y
(cutI) (-,-1 ~
~
r ~f~
~
~ n I)
¥)
( ~ n
I) ~
(7~Y)
(cut) r
r ~
(~6~
~
(-,-,6~
n I)
(76~
"~)
F ~(-~ n l ~ ¥
~ n I) ~
(r ~
y) (CUt3)
(r ~
¥)
F~y Note that all cuts are eliminable by the induction on the grade. We omit the case w h e n the right premise of the cut rule is of the form ( ~
~6)
analogous Case of follows:
=> y
and
case for (m4) o
cfl = (~ ---->~ 5)
as the proof is similar to the
(m2).
For simplicity
assume that
A = $
and
let
~ run as
374
(Bu~ ma)
r~
r ~
(Bum 6)
(ul)
(m4B)
(8u-~ ( ~
(~Bu-~ (~------>~))
r ~
6))~ y
(Cut) F ~
y
It is reduced to 7' :
Sl-proof (cutI)
(u1) r~
OB~--,cO
OBu--,-~) ~
(-~ ~ y ) (cut2)
r ~
(~
B ~
~y)
r ~
Y (s~y)
(ul)
r ~
(IBu-,~) => (r~y)
OBu-~(B)
(cut3)
r ~
(r~y)
r=>y Arguing as usual we can eliminate Analogously
for the remaining rules:
It is not difficult
to conclude
the induction hypothesis norml
all cuts from the above derivation.
that if
RI > i
and
Rr> I
then using
on the rank we transform any derivation
into a
proof•
COROLLARY.
For any formula a the following conditions
i)
~LM,
2)
NS~a,
3)
there exists a cut-free proof for ~.
Problem•
Is it possible
to eliminate
from
are equivalent:
D MS
the rules
(m 3) and
(m4B)?. THEOREM 2.5•
(Interpolation
lemma holds for PROOF.
Let
Craig's interpolation
LM.
~eLM.
for any partition
lemma for LM).
To prove the theorem it is sufficient
to show that
[#,E*] of ~ there exists a formula y such that
375
I)
(~
2)
c n ,
where
y) c LM
and
for any formula
(y ~ * )
~,
E LM
is the set of all sentential
variables
ocurring in ~. The m e t h o d w h i c h we use is due to Maehara the fact that an interpolant constructively
of the formula of the form
obtained from a proof of
By the corollary
MS ~ ~.
and its significance (a ~
lies in
(a ~
~) can be
n).
Now we will construct an interpolant
duction on the number of inference rules,
~ by in-
in a normal derivation
At each stage there are several cases to consider.
According
only new cases arise when ~ is a conclusion of the rule
(miB)
~ of ~.
to 1.9 the or
(mi),
i = 1,2,3,4. We deal only w i t h
(mi) ,
such that
~,
derivation
~ of ~.
Let
~
= k
i = 1,2,3,4
and suppose
the theorem holds, where now
= k + I and the last inference be
Let us consider the following partition
(ml) , i.e.
~ = r',
By the induction assumption we know that for an interpolant ~* 1
=
y.
------>( ~
F*
Hence
-~ 6)
for the partition
(r* ~
Moreover
c .
(ml) we infer that
MS ~ r' ~ It is obvious
and
¥
that
is an interpolant
MS ~ P* ----> ( ~ u - ~
c Using
the rules
MS br'
~
(ur),
(7-~
It is obvious
,
c n <E2>
(ul) we o b t a i n
(¥1uY2)), that
b r'~YlUY2)),
MS
MS ~- (YlUY2)--->
(r*---->n).
:
c n Now u s i n g
the rule
(m2) we
infer
that
(Yl u y2 )
is an i n t e r p o l a n t
for
the s e c o n d p a r t i t i o n . N o w let us assume E = F ~
((~
w e assume
that the last
~ 6) ~
q).
that n is not of the form
have only one p o s s i b i l i t y ,
namely
On a c c o u n t
of the i n d u c t i o n
[r', -~ ~;
F* ~
n]
Y2' r e s p e c t i v e l y .
IF',
(~
rule
(m3).
~ ~);
Let the last
inference
be
E2 = F ~
m ~.
~ = F'
If ~ # F',
(~ ~
we obtain n]
6) and
[~,E*] i.
