Logic: Colloquium Proceedings, 1977

LOGIC COLLOQUIUM '77 Proceedings of the colloquium held in Wrochw, August 1977

Edited by

ANGUS MACINTYRE Yale University, U.S.A. LESZEK PACHOLSKI Wo&w University,Poland JEFF PARIS Manchester Universily, G.B.

1978

NORTH-HOLLAm PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD

0 NORTH-HOLLAND PUBLISHING COMPANY - 1978 All rkhts reserved. No part of thispublication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, elcerronic, mechanical, photocopying, record& or otherwise, without the prior permission of the copyright owner.

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Library of Congress Cataloging in Publication Data

Logic Colloquium, Wro&w, Logic colloquium '77.

Poland, 1977.

(Studies in logic and the foundations of methemetics ; v. 96) Includes bibliographical references. 1. Logic, Symbolic and methemtical--Congressea. I. Macintyre, Angus. 11. Pacholski, Leszek. 111. Paris. Jeff. IV. Series.

Oxford

I 14 MEFlOR I AF1 ANDRZEJ MOSTOWSKI

1913 - 1975

PREFACE

Logic Colloquium ' 7 7 was held in Wroc+aw, Poland from 1st August to 12th August 1977 and was dedicated to the memory of Andrzej Mostowski. The conference was organized and financed by the Mathematical Institute of the Polish Academy of Science in collaboration with the Technical University in Wroc+aw and Wroc*aw University. Additional financial support was received from the Association for Symbolic Logic and from the Logic, Methodology and Philosophy of Science division of the International Union of History and Philosophy of Science. On behalf of the organizing committee we wish to thank the above mentioned insti-

tutions and organisations as well as all the people who by their help made the conference a success. The organizing committee for the Colloquium consisted of Kazimierz Kuratowski (Warsaw) and Alfred Tarski (Berkeley) - honorary co-chairmen, Jens Erik Fenstad (Oslo), Angus Macintyre (Yale), Leszek Pacholski (Wroc*aw)

-

executive chairman

and Jeff Paris (Manchester). The conference was recognized as the 1977 European Summer Meeting of the Association for Symbolic Logic and abstracts of the contributed papers will be included in the report of the meeting in the Journal of Symbolic Logic. The colloquium was mainly devoted to invited lectures on model theory and set theory. There were also a 4 hours course on Silver's machines by J.H. Silver (UC Berkeley) and 5 invited talks on interconnections between probability theory and logic. The invited lectures were given by P. Aczel (Manchester), R.M. Anderson (Yale), J. Baumgartner (Dartmouth College), A.R. Blass (Un. of Michigan), L. Bukovsky (Kosice), G. Cherlin (Rutgers Un.). K. Devlin (Lancaster), J.E. Doner (Stanford), U. Felgner (TGbingen), J.E. Fenstad (Oslo), H. Gaifman (Paris),

F. Galvin (Un. of Kansas), W. GuziCki (Warsaw), P. Hajek (Prague). A. Hajnal (Budapest), L.A. Harrington (UC Berkeley), C.W. Henson (Un. of Illinois), T.J. Jeck (Pennsylvania State Un.) , A. Kanamori (UC Berkeley), A. Kcchris (Caltech), D.W. Kueker (Un. of Maryland). P. Loeb (Un. of Illinois), M. Magidor (Ben Gurion Un.), J.A. Makowsky (West Berlin), R. Mansfield (Pennsylvania State h.),K. McAloon (Paris), M.D. Morley (Cornell Un.), J. Paris (Manchester), S. Roguski (Wroctaw), M. Rubin (Un. of Colorado), G . Sageev (Ohio State Un.),

J.H. Schmerl (Un. of Conneticut), D. Seese (Berlin).

vii

J.A. Sgro (Yale),

viii

PREFACE

S. Shelah (Hebrew Un.), H. Sinmons (Aberdeen), C.G. Smorynski (Westmont), R. Solovay (UC Berkeley), J. Stavi (Bar Ilan Un.), J. Stern (Paris), G. Takeuti (Un. of Illinois), R.L. Vaught (UC Berkeley), P. Vopenka (Prague), A.J. Wilkie (Open Un.),

G. Wilmers (Manchester) and M. Ziegler (West Berlin).

h e present volume constitutes the proceedings of the colloquium and contains

most of the invited talks.

A part of the results of the paper by J.E. Doner,

A. Mostowski and A. Tarski opening the volume were announced by A. Mostowski and A. Tarski in 1949.

The decision to'publish those results was made by the authors

during Professor Mostowski's last visit to Berkeley in summer 1975, shortly before his death.

The paper was presented at the colloquium by J.E. Doner.

Angus Macintyre Leszek Pacholski Jeff Paris

L q i c Colloquiwn

'77

A. Macintyrz, L . Pachobki, J. Paris Rorth-Hollrmd Publishing Canpanu, 1978

OF WELLQRDERING --A METAPnAMEMATICAL STUDY--

THE ELEMENTARY THEORY by

. I .

John E. Doner. Andrzel Mostowski! and Alfred Tarski University of California, University of California, * Berkeley. Santa Barbara. Foreword The elementary (i.e., first-order) theory of well-ordering is, loosely speaking, that part of the general theory of well-ordering which can be formalized in an appropriate language l of predicate logic. To characterize this theory more precisely, we introduce the notion of a well-ordered structure, i.e., a structure 91 formed by an arbitrary nonempty set of elements and an arbitrary binary

relation which well-orders this set, and we stipulate that our theory consists of just those sentences which are formulated in f

and are true of every such struc-

ture 91.

The main purpose of the present paper is to establish some of the principal syntactical and semantical (model-theoretic) properties of this theory. The main results of this paper were obtained by Mostowski and Tarski in the years 1938-1941. They developed the method of elimination of quantifiers in its application to the theory of well-ordering and on that basis established the decidability of the theory and described the ordinals definable by first-order formulas. These results were first announced in Mostowski-Tarski [49]. Subsequently further results were obtained on the same basis, such as the description and classification of all models of the theory. Some progress toward publication was made in 1964 when Doner, then Tarski's research assistant, worked out most of the technical details and proofs. In 1975 Mostowski and Tarski made definite plans to procede with publication. But these plans were disrupted by Mostowski's untimely death in the summer of 1975.

Doner then joined

as a third co-author, and undertook the final writing of this paper in collaboration with Tarski. Doner made new contributions to the mathematical content, specifically results on prime models of the theory of well-ordering, and elementary extensions and substructures of models of that theory. This paper incorporates all the results obtained by the authors up to the present time. Since the appearance of Mostowski-Tarski [49], several methods (usually semantical) other than quantifier-elimination have been developed which can be used for the same purposes and which often prove more efficient. New decidability

*

This paper was prepared for publication at times when Tarski was engaged in research projects sponsored by the U.S.A. National Science Foundation, grants numbered NSF GP-1395, MPS74-23878, MCS74-23878, and MCS77-22913. Doner a l s o was supported under grant number NSF GP-1395.

1

2

J.E.

DONER, A. MOSTOWSKI and A. TARSKI

results have been found, e.g., Rabin [69], of which the decidability of the theory of well-ordering is a simple corollary. Nevertheless, we have decided to present in this paper the solution of the decision problem for the theory of well-ordering in its original form. It seems to us that the elimination of quantifiers, whenever it is applicable to a theory, provides us with direct and clear insight into both the syntactical structure and the semantical content of that theory--indeed, a more direct and clearer insight than the modern more powerful methods to which we referred above. In consequence, the procedure of eliminating quantifiers in a theory may prove valuable as a heuristic source and deductive base for a many-sided metamathematical study of this theory leading to substantial results not involving decidability. while recent research into the complexity of algorithms has deprived the pure decidability results of some of their luster, the interest in other results obtained through the elimination of quantifiers remains unaffected. We believe that the present stuW provides ample evidence for our remarks in this paragraph. Our paper is divided into three sections, $1 - $3.

In $1,we describe the

set-theoretical and metamathematical notation which w i l l be used throughout the paper. Our account of set-theoretical notation will be sketchy, while that of metamathematical notation will be more detailed. At the end of $1 we precisely describe a theory Ir! developed in the formalism 1: of first-order predicate logic and based upon an explicitly formulated infinite recursive axiom set. At the first sight, the theory W may seem to be much more general, and hence much weaker, than the elementary theory of well-ordering. However, it will turn out, rather unexpectedly, that actually the two theories coincide; this will be the content of the Semantic Completeness Theorem for W established at the beginning of §3. $2 is almost entirely devoted to the procedure of eliminating quantifiers.

We start by singling out a special set of formulas in 5 , referred to as basic formulas, and the main result of the section is the Reduction Theorem, a statement to the effect that every formula in C.

is equivalent on the basis of W

to a

formula with the same free variables which is a quantifier-free combination of basic formulas. In consequence, every sentence in 5

is equivalent to a

quantifier-free combination of basic sentences. $3 contains the principal results of this paper concerning the theory Of well-ordering. A l l of these results are obtained with the essential help of the Reduction Theorem from $2. The possibility of applying the Reduction Theorem to these ends arises from the Semantic Completeness Theorem, which is the initial result of $ 3 . The next important results in $3 are the decidability theorems, to the effect that W

is decidable, i.e., recursive, and actually primitive

THE ELEMENTARY THEORY OF WELL-ORDERING --A

METAMATHEMATICAL. STUDY--

3

Fecursive. The main subjects discussed in subsequent portions of $3 are the following: extensions, especially complete extensions, of the theory w ; the variety of all models of w , including prime models of complete extensions of w ; the elementary definability of individual elements, sets of, and relations between elements in models of W . t $1

PRELIMINARIES

The terminology and symbolism adopted in this paper follow, with a number of

*

deviations, the notation of Henkin-Monk-Tarski 1711.

They do not differ essen-

tially from those which are used in most mathematical and metamathematical papers. We suggest that the reader consult Henkin-Monk-Tarski

[711 should he encounter a term which is used here without explanation, and he has any doubt regarding its intended meaning. As we recall from the Foreword, the subject of this paper is a metamathema-

tical study of a theory developed in a language

f

dl1 be carried on in an appropriate metalanguage f

of predicate logic. This

*

,

which we do not present

here as a formalized language (although such a presentation would offer no essential difficulties). f * basis.

is assumed to be provided with a set-theoretical

It is not necessary to describe this basis in all details; it is

sufficient to assume that a l l the familiar set-theoretical notions are available in

*

, and that all sentences which hold in, say, the set theoretical system of

Morse also hold in f

*

.

(In fact, by analyzing the constructions and arguments

in this paper, one readily sees that for most purposes this assumption can be considerably weakened. A presentation of Morse's system can be found in Morse

C651 or, in a more conventional form, in Kelley C551.) The fundamental notions of our set theoretical basis are the notions of elementhood, symbolically

E,

and that of a class. An example of a class is the

empty class, 0, which is the only class having no elements. A class is either a set or a proper class depending on whether or not it is a member of another class. For instance, 0 is a set; the singleton { a ) of a set a , the (unordered) pair { a , b } , and the ordered pair ( a , b ) of two sets a & , are also sets, etc. {a : C ( a ) ) is the class of all sets satisfying the condition

C(a). More generally, {f(a) : C ( a ) ] is the class of all those sets f ( a ) which are the values of the function satisfying C ( a ) . formula

f

correlated with the argument values a

This more general notation reduces to the original one by the

+For reasons beyond the authors' control, Tarski had only limited opportunities to review the final draft of the manuscript, and Doner accepts the responsibility for all details of the text. *Most of the deviations concern the metamathematical symbolism, and are caused primarily by typographical limitations resulting from the manner in which the manuscript of this paper was prepared and reproduced.

b

J.E. DONER, A. MOSTOWSKI and A. TARSKI

We use familiar symbols for various fundamental relations between, and operations on, classes, e.g., E and and

X

C

for inclusion and proper inclusion, u,

-,

n,

for formations of unions, intersections, differences, and Cartesian

products of two classes. The operations izations of n

and u,

u

and

n

on classes are general-

respectively. We set

UA

= Ic : c c B

nA =

{c : c c B

f o r some f o r euery

B

A),

E

B

E

A).

An arbitrary class of ordered pairs is referred to as a binary relation, or simply as a relation. R is called a relation on a class A if R E A x A. We single out various classes of binary relations, e.g., simple orderings ( a l s o referred to as linear orderings), well-orderings, functional relations, or functions, bi-functional relations of bi-unique functions. Given a class A set B we denote by BA the class of all functions from B all functions with domain B

f

E

BA. we may write f

: B

and range included in A.

+A,

into A,

and a i.e., of

Instead of writing

to be read: f maps B

into A.

In this paper we shall deal extensively with ordinals. As is usual in contemporary set theory, we assume the ordinals to be constructed in such a way that every ordinal a coincides with the set of all ordinals which are smaller than a ,

in the sense that they are proper subsets of a. As a consequence,

the relations the


p -the possibility of a disjunct Mxl(O,-) or Nx9(-) with 9 < 11 is excluded by our hypothesis that (I), (ii), (iii) fail. But J a b i Nxs(-) when 9 > p, by 5(iii), 25.

MXl(O,-),

-

-

-

We actually proved more than was stated in Theorem 29; the additional information is captured in COROLLARY 30. If S is a #-consistent sentence then 2 I= S for some

a
L + 1. Simi l a r l y , all formulas Mx,x(y,x), 'X < x, a r e s a t i s f i e d by (pu, p,) when a > L + 1 and no such formulas when u C L - 1. This s i t u a t i o n w i l l be retained i f we choose p/ such t h a t p, > p; > pL-l + ax+'. So we simply l e t E be t h e

'x

s e t of formulas of forms Nx,x(x), by

p,,

invoke (1) t o get

p' = p,-l

and then l e t

E,

< ax+'

-

-I

Nx,x(~) with x' 2

such t h a t

+ ox+' + E;.

p,

<x

which a r e s a t i s f i e d

+&C ~pL-l+ax+l+E,],

One of t h e notions involved i n our next theorem is t h a t of an ordinal definable without reference t o l a r g e r ordinals. define, f o r all formulas

where a

C

and variables

To make t h i s notion precise, we

u,

is t h e f i r s t variable different from X

which does not occur in C,

is a formula i n which u does not occur bound, and all t h e bound variables a r e r e s t r i c t e d t o be < u. An ordinal is intrinsicaZ1y f r e e or bound.

Thus,

'

defiinable if it is definable by a formula ,)'(G THEORIB 38.

with

F v ( C ( " ) ) = {u).

fie foZZ0wing conditions are equivalent:

( i ) a is definable;

tii)

a a. L e t p = a + ow. Inasmuch as p > 'a + y f o r all p, y < aw, Lemma 37 says t h a t F i s s a t i s f i e d i n 3 by some ordinal x. Thus, F is a finite set of axioms for Tk(Za). Finally, suppose Th(9,) i s finitely axiomatizable. Let S be the conSince also junction of its axioms. According to 30, 3 k S for some p < B Zp k HB(-), we have Lla C ff (-), and hence a = fl < a" by 36(i).

( m )and > x, F I-

(II), (III),

"".

B

Our next theorem concerns arbitrary models of W, including those which are not well-ordered. For the purpose of this theorem, we shall relax the normal interpretation of small Greek letters as ordinals and instead let these letters range over arbitrary order types. However, x

and u w i l l continue to repre-

u C S means I C S whenever 71 = u. If 1 is a model of W, we use s to denote its order relation and OI for its first element. If a , b t UnI and a < b , then is the restriction of I to the set sent finite ordinals only.

4

4.

OD

a S O < b ) , and [ a , b ) represents the order type of tia is the restriction of 01 to { C : C C UnI and a r c : , and [ a , - ) its order type. { c :c e Un fi and

We also need some further notions from model theory. hro structures 8 , 4) are etenWnt&Zy equivatent, symbolically I E I, if for each sentence S,

I t S iff $9 t S, i.e., if Thti = T h a A structure 01 is an etementmnj substructure of a structure I (or: 8) is an eZementuty extension of I ) i f I

J.E. DONER. A. HOSTOWSKI and A. TABSKI

46

is a substructure of 8

1 is the restriction of 8 to a subset of its and f h w(UnI), 1 k 4fl if and only if universe) and for every formula F 8

(i.e.,

k qfl. When I is an elementary substructure of 8, then an element

b c Urn is definable in 8 if and only if b c U a and b is definable in elementary substructures contain all definable elements. I is etemsntarizy embeddubZe in 8 , symbolically I < 8 , substructure of 8 .

These relations E

a:

if 1 is isomorphic to an elementary and 4 on structures, which are invar-

iant under isomorphism. give rise in a natural Way to corresponding relations on order types, for which we shall use the same terminology and symbolism, e.g.,

a < p means that every structure of type p has an elementary substructure of type a, and we say then that p is an elementary extension of a. If I 4 8 whenever 8) 3 1, then 91 is a prime modet of Thyl; sufficient for this is that every element of 8 is definable, cf. Vaught [611, Theorem 3.4. DEFINITION 39.

= ww*

where t:

5

+

< 2, X

An order type is canonical if i t

... +

0 "

- Xu +

= w + a* o r

*

+

ha8

the form

... + w2

X2 + w

- 1 + Xo

X < w f o r each x < o, and if t: = 0 ,

then

Xu = 0 f o r a t 1 but finiteZy many x. If TB is canonical, then

THEOREM 40.

I is a p r i m e model of T h 1 .

h.oof. Because of Vaught's theorem, it is only necessary to show that every element of a structure of canonical type is definable in that structure. We begin with the following remark, which applies to any simply ordered structure I : (1)

For if

If a , b c UnI, [ a , b ) < b is definable in 1.

[a,b) = a

formula G i x ) ,

< ow,

then a

Fv(CiX))= {x).

From this, we easily obtain

ow,

and a

is definable in 1, then

is intrinsically definable, i.e., It follows that

8:

defined by a

k G P ) I b 1 . and also

THE ELEMENTARY THEORY OF WELL-ORDERING --A M E T A M A m T I W STUDY-in

Now suppose a i s defined by F

For any formula E,

Fu(E) = { x ) ,

I,

47

and l e t

and ' b

we have

UnI,

c

Now l e t =

X

'F

A

iff

B C $b'l

(F')

(G,)

b'

*

= b,

so t h a t

b is defined i n I by B . This proves (1). Now l e t I be a s t r u c t u r e with canonical TI, s a y

D E UnI

To complete t h e proof, we need only exhibit a s e t such t h a t f o r every

b c h i , t h e r e is an a

D

of definable elements

such t h a t

[ a , b ) ow

COROLLARY 44.

-

-

are congruent modulo ow if either a = $

and there is a

6 such that a = ww

a I p if

6 +

B or $

< ow

or else

6 + a.

ow

only i f a, fi are congruent mohto ow.

Proof. Congruence modulo w" is necessary and sufficient that a , $ be elementary extensions of exactly the same canonical type.

COROLLARY 45. The complete extensions of W which have welt-ordered mods18 are exuctty those of form Th(Sa) with a < ww * 2.

A

Proof. The types of prime models of such complete extensions must have all and A # 0 for only finitely many x. Such a type is some o w ' 5 + y,

<w

with 5

O and t o deduce that Vn-l = Vn. Fix I"-'UEV~-~ T(?-'~-T%)=o,

and solve T"u=ln"v

so writing

~ - 1 U - f i w E w

using Vn = Vn+,.

c vn,

?hen

we may solve w = A t .

%en:

Evn. T"'-'U=W+fi=?n(W'+V)

lhus Vn-l

Lemna 2.6. perfect.

Proof.

C Vn,

as desired.

If K i s a super-stable field of characteristic p>O then K is

Consider the mnomorphism x Corollary 2.4.

+

xp of K,

and argue as above using

Lemna 2.7. Let Do be a stable division ring infinite-dimensional over i t s center. ?hen Do contains a definable division subring D such that: 1. D is infinite dimensional over its center 2. 2. If XED-2 then the centralizer C(x) of x i n D is f i n i t e dimensional over its center.

Proof.

If Do is assuned to be super-stable (which is actually enough for our purposes) then the lennna follows m e d i a t e l y from Corollary 2.4. (Build a sequence of definable division subrings Dn of D inductively satisfying 2 . 7 , l ; i f 2.7.2 f a i l s l e t Dn+l be a division subring of Dn exemplifying the failure of 2.7.2.) 'he same argument applies to stable division rings i n view of the Baldwin-Saxl Lena [l], which gives a descending chain condition for centralizers in stable groups.

10 1

SUPER STABLE D I V I S I O N RINGS

83.

The equation [ a,x]=l

Convention. In this and a l l subsequent sections D is a super-stable division ring satisfying conditions 1, 2 of Lemma 2.7. lhe center of D is called Z. I t w i l l be convenient t o have the following fact concerning endomorphisms of groups, which we write additively with our eye on subsequent notation. The proof is t r i v i a l . Remark 3.1. If h:G + G is a group endomorphism with kernel H and h*:G/H + G/H is the induced endanorphism then: 1. h* is surjective i f f G = (Imh) + H 2 . h* is injective i f f (Imh)nH=(O). Tneorem 3.2. If XED-Z aED to the equation:

has infinite centralizer C(x)

then there is a solution

[ a,xI=l.

(Here [ a,x]=ax-xa

is the additive corntator.)

Proof. Define h:D + D by h(a)=ax-xa. We then have a derivation of C: 1. h(u+v)=h(u)+h(v) (Additivity) (Leibniz rule) 2 . h(uv)=h(u)v+uh(v) W e consider the induced endomorphism h*:D/C(x) + D/C(x)

.

We w i l l show shortly that h* is not injective. that for some dsD we have:

By Remark 3.1 this means

[ d,x]=h(d)~C(x), h(d)#O.

Let c=h(d). Then the Leibniz rule ( 2 ) above yields: [ c-'d,x]=c-'[

d,x]=l.

Thus taking a=c-ld we have [ a,x]=l as desired. Suppose therefore tcuard a contradiction that h* is injective and hence also surjective by Lemna 2.5 (we are viewing C(x), D, and D/C(x) as right vector spaces over C(x), and then h is obviously linear by the Leibniz rule, so that the induced map h* is also linear). By Remark 3.1 we then have (8)

D =

(In1

h) 8 C(x)

.

Let V=Im h, and l e t s:D -c C(x) be the projection map furnished by ( 8 ) . Now i f V is of f i n i t e dimension over C(x) then D is also. By condition 2 of Lemna 2.7 C(x) is f i n i t e dimensional over its center ZC(x), a field, and

102

G. CHERLIN

thus D is f i n i t e dimensional over a subfield, and is therefore also f i n i t e dimensional over its center [2,pp.95-96]. ?his contradicts condition 1 of Lennna 2.7, so we conclude t h at V is i n f i n i t e dimensional over C(x). Now f o r aeV, a#O define a map Ta:V + C(x) by T,(v)=n(va-'). Then T(a)=l, and the kernel K(a) of T is o f codimension 1 i n V. Similarly i f bEK(a) and b#O then K(a)nK@) is of codimension 1 i n K(a). Continuing i n this way we may build a descending chain of definable subspaces of V, violating Corollary 2.3. This is the desired contradiction. Theorem 3.3.

D has ch ar act er i s t i c zero.

Proof.