E* = F* ~
n.
that for the p a r t i t i o n s
our p r o o f
E = Y ~
such that
Consider
By the i n d u c t i o n
and
then we
there are the i n t e r p o l a n t s
completes
(m4) , i.e.
~ and E = F* ~
n 2.
~ = r',
F* ~
(m3) , i.e. case w h e n
(Yl u y2 ) is an i n t e r p o l a n t
which
is only one p a r t i t i o n
= F',~ ~ and
Thus
F* ~ n ]
is
qI ~
hypothesis
[F', -~ ~;
and
There
inference
As b e f o r e w e omit
this
for case of the
7 (~ ~ # F', case.
assumption
Y1
for
there
~). namely Let E 1 = F ~ are
Y1
and
-i ~ Y2
such that VS~ and
r' ~
( ~
71),
~ < r ] ~ > Thus by
(ur),
(ul),
m }-T 1 ~
n, I),
(r*~l) ,
~
~r'
c {~V x ~ ( x ) ~ It is obvious required Problem.
MHS ~ y ~
that
a
Thus and ~ ~)
(str 3) and
(r' ~
~(a)).
does not occur free in y.
Apply-
we infer that: MHS ~
~y~
c n which proves
(F* ~ i ) . that
~y
is the
interpolant. To modify the system
0 }ZIS
in such a way that the rules of
384
inference specific for
MHS
are only
LMH valid.
REFERENCES [ I]
Gabbay, D.
Applications of trees to intermediate logics, J.Symbolic Logic 37(1972), 135-138.
E 2]
Semantic proof of Craig's theorem for intuitionistic logic and its extensions, I and II. Proc. 1969 Logic Colloquium, North-Holland Publ. Co., Amsterdam(1971), 391=410.
E 3]
Craig interpolation theorem for intuitionistic logic and extensions, Part III. J.Symbolic Logic 42(1977), 269-271.
[ 4]
Ldpez-Escobar, E.G.K.
[ 5]
On the interpolation theorem for the logic of constant domains.
J.Symbolic Logic 46(1981),87-88.
A second paper on the interpolation theorem for the logic of constant domains. J.Symbolic Logic 48 (1983), 595-599.
[ 6]
A natural deduction system for some intermediate logics.
[ 7]
Maksimowa,L.
J.of Non-Classical Logic, 1(1982),
Craig's interpolation theorem and amalgamable varieties, Algebra i Logica 16(1977), 643-681.
[ 8]
Sch~tte,K.
Der Interpolationssatz der intuitionistischen Pr~dikatenlogik, Mathematische Annalen 148(1962), 192-200.
[ 9]
[I0]
Proof theory, Springer-Verlag, New York (1977). Umezawa, T.
Berlin-Heidelberg-
On logics intermediate between intuitionistic and classical predicate logic, J.Symbolic Logic 24(1959) 141-153.
TYPES IN CLASS SET THEORIES Rolando Chuaqui
William No Reinhardt Depto of Mathematics, University Boulder,
Universidad
Campus Box 426
Catolica de Chile
Casilla II4-D
of Colorado
Santiago,
CO 80309
Chile
This paper deals with the problem of defining order types for well ordered proper classes. equivalence
This is a special case of the problem of defining
types for an equivalence
relation.
ous problem for set theory with regularity method of introducing
equivalence
We recall that the analog-
is solved completely by Scott's
types as the equivalence
class restrict
ed to its members of minimal rank. It is curious that the situation changes so markedly when we pass to class set theory. somewhat peripheral
to most set theorists,
remarks about our interest
in it.
As this problem may seem
we would like to make some
The problem arises naturally when one
tries to develop the theories of constructible theory
(such as Kelley-Morse-Tarski);
became
involved in the problem.
Reinhardt's
attention
this is how Chuaqui originally
Professor Chuaqui drew the problem to
in 1982, when he was preoccupied with problems
volving the notion of intuitive provability. terested
The problem of developing notations
is analogous
in-
Because of this he was in-
in some aspects of proof theory and in particular
notations° ordinals
classes within a class set
in ordinal
for larger and larger
to the problem of getting types for ordinals.
Since
key ideas often show themselves
more clearly and simply in classical
ings than in constructive
one might hope these problems would illu-
minate ordinal notations; results.
ones,
we were thus thinking largely of positive
We have nothing to say f r o m t h e s e
ordinal notations, the problems
investigations
are related.