Suppose D has ch ar act er i s t i c p>O. W e may also assume D is Yo-saturated. If C(y) is f i n i t e for all YED-2 then Z is f i n i t e and D is algebraic over 2, hence c o m t a t i v e [ 3,p.183], contrary to hypothesis. Hence we may choose YED so t h a t C(y) is i n f i n i t e. By assumption C(y) is f i n i t e dimensional over its center ZC(y), so ZC(y) is i n f i n i t e . Taking acD so th a t [ a,y]=l and defining Dy:ZC(y) + ZC(y) by 4,(u)=[ a,u] , we obtain a derivation of ZC(y) (see L e m 4.1.5) vanishing on Z but not on y. Hence y is transcendental over 2 . W e claini there is an element ycD satisfying: 1. ZC(y) is i n f i n i t e dimensional over Z 2. If XEDZ and ZC(x) C ZC(y) then either ZC(x) is finite dimensional over Z o r ZC(x)=ZC(y). This is an easy consequence of Corollary 2.4 applied to the subfields {ZC(y):ycD}. If xcZC(y), then C(y) 6 C(x) and hence ZC(x) 5 ZC(y). I f xgZ then C(x) is f i n i t e dimensional over ZC(x), and hence: I. ZC(y) is f i n i t e dimensional over ZC(x). 11. C(x) is f i n i t e dimensional over ZC(y). From I we see t h a t ZC(x) is i n f i n i t e dimensional over 2, and then condition e repeat f o r emphasis: (2) on ZC(y) yields ZC(x)=ZC(y). W 2'. I f xcZC(y)-Z then ZC(x)=ZC(y). From I1 and the assumption that D is eO- s at u r ate d it follows that f o r some integer n: 11'. C(x) is of dimension a t most n over ZC(y). &oose x~zC(y)-Z so that C(x) has l ar g es t possible dimension over ZC(y). ?hen also xP~ZC(y)-Z, since i f xPeZ we would have by Lemma 2.6 that xp=yp f o r some ycZ, which yields (x-y)p=O, so x=ycZ, a contradiction. We also have C(x) 5 C(xp) , and so by the choice of x we have C(x)=C(xp) However Theorem 3.2 yields an element acD satisfying:

.

[ a,x]=l.

lhen by the Leibniz r u l e

103

SUPER STABLE D I V I S I O N RINGS

[a,xP] = pxp-l = 0, so aEC(xP)=C(x), and hence [ a,x]=O, a contradiction. characteristic p>O. 54.

Thus D cannot have

Logarithms and Exponentials W e remind the reader that the convention of 83 remains i n force.

Lemna 4.1. Let xfD-2, aED, [a,x]=l. Define h:D + D by h(d)=[a,d]. Dx:ZC(x) + D be the restriction of h to the center of C(x). men: 1. Dx(x)=l 2.

Dx (u+v)=Ex (u) +Dx (v)

3.

Dx (uv) =Dx (u)v+uDx(v)

4.

Dx(u)=O i f f UEZ

5.

Dx:ZC(x)

+

ZC(x)

Let

is surjective.

Proof.

Some prelinunary remarks concerning the definition of Dx are i n order. By Theorem 3 . 3 Z is infinite, so by lheorem 3 . 2 for any ZED-Z a suitable aeD exists. Wo solutions al, a2 to the equation [a,x]=l w i l l differ by an element of C(x), and hence induce the same map Dx on ZC(x). 'Ihus the map Dx is defined and depends only on x.

Conditions 1-3 are already knm. Condition 4 says that C(a)(\i!C(x) 5 2 , so consider any crC(a)T\ ZC(x). 'Ihen aEC(c) so ZC(c) 5 C(a). On the other hand ceZC(x), so C(x) 5 C(c), inplying X(c) f- ZC(x) and XEC(C). (?his argument is used again i n Lemna 7.1.) Now Dx is a derivation of the f i e l d X(x) which by the above vanishes on ZC(c) but not on x. Hence x is transcendental over ZC(c), and thus C(c) is i n f i n i t e dimensional over ZC(c). By condition 2 of L e m ~2.7 t h i s forces CEZ, as desired. I t remains t o prove condition 5. To see that Dx:ZC(x) + ZC(x) we apply the Jacobi identity:

First take c=x, bEC(x) to see that h:C(x) + C(x). ?hen take CEC(X), beZC(x) to see that Dx:ZC(x) + ZC(x). To see finally that the map Dx:ZC(x) + ZC(x) is surjective, apply Lenma 2.5, recalling that Z is infinite. I t suffices therefore to show that z 5 $[ZC(X)] for a l l n. For ZEZ, it is and zxn/n!eZC(x), as desired. t r i v i a l that z=D:(zxn/n!)

I04

G. CHERLIN

Definition 4.2. Let XED-2, zE2, 1. log x = {YEZC(X):DX(y)=x-l} 2.

xZ = {yfZC(x):y#O and DX(y)-zyx-'f.

Lemm 4.3.

Let XED-2, ZEZ. 1. log x is an additive coset of 2. 3.

xz

If

2. is either empty o r a multiplicative coset of z l , z 2 ~ Z and Y E X ' ~ then e x 5%

Yz2

.

2".

Proof. -

1. log x is nonempty by Lemna 4.1.5, and our claim follows by Len2. Our claim is that f o r y1,y2EZC(x), both nonzero solutions t o

("1

4.1.4.

D X O = zyx-',

we have y1/y2~2. By Lemna 4.1.4 it suffices t o show t h a t Dx(y1/y2)=0, which follows easily from (*) and the Leibniz rule. 3.

Easily [ z ~ ' a ~ y - ~ , y ] = l . If UE'; then certainly u90, u~2C(y)5 ZC(x) To verify the f i n a l clause of Definition 4.2.2 we compute [ a,u]=zl[ z ~ 1 ~ ~ 1 , u ] y x ~ 1 ~ z l ( ~ z u y ~ 1 ) y x ~ 1 = z las z 2 desired. ux~1,

We can improve Lema 4.3.2 a t this point, showing that xz is always nonempty f o r certain z. Definition 4.4. Let Z0 be the minimal definable subfield of D. of Z0 follows f r o m Corollary 2.4 i n view of Theorem 3.3.)

(The existence

Lemma 4.5. If A is a nontrivial definable additive subgroup of D contained i n Zo then A=Z0.

Proof.

Let R be the idealizer of A i n Zo, R={ZEZ~:ZA ZC (xa) * , Furthemre since ZC(xa) is commutative h is a multiplicative endomrphism. We need to see that Z* is the kernel of h, i n other words that C(x)n ZC(xa) c Z. Looking a t the proof of Lennna 4.1.4, the f i r s t part shows that for ccC(x)~\ZC(xa) we have: ZC(c)

5 C(x)nZC(xa)

The rest of the proof of Lemma 4.1.4 on ZC(c). Now

and

xaEC(c).

also applies once we show that Dxa

vanishes

[-log x,xa]=l, so Dxa

may be defined as the restriction of

d

+

[-log x,d]

to ZC(xa). Since log xcZC(x), clearly Dxa * Thus ker h-Z. Consider the induced endomorphism: h* :ZC (xa) */ Z*

+

vanishes on ZC(c),

ZC (xa) */ Z*

as desired.

.

I t follows from Remark 3.1.2 and the assumption M(x)=l that h* is injective. We claim that h* is surjective. Let G-ZC(xa)*/Z* and T=Im h*. Since h*:G + T is a definable isomrphism, G and T have equal Shelah degree, and thus the index [G:T] must be finite. But G is divisible, so this forces G=T, as desired. Since h* is surjective we can solve h*(c/Z*),x”a“/Z* for C, so that for some ceZ* we have:

.

x-1c-1xcqx -1a -1 ‘his proves the l e m .

Lema 7.3. If XED-Z x+z are conjugate.

Proof.

and for a l l zeZ bl(x+z)=l then for a l l ZEZ x and

Solve [a,x]=l for a. By Lennna 7.2 some Since also [a,x+z]=l for ZEZ, cza-l are conjugate for some c Z E ~ . Setting and ~ ( z (x+z) ) are conjugate. Our claim is then that c(z)=l .for a l l c(z)+l for some z, l e t a=zc(z)/(c(z]-l).

x and tla-’ are conjugate for the same lemna shows that x+z and c(z)=

Pi.

Now,

although it is a consequence of CH, MA i s consistent with

%,

>

2"

and it is only in t h e presence of 2"

becomes a powerful assumption.

>

sl

t h a t MA

So t h i s method of obtaining

independence r e s u l t s always leaves open t h e p o s s i b i l i t y t h a t

V = L can be weakened t o CH (a very considerable weakening). An obvious way t o resolve this problem i s t o formulate a Martin's

Axiom type principle which has many of t h e consequences of MA

+

2"

> H i but is provably consistent with CH.

In t h i s

paper we consider t h e progress made on t h i s problem to date.

1. INTRODUCPION The Souslin Problem asks i f every l i n e a r l y ordered s e t X , without end-points, which i s connected i n t h e order topology and which s a t i s f i e s the countable chain The Souslln Hypothesis (SA) i s the

condition, is isomorphic t o the r e a l line.

assertion t h a t the Souslin Probiem has a positive answer.

SH can be formulated in terms of trees. Set

X

A

tree i s ( f o r us) a p a r t i a l l y ordered

= ( T , f ) such t h a t f o r each x Q T, the s e t j = fycTIy<xl i s well-ordered by h', of MA.

2" >

HI, MA

a l s o implies SH. Now, i n o r d e r t o deduce SH from MA one needs 2n0 >

gl.

Moreover, in t h e Solovay-

Tennenbaum model of ZFC

+ SH,

i s false i f

Indeed, Jensen has shown t h a t -,SH follows from a

cH

holds.

combinatorial p r i n c i p l e ,

0 , which

So t h e r e remains t h e p o s s i b i l i t y t h a t SH

c l o s e l y resembles CH, and which follows from

Nevertheless, CH does n o t decide SH, as Jensen a l s o c o n s t r u c t e d a model

V = L. of ZFC

CH f a i l s .

+

u3H

+

SH.

Thus, although MA

+

2''

>

x,

decides SH one way and V = L

decides i t t h e o t h e r , CH does n o t a f f e c t SH e i t h e r way. statements known t o be decided one way by MA

+

2n0 >

(e.g. t h e Whitehead Conjecture on a b e l i a n g r o u p s ) . Jensen c o n s t r u c t i o n of a model of ZFC

+ GCH +

H,

B u t t h e r e are many other

and t h e o t h e r way by V = L

Can some g e n e r a l i s a t i o n of t h e

SH be used t o show t h a t CH does n o t

MARTIN'S AXIOM VERSUS THE CONTINUUM HYPOTHESIS

115

I n f a c t , can w e o b t a i n from t h e Jensen c o n s t r u c t i o n a MA

decide t h e statement?

type p r i n c i p l e c o n s i s t e n t w i t h CH, much a s MA i t s e l f came from t h e SolovayI n t h i s paper w e survey t h e p r o g r e s s on t h i s problem t o

Tennenbaum argument? date.

2. AN OBVIOUS ATTEMPT FAILS

Now, as w e have s a i d , MA

We a r e s e e k i n g an MA-type p r i n c i p l e c o n s i s t e n t w i t h CH.

is o n l y a powerful assumption i n t h e presence of 2 fi0 > any poset and 3 is a countable set of dense s u b s e t s of set.

(Hence CH i m p l i e s MA.)

t h a t we can o b t a i n

s,. p,

So t h e power of MA + 2"'

3-generic sets when

>

The p o i n t is, i f then

p has an

HI lies

3' is uncountable.

is

p

3-generic

i n the fact

This l e a d s t o t h e

following v a r i a n t of MA. If

K

is an i n f i n i t e c a r d i n a l , l e t M A ( K ) mean t h a t i f

a t most

K

which s a t i s f i e s t h e

subsets of

then

p,

p

C.C.C.

+

is a p o s e t o f c a r d i n a l i t y K

dense

has an 3 - g e n e r i c set.

n u s M~(H,) is t r u e , and MA is e q u i v a l e n t t o (vw < a p p l i c a t i o n s of MA

p

3 is a c o l l e c t i o n of a t most

and

2"

For most

zLl0)MA(K).

is a l l that is required.

> til, MA(fl,)

So

l e t us t r y t o

f i n d our new axiom by looking f i r s t a t M A ( H l ) . Well, M A ( $ , )

*

ZHo >

g,,

a s is w e l l known, t h e reason being t h a t t h e u s u a l p o s e t

f o r d e s t r o y i n g CH s a t i s f i e s

C.C.C.

so l e t u s t r y t o r e p l a c e

C.C.C.

i n MA(fi,)

by

some o t h e r c o n d i t i o n on p o s e t s which excludes t h i s example, and t h u s s t a n d s a chance of being c o n s i s t e n t with CH.

I t is clear t h a t t h e c o n d i t i o n w i l l have t o

exclude a l s o any p o s e t which c o l l a p s e s F i r s t l y , w h a t i f w e amend M A ( $ , )

mind. that

HI.

Two p o s s i b i l i t i e s s p r i n g a t once t o

by r e p l a c i n g

C.C.C.

by t h e requirement

g be a-closed (i.e. every countable decreasing sequence has a lower bound)?

Well, w e then o b t a i n an "axiom" which is a theorem of Z F C , and is t h u s of no use t o us.

p

The o t h e r candidate?

Replace

C.C.C.

i n MACK,) by t h e requirement t h a t

b e 0-dense (i.e. t h e i n t e r s e c t i o n o f countably many dense i n i t i a l s e c t i o n s of

i s dense).

i n [Dell and CDe21. SH.

p

Demonstrating unsurpassed modesty, w e denoted t h e r e s u l t i n g axiom DA DA

is c e r t a i n l y n o t a t r i v i a l assumption:

(This is immediate.)

DA is a l s o c o n s i s t e n t with ZFC.

indeed DA implies Indeed, t h e

following is proved in [ D e l l .

2.1

Theorem

Unfortunately, DA does n o t s o l v e our problem. themselves d e s t r o y C H , we n e v e r t h e l e s s have:

Although a-dense p o s e t s do not i n

K.J. DEVLIN

116

Theorem

2.2

DA + 2xo > t i , . This i s a l s o proved i n [Dell, a s h o r t e r proof being given i n [ O e Z l .

Let u s a l s o

remark t h a t C u r t i s Herink has shown t h a t DA is, as one would expect, s t r i c t l y weaker than So

MA(K,). Now l e t us t r y t o f i n d a decent axiom.

much f o r "obvious attempts".

3. AXIOM SAD

The axiom SAD, discussed i n [ADS],

arose o u t of consistency r e s u l t s of Avraham-

Shelah and of Devlin, using the Jensen technique of LDeJo].

I n o r d e r t o State

t h e axiom, we need some terminology.

We have defined i n 51 t h e concept of a normal tree.

z

f u l l subtree of

(ii)

is a normal tree of h e i g h t h t (_T)

(iii) i f x

E

S

-a

L e t ;E be a normal t r e e .

A

such t h a t :

and y

and

E

X

' Ye

i

then Y

E

2.

a r r a y o f f i l t e r s is a c o l l e c t i o n

such t h a t D ordinal}.) Let

z

S is an i n i t i a l s e c t i o n of 2;

(i)

An

is a substructure 2 of

arf

is a countably complete f i l t e r on ma.

x be a normal tree o f h e i g h t w,

inclusion.

Let D = { D

a,f is a p p r o p r i a t e f o r D i f f : (i)

la

E

R E f

such t h a t Ta E

$1

be

5 w"

(R = {a c .,[a

is a l i m i t

and the ordering of ;E is

a r r a y of f i l t e r s .

W e say t h a t

i f a c R and f c ;Era, t h e r e is A c D such t h a t whenever h c A, h a,f then (VC < a) ( h h c T) + h c T;

(ii) if a

E

set A c

n

2 f,

and W 5 Era is a f u l l subtree of x r a , then f o r any f E w and any t h e r e is h E A, h f , such t h a t (WE < a) ( h f c E W).

Let SAD denote t h e conjunction of t h e following statements:-

X

117

MARTIN'S AXIOM VERSUS THE CONTINUUM HYPOTHESIS (ii) Every constructible cardinal is a cardinal, and f o r every cardinal L Cf(K) = Cf ( K ) ;

K,

( i i i ) Every countable sequence of ordinals is constructible;

I f D is a constructible array of f i l t e r s , then every t r e e which is

(iv)

appropriate f o r D has an wl-branch. In [ A D S ] , the following theorem is proved:

3.1

Theorem

If ZF'C is consistent, so too is ZFC

+ SAD.

A s an i l l u s t r a t i o n of the use of SAD, l e t us prove t h a t i f SAD holds, then

(This gives a new proof of Jensen's r e s u l t t h a t

must f a i l .

0

0

does not follow

from CH.) Theorem

3.2

SAD -+

-0.

Suppose otherwise.

Proof:

Let < f a l a e

and whenever f c wwl the s e t (a

E

n>

be a 0-sequence

:

that is, fa c wa

W e obtain a

n l f r a = fa) is stationary.

contradiction by using SAD i n order t o find an f c wwl such t h a t f r a # fa f o r any

a

n.

E

(Of course, we have taken here the formulation of

0 most

convenient f o r

our proof. ) Let

x = If

tree.

z: w i l l

of

e wsI(va c

n

n [dcnn(f)+l1)[fra # f a l l .

I t is e a s i l y checked t h a t

provide us with our required counter-example.

proving t h a t

9, =

f g c walg # h).

then

n A ncw hn

# 0.

form

91, h E

wa,

f

E

us),

forms a Any wl-branch

wl.

so we a r e reduced t o

is appropriate f o r some constructible array of f i l t e r s .

We now place ourselves i n L.

(Thus Da,f

Under inclusion,

2 is a normal t r e e of height

Let a

C

n,

f

E

2.

For each h

E

wa,

By a simple diagonalisation argument, i f hn

E

let wa,

n=1,2,.

..,

So, the family of a l l countable intersections of sets of t h e

generates a countably complete f i l t e r .

depends only upon a i n t h i s simple example.) a f i l t e r array.

W e now return t o the r e a l world, and shav t h a t

Let D

a,f

be t h i s f i l t e r .

L e t D = (D

a,f

x is appropriate for D.

l a e n &

K.J. DEVLIN

n,

Let a

f

Tra.

E

Now, A

h # fa, so (VE f o r which fa c ua, such t h a t f o r any f c wwl, f a c Elfra = f a ) i s stati:nary.) weaker than t h e assertion t h a t

This shows t h a t

SACS) implies y o ( S ) , a f a c t which we prove below

Of SA.

0 is

strictly

M E ) holds f o r every stationary s e t E =5l,

because

as an i l l u s t r a t i o n of the use

K.J. DEVLIN

I20 Theorem

4.2

SA(S) + +(S). Proof: =

2”’

I f 2”

s,.

> 8 , the r e s u l t is immediate, so assme otherwise:

L e t < f a l a c S> be a

finding an feww1 such t h a t f o r no a Let

= If c w3l(Va

X

S n (dom(f)

E

So if

;is

c l e a r l y has an extension i n if 6 c

R

- S, then

1 ) ) ( f r a # fa)}.

I f b i s an wl-branch of

a t r e e of height w 1 . counterexample.

s does f r a = fa.

E

+

thus

We obtain a contradiction by

O(S)-sequence.

x,

Well, every element of

S-good we a r e done.

T, and by

CH a t most

c e r t a i n l y every 6-branch of

Ordered by inclusion, T, is

f = Ub w i l l be the required

8 , extensions i n T,.

X

Moreover,

TI6 w i l l have an extension on

T6

( i t s union). Now l e t < X a ( u < ol> be a s t r i c t l y increasing, continuous sequence of countable

s e t s with union T.

Let

Since <x la<wl> is continuous, C is c l e a r l y closed and unbounded i n wl. 6 c C n S, f c T r 6 , y = h t ( f ) . h(y) T*

# f6W. =

{g

E

Pick h

E

Ty+l

such t h a t h

3

Let

f and

Let

T[’61g 5 h o r h

5 gl.

Then T* c l e a r l y s a t i s f i e s a l l t h e requirements of condition 5 . i n the S-good definition. The proof is complete.

D

References

U. Avraham, K.J.

Devlin, S. Shelah.

+

Consequences of

K.J.

Devlin.

2’0

> gp

The Consistency with CH of Some

t o appear.

An Alternative t o Martin’s Axiom.

Hierarchy Theory, Springer Lecture Notes 537, 65 On Generalising Martin’s Axiom. Variations on 0 & H.

.

Johnsbriten.

Notes 405.

-

S e t Theory and 76.

t o appear.

t o appear. The Souslin Problem.

Springer Lecture

MARTIN'S AXIOM VERSUS THE CONTINUUM HYPOTHESIS [DeShll

K.J. 2'''

Devlin P S . Shelah. < 2'1.

A Weak Version of

0

12 1

Which Follows From

t o appear. A Note on the Normal Moore Space Problem.

EDeSh21

to appear. [Jel

T. Jech.

[Shl

S.

Shelah.

to appear.

Trees.

Journal of Symbolic Logic 36 (19711, 1

-

14.

Whitehead Groups May Be N o t Free, Even Assuming CB, I P I1

Logic Colloquium ' 7 7 A. M a i n t y r e , L . Pacholski, J . P a r i s (eds.) 0 North-Hotland Publishing Canpang, 1978

ON THE TIGHTNESS O F PRODUCT SPACES

J. Cerlits and A. Hajnal (Budapest)

8 I . INTRODUCTION Let R be a topological space,

X E R,

A C R. Put

a ( x , ~ ) =min(IB1: B C A

AXEB};

r(R) = sup (a(x, A); x € 2c R}.

'

r(R) is said to be the tightness of the space R. The aim of this paper is t o investigate what can be

the tightness of the topological product of two spaces of tightness w .

The first example of two spaces X,, ,X , with t(x,)= w (i < 2), r(Xo X X , ) > w was given by A.V. Archangel&: [I]. The spaces he found are even Frkchet - Urysson spaces. (A space X is a Frkchet - Urysson space if each limit-point of a set A C X is the limit of a convergent sequence from A). Recently F. Calvin proved that, assuming Martin's Axiom, there are two Frkchet

- Urysson spaces

X o , XI with f(Xo X XI) = 2 w .

Although we are not able to describe completely the cardinals which are the tightnesses of such a product, our results show that under certain set theoretical assumptions there are very hrge such cardinals. For example we prove that if the Axiom of Constructibility holds then for each cardinal K there are two Frkchet - Urysson spaces such that the product space has tightness K . As K. Kunen kindly informed us he proved this and Theorem 1 independently and somewhat earlier. We include these results for the sake of completeness only. Assuming that the continuum is relatively small we can prove that there are many cardinals under the fust measurable cardinal which can be the tightnesses of the product of two spaces with tightness w . Especially C.C.H. implies this for all cardinals less than the first measurable cardinal. (See Theorem 14 and Corollary 17 for a more precise formulation). We shall use the usual notation of set theory and topology; all undefined terms can be found in [21,

131 or 181.

a,0,[,r) denote ordinals, K , A, p denote cardinals (i.e. initial ordinals). (Y + 0 denotes ordinal addition and for a cardinal K K + denotes the immediate cardinal successor of K , exp°K stands for K and e x p " + ' ~= exp"(2") for all integers k. If [,r) are ordinals, [, denotes the set Of functions mapping [ into r). For a set X , I XI denotes the cardinality and Y ( X ) the power set of the set X. Where K is a cardinal, [XI', [XI'" and [ X I c K denote the set of all subsets of cardinality K , of cardinality < K and of cardinality < K of the set X , respectively. Concerning the definitions and the simplest

123

124

J. GERLITS and A. W N A L

properties of the so called "large cardinals" mentioned in this paper we refer the reader to F. Drake's d denotes the closure of the set A in a topological space. If E is a space, E, denotes the G,-topology in E; i.e. the G,-sets in E form a base for the topology of E, . For a set H D(H) denotes the discrete space on the set H; D(2) = D , D(w)= N.

book [2].

In the paper we shall repeatedly refer to the product spaces (D"),and (N"),;note that for infinite K'S they are homeomorphic (both are homeomorphic to (D(2")"),.)

In the topological produd E = ll {Ei; i E I) of the spaces Ei the set U = Il { U i ; i E I) is said to be = { i E I ; Ui + EiI is finite. a basic open 'set iff Ui is an open set in E, for i E I and Supp (0 The space E is called K-compact if each open cover of E has a subcover of cardinality < K . If K is regular and the weight of E is K then this is equivalent to the property that each subset of cardinality K has a complete accumulation point in E. 52. THERESULTS

For a set S, if I C [S]' on S.

"

is an ideal, put

3 = { A c S; [A] w is the rust strongly compact cardinal then an arbitrary product of &-compactspaces in the &topology (indeed, in the X-topology) is X-compactJ Our next result generalizes the theorem of

For f,g E

w put f 4 g if

v,;

F. Galvin mentioned in the introduction.

An) < g(n) for all but finitely many integers n.

A sequence t < K ) is said to be a pseudo-scale of " w if K is a regular cardinal, f, 4 f,, for E < q < K and given any f € w f, 4 f does not hold for each [ < K .