The positive results we have obtained are In particular,
the method of defining order
types appears to lead to a more conceptual
development
classes than any we know of in the literature°
Also,
of constructible in conjunction with
ideas of Manuel Corrada it leads to a nice characterization part of the impredicative
theory of classes
of the set
(ioe. Kelley Morse-Tarksi
set
The negative results mentioned here were pointed out to us by
several people at the symposium, Magidor,
concerning
but it does still seem to (at least one of) us that
modest but of interest.
theory).
sett-
and W. Mareko
especially
Steve Simpson,
Menachem
386
Suppose that we are given an e q u i v a l e n c e r e l a t i o n X ~ Y We permit ~ to involve p a r a m e t e r s Z.
on classes.
The q u e s t i o n then is w h e t h e r there
is a term r (which may depend on Z and on other parameters U as well)
so
that I)
VZ~u(T(X)
We can ask for more,
: ~(Y) ÷÷ X ~ Y).
that T be a selector:
w h i c h says that if ~ is an equivalence, or Type
~(X)
~ X.
We call the a x i o m
then i) holds,
"~ is an
~-type"
(~,~) or even "T" for short.
Surprisingly,
it appears to be an open p r o b l e m w h e t h e r there is such a
for the e q u i v a l e n c e w h i c h holds b e t w e e n two o r d e r e d pairs w h e n they c o r r e s p o n d to the same u n o r d e r e d pair.
That is, it appears to be un-
k n o w n w h e t h e r u n o r d e r e d pairs can be defined in class set theory. On the other hand, it can be shown using methods of L~vy d e v e l o p e d for i n v e s t i g a t i n g questions of definability,
that there are models of set
theory in w h i c h no such T can be given for the r e l a t i o n of order isom o r p h i s m for well orderings This happens
(i.e. well order types cannot be defined).
in the model o b t a i n e d by collapsing all cardinals b e l o w
the second i n a c c e s s i b l e to the first inaccessible° sidered i n d e p e n d e n t l y by L~vy and Rowbottom) o
(This model was con-
We are i n d e b t e d to Rich
Laver for the following argument. Let M be a m o d e l of ZF w i t h two inaccessible cardinals
Km
U n-
it is well known that if the series z
n
converges,
~
(un)
then,
of zero-measure there is a
lim sup (Un) is of zero-measure; coded in the ground model.
G~ set of zero-measure
sense that it has the same code). tion of the term T,then B belongs exists a condition p
iF T~
p
n n~
of
•
B
thus, B is a G~
set
Now, in any generic extension
w h z c h is built like
B
(in the
We claim that if ~ is the interpretato
B.
If this is not the case,
and an integer
m
there
such that
Uno
This means (3)
Vn~m
b I~ T does not extend s
Now, we pick an open neighborhood fixed;
V
appears
for some q
q~p
n
in the basis
infinitely many times in the sequence
can pick an integer get,
V of p
n ~ m for w h i c h
V n = V.
that we have (Vn) so that we
From property
(2) above, we
:
I~ T extends
sn
but this contradicts
(3).
appears
extension,
in a generic
So we have shown that any real B, w h i c h belongs
to some G~
set of zero measure
coded in the ground model M, hence is not r a n d o m over M. 3. LOCALLY COHERENT TOPOLOGICAL ORDERED SETS. 3.1
We now study the closure properties
of the class of coherent
topological theorem
ordered
1.3.
PROPOSITION. satisfies PROOF. basis
sets.
This wili be useful
Any separable,
locally
If (Vk) is a sequence •
, then,
Vk, hence,
3.2
of coherent
pairs
(k,f) with
where
fFk denotes
(£,g)
of
~
< (k,f)
is a coherent have proved LEMMA.
used to add a new element
f c~;
if
the ordering
£ exceeds
basis;
of
m~ domi-
UD consists
of
on UD is given by:
k, f~k = g~k and ~n f(n) ~ g ( n ) of f to integers
induced by the usual
:
topology
furthermore,
T is a mapping
for some j.
and (p,k,o) whenever
~ UD is defined by
(q,£,~)~(p,k,T)
ordered
to the integers
i T = o.
400
q I~ Vn
o(n) ~ ~(n)
q l~ Vi