J. GERLITS and A. HAJNAL

I26

It is well-known that Martin's Axiom implies that the cardinality of any pseudo-scale for

Theorem 3. I f

K

Q

the cardinal o f a pseudo-scale for

Roof. Put H = (w X w ) X

K,

and let

v,; [
0,

For any

T h i s i s imnediate:

,

Fn

{a} >

then

K

a
= < M

sequence i s n o t necessary.

i

v a r i o u s embeddings

a 1

a < a2 >

-

, so

that "starring"

the natural

t h e a b s o l u t e n e s s c o n s i d e r a t i o n s about t h e

Also.

s t i l l apply.

( i i i ) Conclude t h a t

T

and as b e f o r e . complete t h e p r o o f w i t h two a p p l i -

U

E

cations o f normality. #-indescribable

For t h e n o t i o n o f a ing [B],

l e t us c a l l an

, and

!lI

I=

.

@

,

< VK

,

E

R >

X

I=

&

K

@

, then

Levy showed t h a t f o r each

d=

c_

{ X

I

K

K

-

X

cardinal,

I#-indescribable t h e r e i s an

see L 6 v y [ L ]

iff B

E

i s not

n!-indescribable

is

)

(improper j u s t i n case

K

nX-

i s not

K

The n e x t c o r o l l a r y i s t h e e x a c t analogue o f 7.2

p r o o f i s e s s e n t i a l l y unchanged;two

@

m,n > 0,

i s a normal K-complete f i l t e r o v e r indescribable).

follow-

, E , RflU D >

< V5

so t h a t

X

. Again REV,,

whenever

i n [B],

and t h e

f i l t e r s a r e s a i d t o be c o h e r e n t i f t h e f i l t e r

generated by t h e i r u n i o n i s a ( p r o p e r ) f i l t e r . C o r o l l a r y 7:

For any

n > 0,

i s Vopenka-n-ineffable

K

iff

H$ and Fn

are coherent.

If

Proof: extending is K

Fn

.

K i s Vopenka-n-ineffable,

Also,

ilh-indescribable

-

X

i s not

E

Gn

.

Thus,

.

(see 7.1 o f [ B ] )

Ill-indescribable

.

r e l a t i v i z e d v e r s i o n o f Theorem 4, X

Hi and

then

i t i s w e l l known t h a t i f

Fn

Then K

-

c_

K

Suppose now t h a t K

X

i s a (proper) f i l t e r

Gn Y

-

X

i s n-ineffable, X

c_

K

is in

i s not n-ineffable,

then

Cn

. .

Y

Hh, i . e .

and by t h e

i s not Vopenka-n-ineffable

a r e c o h e r e n t , b o t h b e i n g extended by

-I

Hence,

152

A.

For the converse,

f i r s t note that

Hh

i n d e s c r i b a b l e , else e i t h e r

I a


>

K

Vopenka-n-ineffability o f

D = { a
- 1 .

Then UE n

0-

ward cofinal, d. IKI; for

one

be such that the 1; -definable points are down0

, we

of these points, say . a

have

Mo be such that the Eo -definable points are M I= Cons(Tn + A- ) . Let M' 9 "0 0 . a no longer downward cofinal and let co E IMAI be an infinite integer which is less that all 1; -definable non-standard points in MA. Applying Theorem 3.1, 0 let M1 be a T;I -elementary end extension of M' satisfying A in which c0 is 0

>

9

1; +l-definable from the parameter ao.

>

However,

0

is 2;

.a

MA

> Mo and the point

in

R,.

"0

-definable in Mg; thus co is E:o+l-definable

Moreover, the

0 Z;

-definable points of M and 0

n

points of

M1.

We then have M

El coincide.

< "0

M1

Let M consist of the definable 1

and the :E

-definable points are not 0

180

K. Mc ALOON

downward cofinal in M1; furthermore, M1 I = A

and

M1 is pointwise definable.

M,

Iterating this process, we can define a triple sequence

, T+l of ... I n k 5 ... -

countable models of A and a sequence of integers no (nl such that

(%) * E:k(M,+l) where, k in general, we let Eo(M) denote the set of Eo-definable points of a model M P P < % and the [ii) if b is not a strong cut in %, then no c nI < M,

each

(i)

is pointwise definable and :E

...

- 9 II,+~%

t&-definable then M,+l

points are downward cofinal in =

and

if IN is not a strong cut in M,, then

(iii) standard

9; if

ck

q

(iv) if N is not a strong cut in M,, extension of in which ck is E:k+l-definable

M,+l

We then set Mu

M, and

M

I=

ConsR(Tn

+ A)

.

(Fere ConsR(Tn

+ A)

abbreviates

fxCons(Tn

+ Ax).)

3.5 Theorem Let M be a model of A in which A is represented by a nO-formula I- ConsR(Tn + A). Then for every c E lMl, there

R(u,vl,. ..,vS) such that M

end extension M' of M such that (i)

is a R:-elementary

A:+l-definable

M' 1=A

in M' in a way that persists to R:-elementary

(iii) M' \ = ConsR,(T

+ A)

for some n:-formula

A in M and in M'.

R'(u.v1.

,

(ii)

c

is

extensions, and

...,vt)

representing

I=

ConsR(Tn + A ) for some By Lemma 6.5 of IMcl, we can suppose that M li non-standard n:-minimal point k in M. Then by the methods of proof of Theorems

Proof

6.1 and 6.2 of lMcl and of Theorem 3.1 above, the result followa.

In order for all points c E \MI to be Z0 n+l -definable in some IIielementary extension some hypothesis on M such as satisfying Cons R (Tn + A) is

3.6 Remark

necessary; for example, with n P; and M is an extension of M

-

1, if Mo is an existentially complete model of

satisfying P

, then there are points in M which

can never be Ei-definable in an extension of M Satisfying P.

DIAGONAL METHODS AND STRONG CUTS IN MODELS OF ARITHMETIC

181

3.7 Corollary Let M be a countable model of A in which A is represented by a

.

...,

v ) such that M 1 ConsR (Tn + A) no-formula R(u,vl, Then there is a noelementary end extension M' of M satisfying the lli+2-consequences of A in which every point is Z:+l-definable.

3.8 Remark Ir, IBI it is shown that every countable model of XF a poi.?.twise definable end extension satisfying XF.

+V

=

HOD has

Corollaries 3.3 and 3.7

are analogues of this result for Peano arithmetic.

BIBLIOGRAPHY 1B1

J . Barwise, Infinitary methods in the model theory of set theory, in

IF1

S.

IJl

F.M. 49 (1960) 35-92 C. Jockusch Jr., Ramsey's theorem and recursion theory, J.S.L. 37

IKI

(1972) 268-280 L. Kirby, Initial segments of models of arithmetic. Thesis. Manchester.

IL,PI

L. Kirby and J. Paris, Initial segments of models of ?earJo's ax!.oms,

ILI

in Springer-Verlag Lecture Notes in Mathematics, vol 619, 211-226 H. Lrssan, Models of arithmetic, Thesis, Manchester, 1978

IM,PI

G. Mills and J . Paris, Closure properties of countable non-standard

III,RI

A. Manaster and J. Rosenstein, Effective matchmaking, recursive theoretic aspects of a theorem of Philip Hall, Proc. London Math. SOC. 25

Logic Colloquium 69, North-Holland 1971, 53-66 Feferman, Arithmetization of metamathematics in a general setting,

1977

integers, to appear

(1972) 615-654 IMc 1

K. Mc Aloon, Completeness theorems, incompleteness theorems and models

[Mi( IMonl

L. Mirsky, Transversal Theory, Academic Press. New York 1971 R. Montague, Semantical closure and non-finite axiomatizability, in

IP 1

J. Paris, Independence results for Peano arithmetic using inner models,

of arithmetic, to appear in T.A.M.S.

Infinitistic Methods, Pergamon 1961, 45-69 to appear in J . S . L . IS1

E. Specker, Ramsey's theorem does not hold in recursive set theory, Studies in Logic and the Foundations of Mathematics, North-Holland 1971

Logic Colloquiwn '77

A. M a d n t y m o , L . Pacholski, J. Paris (ed8.I 0 North-Holland Publishing Canpany, 1978

QUANTIFYING POSITIVE

vs

OVER

COUNTABLE

STATIONARY

SETS :

LOGIC

J.A.MAKQWSKY 1I.MATHEMATISCHES INSTITUT DER FREIEN UNIVERSITAT BERLIN, BERLIN

Introduction ~The study of extensions of f i r s t order p r e d i c a t e c a l c u l u s i s by now over twenty years o l d and many papers have been published i n t h e f i e l d . But t h e s u b j e c t seems t o l a c k a program o r an ideology. Such an ideology is u s u a l l y provided by

-

p a t t e r n s of p o s s i b l e theorems hard open problems and hard theorems

- applications

t o o t h e r f i e l d s of mathematics o r o t h e r sciences.

For f i r s t o r d e r p r e d i c a t e c a l c u l u s t h e f i r s t and second aspect overlap: Completeness and compactness theorem, LBwenheim and Hanf number c a l c u l a t i o n s , c a t e g o r i c i t y theorems and d e f i n a b i l i t y theory g i v e a good frame, and a p p l i c a t i o n s i n a l g e b r a and non-standard a n a l y s i s a r e w e l l known. For extensions of p r e d i c a t e c a l c u l u s a f i r s t t r y i s t o mimick c l a s s i c a l m o d e l theory, but we a r e l i m i t e d by Lindstrom's theorems which c h a r a c t e r i z e LU.

Fewer hard

theorems e x i s t and hard open problems are s c a t t e r e d without much coherence. In t h i s paper I w i l l survey some r e s u l t s on countably ccmpact extensions of Lww which have grown out of j o i n t work with S.SHELAH and J.STAVI [MSS,MS2]. Let u s s t a r t with a hard problem (which occurs i n FRIEDMAN'S l i s t [ F r ] ) : PROBLEM

Is t h e r e a proper extension of Lw,

which i s countably compact and satis-

f i e s Craig's o r Beth's d e f i n a b i l i t y theorem ? L

w

(Ql ), t h e l o g i c with t h e a d d i t i o n a l q u a n t i f i e r " t h e r e e x i s t uncountably many",

i s countably compact, but does not s a t i s f y any d e f i n a b i l i t y theorem as u s u a l l y s t a t e d . Analyzing t h e counterexamples v a r i o u s ways of extending LM(Q,) died i n t h e l i t e r a t u r e , i n p a r t i c u l a r i n [MM]

where stu-

and i n [MSS] which culminated i n

t h e l a t t e r with t h e c o f i n a l l y i n v a r i a n t weak second order l o g i c . I w i l l not d e f i n e

183

J.A.

I84

MAKOWSKY

t h e s e n o t i o n s h e r e , b u t i l l u s t r a t e them by p a r t i c u l a r l y n e a t examples, L;+,

and compare them w i t h Lww(aa)and Lu,,(aa)

Lk and

due t o SHELAH [ S H 1 1 and s t u d i e d

f i r s t e x t e n s i v e l y i n [BKM]. When I was f i r s t c o n f r o n t e d w i t h [BKM] it looked as i f a l l of t h e r e s u l t s f o r

Lk

where superceded by r e s u l t s f o r L w ( a a ) . But a more c a r e f u l a n a l y s i s o f t h e two l o g i c s shows t h a t d e i r model t h e o r y d o e s n o t e n t i r e l y look t h e same: The most s t r i k i n g examples b e i n g t h a t f o r Lw(aa)

t h e Lowenheim-Skotem-Tarski

theorem i s

n o t p r o v a b l e b u t c o n s i s t e n t With ZFC whereas f o r LElu it is simply t r u e . (Theorems

4

and 5 ) A Feferman-Vaught-type

theorem f o r L k which does n o t hold f o r Lww(aa)

( P r o p o s i t i o n 9 ) i s a n o t h e r example. P a t t e r n s o f theorems a r e p r o v i d e d by t h e theorems t r u e f o r L,(Ql)

[Ke,Sh 21, i n

f a c t t h e main p o i n t i s t h a t many o f t h e t e c h n i q u e s developed so f a r f o r Lw(Q1) t h e o r y [SH

41

of which t h e Lwl,,(Q)

b e a u t i f u l example. For L,(aa)

c a t e g o r i c i t y theorem i s t h e s i m p l e s t and most

such a t h e o r y seems n o t t o work, b u t t h e r e i s a

f a i r chance t h a t it can be developed f o r L;,,,. have o n l y

one way

can

SHELAH h a s developed a s t r u c t u r e - non s t r u c t u r e

b e extended to. L L . For Lu,u(Ql)

The i n h e r e n t d i f f i c u l t y i s t h a t we

of c o n s t r u c t i n g models, a d i f f i c u l t y which l i e s a l s o behind con-

j e c t u r e 1 4 . So t h e p a p e r w i l l s u r v e y t h e model t h e o r y of L$wand

I,,,daa)

as f a r as

t h e y a r e p a r a l l e l and e x h i b i t t h e i r known d i f f e r e n c e s . A t t h e end a modest a p p l i c a t i o n t o t o p o l o g y w i l l b e g i v e n . The program may b e summarized a s f o l l o w s : Approximate second o r d e r model t h e o r y by compact f r a g m e n t s w i t h a manageable s t r u c t u r e t h e o r y of i t s models and w i t h a c o h e r e n t d e f i n a b i l i t y t h e o r y . P r o o f s w i l l b e pres'ented i n [Ma 21. The p a p e r more o r l e s s r e p r o d u c e s my h a l f hour a d d r e s s d e l i v e r e d a t t h e colloquium.

1 . S t a t i o n a r y and p o s i t i v e l o g i c

Let L b e a f i r s t o r d e r language x,y,z... U,V,W

be a countable s e t of individual v a r i a b l e s

... a

c o u n t a b l e s e t of u n a r y p r e d i c a t e v a r i a b l e s .

What we want i s t o q a a n t i f y o v e r b o t h o f them. Without f u r t h e r r e s t r i c t i o n s on t h e q u a n t i f i c a t i o n of p r e d i c a t e s r a t h e r l i t t l e can b e s a i d about t h e r e s u l t i n g l o g i c . So we l o o k f o r r e s t r i c t i o n s , which make t h e r e s u l t i n g model t h e o r y workable. F i r s t we r e s t r i c t t h e r a n g e of t h e p r e d i c a t e v a r i a b l e s t o c o u n t a b l e s e t s . Furthermore we p u t a s y n t a c t i c r e s t r i c t i o n on t h e v a r i a b l e s t o b e q u a n t i f i e d which i s i n s p i r e d from t h e s u c c e s s f u l approach t o t o p o l o g i c a l model t h e o r y due t o McKEE, GARAVAGLIA and ZIEGLER i n d e p e n d e n t l y ( f o r a s u r v e y c f . [Ma 1 1 ) .

QUANTIFYING OVER COUNTABLE SETS

I85

The formulas o f Lp ( p o s i t i v e l o g i c ) a r e d e f i n e d i n d u c t i v e l y : ww

( i ) L-atomic formulas ( i n c l u d i n g x = y ) and U ( x ) a r e f o r m u l a s ( i i ) i f cp,$

a r e f o r m u l a s so a r e

(PA$,

cpvq, T c p , i'xcp and VXQ

( i i i ) i f cp i s a formula and U does n o t o c c u r n e g a t i v e l y i n cp (we w r i t e c p ( U + ) ) P

t h e n 3Ucp is a formula and U d o e s n o t o c c u r p o s i t i v e l y i n cp (we write c p ( U - ) ) t h e n Wcp i s a f o r m u l a .

The f o r m u l a s o f Lp

W3 W

(iv)

)L:(

a r e d e f i n e d by t h e a d d i t i o n a l c l a u s e .

I f 0 is a c o u n t a b l e ( i n f i n i t e ) set o f f o r m u l a s , t h e n

M @ and

w @ a r e formu-

l a s , p r o v i d e d o n l y f i n i t e l y many v a r i a b l e s o c c u r f r e e i n @. If we r e p l a c e ( i i i ) by P

( i i i ) a aIf cp i s a f o r m u l a , so a r e aaUcp and

statUcp

we o b t a i n Lww( a a ) , LWIU(a a ) o r L ( a a ) r e s p e c t i v e l y . ( s t a t i o n a r y l o g i c ) mw The s a t i s f a c t i o n f o r Lp

WlW

and L

Ul

( a a ) i s defined for L-structures a s usual l y with

the additional inductive clauses (p)

at=3Ucp(U+)

i f f t h e r e i s a c o u n t a b l e s e t X S A such t h a t < U , D

I=

w(X)

( a n d hence by p o s i t i v i t y i f Y 2 X is c o u n t a b l e t h e n ,'1,.

Define Lw(aa) as Lw(aa) but change t h e ( a a ) s a t i s f a c t i o n c l a u s e t o

&I=

aaW i f f { X S A l

?


t o show

4 and Leunna 3, ' d y

< t e(u.

x < a ( 3 u < t e(u, x),,il

S i n c e w e c a n i g n o r e t h e bound

Proof.

-7u

i s e q u i v a l e n t to a

x < a

A

i n d u c t i o n h y p o t h e s i s w e may a p p l y

P-

- 1 J v e ( v , x))].

we g e t

BCn+l,

Proposition 6.

o

(u =

v

0 E

nn.

By P r o p o s i t i o n

formula: a p p l y i n g

X

ICn

to

i t gives t h e required l e a s t element.

C o n v e r s e l y assume such t h a t

ie(a).

Ln,,

S i m i l a r l y ( b u t more e a s i l y ) , LEn

*+

BE

argument. Proof.

Let

n 2 1

For

.

t h e r e exists

-4 x i $ ( x ) . P i c k e ( a - 1) i s f a ) . A

IC

Kn

the least

a

follows.

IE

and t h i s s u f f i c e s .

n'

f

which i s n o t a

P- + I X n - l

[ T h i s was f i r s t p r o v e d by P a r s o n s [ 2 ] u s i n g a p r o o f t h e o r e t i c a l

We s h a l l g i v e a d i r e c t model t h e o r e t i c p r o o f . ] P

M

c o n t a i n n o n - s t a n d a r d E n - d e f i n a b l e e l e m e n t s and d e f i n e

F i r s t notice that and and h e n c e i s i n

+

and

En

or

model o f

Then e i t h e r

E

a = 0

P r o p o s i t i o n 7.

Kn

$(x)

a

Kn.

E

{x

Kn

=

Kn

2

Kn.

6

M

I

x is C - d e f i n a b l e i n

.-I 2

M: f o r s u p p o s e M

Then t h e l e a s t s u c h

I t i s now e a s y t o see t h a t

z

is

Kn

MI.

e(z, a )

where

Cn-definable LEn-l

0

in

E

M

and h e n c e

P- + I X n m 1 . To show t h a t

BEn

fails in

Kn

let

7w

Y(e,w,x)

be a u n i v e r s a l

'n

nn-,

En-COLLF,CT'ION SCHEMAS I N ARITHMETIC formula where

. I e
M 1 s i ( b , %).

qi(b,

such t h a t f o r a l l

Then by r e c u r s i v e s a t u r a t i o n we can f i n d b < a P + Yn + 9(b,

i

M

n

n1

E

1 n(b, % I .

'

b'

can be s u c c e s s f u l l y found and the r e s u l t follows.

n+l

Remarks 1). Let

(x < a

A

be a countable model of

M a

ni(x. b)

I

i

E

BE1 + B1

N ) i s a r e c u r s i v e s e t of

Ill

satisfiable in

H then i t i s s a t i s f i a b l e i n M.

we see t h a t

i s an i n i t i a l segment of a model of

M

The converse appears t o almost hold. model of ? and

P) then M

BE1

happens when

NkfM

+ B1 NtcfM

Theorem C

2).

Theorem C '

Let

Iln+l

M

and

M

C

P.

i s an i n i t i a l segment of a

What

is not c l e a r . r e a d i l y g e n e r a l i z e s t o give:-

T be a theory.

consequences of

H. Lessan, Ph.D.

Then using t h e proof of

s a t i s f i e s the above s a t u r a t i o n property.

Then

:i

T

M

P

such t h a t

i f and only i f

P) i s c o n s i s t e n t . References

[l]

formulae f i n i t e l y

( i . e . N is confinal in M from above i n t h i s model of

elementary i n i t i a l segment of a model of (= the

If

such t h a t i f

Thesis, Manchester 1978.

M

is a

T + BEn+l

Zn

+ Bn+l

Cn-COLLECTION SCHEMAS IN ARITHMETIC [2]

209

C. Parsons, "On a Number Theoretic Choice Schema and its Relation to Induction".

Intuitionism and Proof Theory, North-Holland 1970.

Kino, Myhill L Vesley.

Ed.

L q i c Cottcquiwn '77 A. Macintyre, L . Pachohki. J . Parie (edS.1 0 North-Holland Rcbliehing Canpan#, 1978

INNER MODELS, ORDINAL DEFlNABlLlTY

AND THE AXIOM OF POWER SET by Stanis+aw Roguski and Andrzej Zarach (Wrocfaw)

Introduction.

A formula " ( r )defines an inner model f o r a s e t theory

T i f i t 1s

provable i n T t h a t : (1)

(y)(z)(y€2 &

(2)

(a)O(a) ,

(3)

A"

"(2)

+

"(Y)) ,

f o r every axioms A o f T.

T i l l now two inner models f o r ZF, the c l a s Z o f a l l c o n s t r u c t i b l e sets and the class HOD o f a l l h e r e d i t a r i l y o r d i n a l d e f i n a b l e sets, were i n t e n s i v e l y investigated.

In

C51

M y h i l l and Scott have defined the c l a s s L 1 o f s e t s c o n s t r u c t i b l e i n

the second order language and they have proved t h a t Z F t A C

t L 1 =HOD.

By L" we

s h a l l denote the c l a s s o f sets c o n s t r u c t i b l e i n the language o f order n t l ( i n 1 2 p a r t i c u l a r L o = L ) . I t i s obvious t h a t ZF I- L c L c L 5 ... 5 L " c SHOD and every

...

.'

Ln i s an inner model f o r Z F Adam Krawczyk

(more f o r m a l l y : ZF

from Warsaw

I- ALn

f o r everylaxiom A o f ZF ).

has r e c e n t l y defined a c l a s s L w l . I t i s defined

as f o l l o w s . L e t Def ' ( A ) denote t h e f a m i l y o f subsets o f A d e f i n a b l e i n 4 , E A > 1 by r1 formulas o f the second order language w i t h elements o f A as parameters.

for l i m i t h

He has shown t h a t Ln' ZFtAC

1- L

' =llOD.

A1

i s an inner model f o r ZFC and he has asked i f

We do not know an answer t o t h i s questiqn yet. b u t i n a l l

models f o r Z F t Y = H O D t h a t have been obtained by f o r c i n g L B 1 =HOD holds. D e f i n i n g each o f the inner models mentioned above requlres d e f f e r e n t set theory. The theory EF ( e x t e n s i o n a l i t y , sum. existence o f .$ and w . r e g u l a r i t y f o r sets, Ao-replacement and A - t r a n s f i n i t e i n d u c t i o n schemata

-

see [ 1 4 1 f o r d e t a i l s )

s u f f i c e s t o c o n s t r u c t L and t o prove i t s fundamental p r o p e r t i e s . I n a d d i t i o n t o E F , we o n l y need nl-replacement

21 I

and n l - t r a n s f i n i t e

i n d u c t i o n sche-

2 12

S. ROGUSKI and A. ZARACH

mata t o c o n s t r u c t t h e c l a s s Ln'.

The w e l l known argument i n v o l v i n g t r u t h d e f i n a b i -

l i t y f o r En f o r m u l a s , w h i c h was p r e c i s e l y d e s c r i b e d f o r EF i n C141, shows t h a t i n

ZF- ( Z F w i t h o u t t h e power s e t axiom) one can d e f i n e DeflSk(A) o p e r a t o r . A Deflqk(A) i s t h e f a m i l y of a l l subsets o f

definable i n

a,€,>by

f o r m u l a s ( w i t h parameters f r o m A ) h a v i n g a t most k - q u a n t i f i e r s .

For e v e r y

second o r d e r

But we cannot t a l k

k U Oefl' ( A ) i n ZF-, because t h a t would g i v e a t r u t h d e f i n i t i o n f o r k iW0., aty.z))

M + = ( ~ E w : cp Y

We want t o g i v e a s i m i l a r theorem i n n'l

v(g,x)1

f i n i t e sequence o f o r d i n a l s , we have t h e f o l l o w i n g

THEOREM 4. ,The theory T f T o t

L

We s h a l l p o i n t o u t a formula

the language o f Z F ( w i t h o u t parameters), such t h a t i f M-c = { x :

v(*,.)oof for

I n view o f the absoluteness Lemna f o r A = L t h e r e i s

I t i s i l l u s t r a t e d by t h e f o l l o w i n g example.

lute. Let

-

(see

(x€OD).

( Z F t V = HOD) n E2 Z ZFC n L2

Corollary:

[TI.

t o v e r i f y t h a t t h e p r o o f from

too. L e t us remark t h a t (

C31)

d

Z F t V = A and ZFC agree on n2-sentences.

For t h e c l a s s HOD t h i s theorem i s proved i n (for

I-

ZFC

++

i s c o n s i s t e n t r e l a t i v e t o ZF.

ZF-. I n t h i s theory we can speak o f the c l a s s

implies the existence o f a definable well-ordering o f the universe

o f type On. Now t e t I'(a.x) be a formula enumerating t h e universe.

I f ADtl

= TC{y: (3a),Dr(a,y))

and A A = a Y h Aa f o r l i m i t A , then ul : @ € O n > i s an R - h i e r a r c h y o f t h e universe. D

Therefore HOD i s e x p r e s s i b l e i n Z F - + V = L n l Let T A - Z F - t V = L T :

+

l

and moreover V = H O D .

"every s e t has a Hartog's number"

per class". L e t s h a l l p o i n t o u t a formula

'+(*,a)

o f the

N + = (x: Y(h,,x)} then we have t h e f o l l o w i n g

LZF

-

+

"the continuum i s a p r o -

language such t h a t i f

INNER MODELS, ORDINAL DEFINABILITY AND THE AXIOM OF POWER SET O f course i n

2 15

ZF, t h e formula runk(x)

, 8=M[bo,8,h] ;

N i s o f t h e form

M01Zl.

rn [lll t h a t P ( p k ) n N = P ( p k ) n M o f o r k < w , N k L 1 = L 1 b o l and =cjLLCA' I n p a r t i c u l a r N C pk has a good s u b s e t ++ M" C wk has a good

I t i s shown c/

=cF.

s u b s e t . By i n d u c t i o n one can p r o v e t h a t (L::)Mo= enough t o show t h a t

pi'

(La')

N

for a<w

0.0

. Now,

I f the universe i s c o n s t r u c t i b l e from a subset o f

On t h e n t h e u n i v e r s e i s a gene-

r i c e x t e n s i o n o f HOD ( s e e 1121). From t h e p r o o f o f t h e Vopenka-Hajek

121)

it is

boELpi; (in N ) .

i t follows that the appropriate

theorem ( s e e

n o t i o n o f f o r c i n g has c a r d i n a l i t y

H

5

'2

',

p r o v i d e d t h e u n i v e r s u m i s c o n s t r u c t i b l e from a s u b s e t o f w. From t h a t and t h e assumption L r A l

H 'K

>Z2 O=w2

k

L = HOD i t f o l l o w s t h a t i f

( i n L C A l ) then L [ A l

/=

K

i s a r e g u l a r c a r d i n a l i n LCAland

does n o t have any good subset".

I ~ K

Cohen f o r c i n g does n o t add any good s u b s e t t o r e g u l a r a l e p h s f o r c i n g Lemnas g i v e n i n McAloon's paper ows t h a t k E b , From

++

F?

( 6 ) and ( L : ' ) A p

!=

pk

141

> w.

From t h e

and t h e above c o n s i d e r a t i o n s i t foil-

has a good s u b s e t .

= [L")"

for

a < w 6 we o b t a i n

bo"L"')''.

P ~ o u j jof lheorcrn I and 2hcor.em 3 :

( 5 ) and AC we have L C K l r L n ' ~ i I O D . # o ~ + ~ )By. [81 a model iV i s c o n s t r u c t e d such t h a t N k i i O D = L [ K I ' = M w h i c h p r o v e s Theorem 1 . I n [ 7 I a model lu' i s c o n s t r u c t e d such t h a t f o r a g i v e n L2-sentence Q N k 8 I Y I=:iUD = L r h l = V i .e. N k Q t Y = LT'. Let

h = { y : 2oy+l

L e t ,'I be a c.s.m. f o r Z F C . I n fl B and irOD ='I. T h e r e f o r e [ L n l ) and

We assume t h a t t h e r e a d e r i s f a m i l i a r w i t h c o h e r e n t n o t i o n o f f o r c i n g , cohe-

r e n t and c o n t i n u o u s n o t i o n o f f o r c i n g , E a s t o n - l i k e n o t i o n o f f o r c i n g ( i . e .

INNER MODELS, ORDINAL DEFINABILITY AND THE AXIOM OF POWER SET

m5,pc - d e n s i t y c o n d i t i o n and McAloon's paper

6 n E : CEOn> cofinal w i t h

217

On), and lemmas used i n

C41.

Some theorems connected w i t h the above mentioned n o t i o n s a r e g i v e n i n t h e r e a r e g i v e n t h e i r versions f o r s t r u c t u r e s o f type <M[GI,E.M.C>.

C131.

In

C71

In particular

we w i l l use the f o l l o w i n g f o r c i n g lemmas:

A)

Let

be a c.s.m.

M

f o r ZFC and l e t C = < C . < >

be an E a s t o n - l i k e n o t i o n o f f o r -

k ZFC.

I f G i s C-generic over M then <MCGl,f.M.G>

cing.

I f C s a t i s f i e s the rn C,Oc - d e n s i t y c o n d i t i o n ( i n M ) then

B)

Let

M be a c.s.m.

f o r the Z F - t A C

+

c o l l e c t i o n scheme ( A C denotes here the

p r i n c i p l e o f Zermelo) and l e t C be a continuous n o t i o n o f f o r c i n g .

I.ZF-+AC

r i c over M then <MCGl.E,M,G>

C)

M be a c.s.m. f o r ZFC- + c o l l e c t i o n scheme and C be coherent n o t i o n s o f f o r c i n g ( i n M). L e t G be C e C - g e n e r i c over M

Product lemma f o r classes: L e t

let C

I' 2 Then G = G 1 x G 2 is

I f C i s C-gene-

t c o l l e c t i o n scheme.

.

C2 i s C2-generic over <MEC,l,f.M,G

C1 -generic over

M and G2 i s C2-generic over

> 1

and

1 2 MIClloMCGZI

=M. I f G,

(MCCII.E,M.GI> then G , x C 2 i s

C, e C -generic over M and M C C , l C G 2 1 = M C G l x C 2 1 .

2

A p r o o f s i m i l a r t o t h a t f o r s e t s from [91 i s given i n [151.

Proof of Theorem 4: L e t M be a c.s.m. ~ = O { Ua+):

f o r Z F t V = L . We s h a l l work i n s i d e M.

Let

aeon}

p i s a c o n d i t i o n i f p i s a f u n c t i o n such t h a t dorn(p) 5 U {a} x w a & aE0 ( 6 1 X W ~ E C ~ x~w C( ){) ~ and ) I{,i>Ep: P < A ) I < a A f o r i n a c c e s i b l e

Definition 1 .

(CI0(p wA '

P ( ~ =) { E p : p

5

a)

,

p(a) =

P-P(a)

*

The c l a s s o f c o n d i t i o n s i s ordered by reversed i n c l u s i o n .

If X i s an M-definable c l a s s included i n on]. 3. L e t J be an a r i t h m e t i c a l p a i r o p e r a t i o n

D e f i n i t i o n 2.

dorn(p) C X

x

Definition

0 then

C x = (pEC:

f o r n a t u r a l numbers and J

be the corresponding p a i r operation f o r o r d i n a l s . Then J ' ( a . p ) = J ( a , P ) + 2 f o r

a E 0 & p<w,

and

3v) = {

E ( a ) u i}

0

otherwise

if

y=J'(a,D)

.

S. ROGUSKI and A. ZARACH

2 18

I f X c O i z 1Y-definable and G i s C - g e n e r i c o v e r M , t h e n C = G n C i s Cx X X I= ZFC t cf =c? + G C H .

Lemma 1.

-

- g e n e r i c o v e r M and (M[Cx] .E,M.GX>

I f Y c O i s M - d e f i n a b l e and X n Y = 0 t h e n C @ C zCxuY and M [ G l n M I G y l = L . X Y X For every 5 C s a t i s f i e s the Proof: L e t D be t h e e n u m e r a t i o n o f 0 and m = w X .E i n a c c e s i b l e . By A) and C) Lemna m c J B g and uh+,,h d e n s i t y c o n d i t i o n s i f wh

-

-

-

i s proved. D e f i n i t i o n 4.

Let A={:

E(a)cAl.

B={EA:

and

(UG)(a,D)=O}

By a McAloon's argument we o b t a i n Main Lemma: Remark I .

LCBI I = U ~ , ( ~ , ~has ) a good s u b s e t I f M c N c M [ C l and N ZFC t h e n cfN=cfL and N l=GCH. I t f o l l o w s t h a t EB

c-t

and J ' a r e a b s o l u t e i n i n t e r m e d i a t e submodels. Now l e t S = ( O ) u (a: w = a } and l e t sa be ath element o f S. L e t Sa

F a c t 1.

Sans = 0 f o r a#@ ;a E S

= s a t l- s a .

.

a E S t+ w E S p-, Y Y Y a y D e f i n i t i o n 5. I f a = < a ,a,,...,a > i s a f i n i t e sequence o f o r d i n a l s t h e n f o r e v e r y

iu)=u ,

ordinal Y

P

Y-r={p(y):

yEOn}

++

J'(a.p)ES

P~+~(Y)=J(~~_,,P,(Y))

X+ =

u S

and

YEY;:

o s k < n and P ( Y ) = P ~ + , ( Y ) -

R-r = ( < n , p > E R :

nEX;).

By Remark 1 we o b t a i n Lemma 2 :

~B;

*+ LIB;]

t o check t h a t cp(:,x)

k

uJl(n,D) has a good s u b s e t . Now i t i s n o t d i f f i c u l t

z x E L [ B - ] i s t h e r e q u i r e d f o r m u l a and by

(7) L[R+I /=

V=L':.

Proof of Theorem 5: L e t P be t h e weak p r o d u c t o f On c o p i e s o f Cohen's n o t i o n o f f o r c i n g . P s a t i s f i e s 6.c.c.

and P i s a c o n t i n u o u s n o t i o n o f f o r c i n g ( s e e

a b s o l u t e w.r.t.

ZF-.

I f N is a c.s.m.

I = Z F C - + c f =cf

An,D={aEp

I(Ep:

a < D ) I < uD f o r r e g u l a r o

=> C < w

R n < ua+ I

and

D'

Q i s o r d e r e d by r e v e r s e d i n c l u s i o n . I f X i s an M - d e f i n a b l e c l a s s t h e n PX = { p E Q : d o m ( p ) cX

x

On)

D e f i n i t i o n 2. For p € Q Definition

3. C = Q @ P

p(D)={Ep: a i s not a cardinal"

++

X

I=ZFC-

X

+

M

and

"continuum i s a

aEX.

a+1 I n MCGI we d e f i n e a c l a s s B i n the f o l l o w i n g manner:

D e f i n i t i o n 5. A

={: _

I

UG'(a,p) = y }

, A 2 =(:

B I = { < a , P , y > E A l : H ( a ) s A I u A p } ; B = B I u.4 2 '

O f course L[B1 i s a model f o r ZFC- s i n c e MCGI I=ZFC-. we o b t a i n Main Lemma:

zEB

L e t 6 =H(z).Then

++

UG"(a,n) = O } By standard f o r c i n g arguments

L LCBI ( = " o ~ +i ~ s not a cardinal".

Let be a f i n i t e sequence o f o r d i n a l s . Then for z = < n , P , y > o r D e f i n i t i o n 6. z= zEBz E B & nEX+. L S i m i l a r l y as Main Lemma one can prove t h a t f o r 6 = H ( z ) z E B + LCB-tl I="o6+ 1

if

++

Hence {in: UC"(n,n)

i s not a cardinal".

= O ) : n€X;)

i s a proper c l a s s i n LCB+l l

a

i f follows t h a t LCB+II=V=LT1. + The absoluteness o f t h e above c o n s t r u c t i o n i m p l i e s Theorem 5 i f we p u t B ( a . r ) r

o f subsets o f o. By

(9) from

p r o p e r t i e s o f L"'

I= "every s e t has a Hartogs' number". L I=" w2a+l i s a c a r d i n a l " f o r every a i . e . i n LCB-t] t h e c l a s s o f c a r d i n a l s i s c o f i n a l w i t h On.

E

rELCB;l.

We must check o n l y t h a t LCB;]

This i s e v i d e n t s i n c e LCB;lsMCGl

Proof of

and M C C I

Theorem 6:

D e f i n i t i o n 1. F " = ( p : Func(p) & d o r n ( p ) c O n x w l & r g ( p ) ~ ( O , l ) & Ipl < a l ) P'

is

P'

ordered by reversed i n c l u s i o n . C ' = Q a P p ' . Since Q and P'

a r e ol-closed

Lemma 1. -

be a c . s . ~ . f o r 2 F t V . L

Let

M

I=ZFC-

UCCl=WCCl.E,M.C> proper c l a s s "

+

+

we have

IbL i s an aleph"

5

and l e t G be C I - g e n e r i c over M . Then

"the continuum i s a c o n s t r u c t i b l e s e t "

*+

+ "P(w,) is a

5Envg is limit.

I n MCGl we d e f i n e a c l a s s R l i k e i n the p r o o f o f Theorem 5. L H and i , we r e p l a c e w , by a,. Then we o b t a i n a c l a s s B d e f i -

I n the d e f i n i t i o n o f

nable i n MCGl such t h a t B i s d e f i n a b l e i n LCBI by a formula o f

LZF

w i t h o u t para-

S. ROGUSKI and A. ZARACH

220 meters. Hence

LCBI

[=ZFC-

Hartogs' number"

+

the continuum i s a c o n s t r u c t i b l e s e t "

+ "P(o,)i s

a proper c l a s s "

+

+ "every

s e t has a

Y=HOD.Now l e t c be a Cohen r e a l

over L i E l (Cohen's n o t i o n o f f o r c i n g i s a c o n s t r u c t i b l e s e t so i t i s a s e t i n

LCBI). B is d e f i n a b l e i n L C B l L c l e x a c t l y as i n LCB1. By a Levy's argument C31 t h i s = L [ B I . Df course L [ B l [ c l I=ZFC- + "the continuum i s a s e t " i m p l i e s t h a t HODLcB"cl + "every s e t has a Hartogs' number" + ' I P(o1 i s a proper class" + H O D # V = = L[B.P(w)l and {La[B.P(o)l:a E O n } i s an R-hierarchy o f t h e universe.

References :

[l]

J.Barwise and E.

vo1.8

pp.

F i s c h e r , The Shoenfield absoluteness Lemma, I s r a e l J. Math.

329-339. 7970.

G r i g o r i e f f , Intermediate submodels

[2] S.

and generic extension in set theory,

Annals o f Mathematics ~01.101, No

[3]

A.

3, 447-490. in aziomatic set theory II, H o l l a n d 1970, pp. 129-145.

Levy, Definability

- H i 1 lel, N o r t h

[4]

Jerusalem

1968,

ed. Y.

Bar

K. McAloon, Consistency r e s u l t s about ordinai! definability, Ann. Math. Logic

2 (1971 pp.449-467.

( 5 1 J. M y h i l l and D. S c o t t , Ordinal definabitity, i n Axiomatic s e t theory, P a r t 1. ed. D. S c o t t

[6] S.

AMS

vol. X X I I No 12,

[7] S.

(1971) pp 271-278.

Roguski, Hartogs'

nwnber and uxiom of power set, B u l l . Acad. Pol. S c i . ,

1974,

pp

1191-1192

Roguski, Extensions of models f o r ZFC to models for ZF+V=HOD with applica-

tions, i n Set Theory and Hierarchy Theory, B i e r u t o w i c e 1975, ed. W.Marek, M.Srebrny and A.Zarach,

[8] S.

Springer

J.R.

D.

[lo]

1976, ed. Verlag 1977.

241-248,

A.Lachlan,

Springer V e r l a g

1976.

i n Set Theory and Hierarchy Theory,

M.Srebrny and A.Zarach,

1971 )

pp

LNM

619.

pp

251-256,

P a r t 1, ed.

357-381. 1

2 . Szczepaniak, The consistency of the theory ZF+L #HOD,

LNM

619

pp

285-290,

1977

Z. Szczepaniak, 0 konstruowrlnoSci w jpzykach wyiszego rzpdu, Doctoral Thesis, Wrodaw

[12] 1131

pp.

Shoenfield, ltnramfied forcing, i n Axiomatic s e t theory. S c o t t AMS f

Springer Verlag

1111

537

Roguski, The theory of the class HOD,

Bierutowice

[9]

LNM

1977.

P.Vopenka and P. Hajek, The theory

of semisets, North-Holland 1972.

A. Zarach, Forcing with proper classes, Fund. Math.

LXXXl

(1973) pp 1-27. [ 141 A. Zarach, More constructively about the constructibility, LNM 619 pp 329-358, Springer Verlag 1977. [15] A. Zarach,Lemat produktowy d l a klas, Raporty I n s t . Matem.Polit.WrocZ. 1977.

Logic Collcquiwn '77

A. MacinLyre, I;. Pacholaki. J. Paris (edn.) Worth-Holland PubEishing Conpang, 1978

AND COMPARABILITY GRAPHS

S~-CATEGORICITY

James H . Schmerl (1) Department o f Mathematics U n i v e r s i t y o f Connecticut S t o r r s , Connecticut 06268

In t h e paper")

which we p r e s e n t e d a t t h e WrocZaw conference, w e c o n j e c t u r e d

(Conjecture 3.4 o f [ 5 ] ) t h a t f o r any countable,

( A , < ) there is a l i n e a r order

set

(A, 2

(A,

i s a s t r o n g path

We w i l l give a proof by induction

i s even and t h a t t h e lemma is t r u e f o r a l l even

. ..,a m-1 )

i s a s t r o n g path.

- i i s odd,

of

WO-CATEGORICITY AND COMPARABILITY GRAPHS i s not a s t r o n g pseudo-cycle.

k (iv) j 7 k

For each two nuclei c H, b z H, b = A {vbk ua: a c F, b = A { wb : a c H , b

= \/{ubhva: a = A { wb : a

j , k on the frame H,

= j(a)), = j(a)), = j(a)), = j(a)> k(a)).

Given a frame H how do we determine the assembly N H , in A few specific answers are particular for a space S what is NOS? known here, but not much. At the moment the only known general result is the following, which is due to Macnab. 8. THEOREM. For each finite frame H the assembly 1.IH is the boolean closure of H.

The assembly of a frame is itself a frame, so this too has an This second assembly gives us a third assembly, which in assembly. In other words each frame H gives turn gives a fourth, and so on. us a transfinite tower of assemblies

( F o r each limit ordinal a the assembly NaE is defined as the limit of the earlier assemblies, where this limit is computed in the category of frames, and this is not the same as the set theoretical limit. ) Isle check easily that the canonical enbedding H EIH is an Thus the above tower stops isomorphism exactly when H is boolean. if and when a boolean frame is reached. Elow such a tower sits on top of each topology, so what is the significance of the length of this tower? It is by no means clear which properties of a space determine the 1enp;th of its tower but it seems to be something to do with the pathclogy of the space (with shorter towers corresponding to more pathology). +

9. TJiEOREI.!. ( a ) For each To space S the assembly NOS is boolean if and only if S is scattered.

244

H. SIMMONS

(b)

There are spaces S such that !J20S is boolean but NOS is

not. (c) Let If be the usual topology of the reals. ordinal & is MUH boolean.

Then f o r no

Beazer and Macnab C13 have characterized those frames H for which MH is koolean, and 9(a) is a topological version of this result. The observation 1O(c) is due to Isbell. There are frames which are not topologies, nevertheless each frame has a canonically associated space (provided we alloh, the empty space). The construction of this space is analogous to the construction of the stone space of a distributive lattice. 3 Ey a point of a frame H we mean a frame rr.orghisnp : I! (where 3 is the two element frame). Let IlFi be the s e t of points of I ? , and 'for each x c H , l e t

-.

The proof of the followinr can be extracted from C 3 ; Appendix1 and C 7 ; pages 7 and 81. 10. THT0RF:E.l. For each frane I! the se.t t o ? o l o p y on T[H. With this topology the nap surjective frame norphism. The space IIH is 0 s there is a ,space S and frane morphisci I I S IIti sucli thzt t n e induced diaCram +

-+

F!

-

I!}

is a r l : I! + O!lH is a sober, and for each unioup continuous map

(q(x) : x

E

0s

commutes. We call lIIf the point-ssace of H. Xotice that f o r each space the point-space of 0 s is not necessarily S. In f a c t IIOS is the sober version 1 :; of S . Consider now tiie point-spaces of a f r a m an:; its assembly. It turns out that these two s?aces have exactly the same ? o i n t s h i t 2ax-y .'.jfivre.ntt 0 - n ; ~ p i - s . S

A FRAMEWORK FOR TOPOLOGY

245

f-cqcrnber no:< t h a t f c r each s p i c e S t h e f r o n t s p a c e FS i s t h e set S t o p o l o r i z e d by t h e s m a l l e s t t o p o l o p y i n which e a c h S-open s e t

i s F'S-clo?en. F o r e a c h f r a m e H t h e n a p n : Fllf!

l?. ?l!EO:<Ek$. homeomrr h i sm.

+

lI!lE

is a

This c o n s t r u c t i o n g i v e s us sone h o l d on the f i r s t and s e c o n d a s s e n b l i e s of a f r a m e . F o r l e t S S e t h e p o i n t - s p a c e of t h e frame E s o , by 1 2 , t h e p o i n t - s p a c e s o f WH and Ii'H t h e c a n o n i c a l diagram

OS

a r e FS and F'S.

Ploreover

- OFS

OF'S

commutes. Unfortunately t h e obvious e x t e n s i o n of t h i s diagram a c h i e v e s n o t h i n g s i n c e , for e a c h s p a c e 9 , F 2 S F'S. In other w o r d s , for e a c h o r d i n a l a 1. 2 t h e p o i n t s p a c e of ?iaE i s F'S. Suppose now w e s t a r t f r o m a t o p o l o g y [I = 0 s . Ve Itnow t h a t IIFI is j u s t t h e s c b e r v e r s i o l : Z S of S and so II?!!? i s j u s t Y E S . Thus, as a b o v e , w e h a v e a commuting d i a p r a m

NOW

he s p a c e FZS is n o t q u i t e t h e same a s FS, b u t is c

nely

r e l a t e d t o TS. I n f a c t ' w e c a n modify t h o a b o v e c o n s t r u c t i o n t o obtain the following r e s u l t . 1 3 . T:IEOREY.

morphrsm

0 :

I!OS

+

For each s p a c e S t h e r e i s a unique frame OFS s u c h t h a t

246

H. SIMMONS

commutes.

For e a c h j

c

NOS,

o ( j ) = U(j(U)

- U:

U

c

OFS)

and o ( j ) i s t h e l e a s t F-open s e t A s u c h t h a t j

5

[A].

T h i s n o r p h i s m u e n a b l e s u s t o d e t e r m i n e t h e r e R u l a r and t h e b o o l e a n n u c l e i on 0 s . For each nucleus j of t h e space S , j i s r e g u l a r

14. THEOREM.

e x a c t l y when j = C o ( j ) l and o ( j ) is F - r e g u l a r o p e n , and j i s b o o l e a n e x a c t l y when j = C o ( j ) l and o ( j ) i s r - c l o p e n . This nethod

dl50

gives u s a c h a r a c t e r i z a t i o n of t h e canonical

b o o l e a n i z a t i o n of NOS, which l e a d s t o a p m e r a l i z a t i o n of [ 3 ; Theorem 6 1 . T h i s g e n e r a l i z a t i o n was o b t a i n e d o r i g i n a l l y by Macnab u s i n g d i f f e r e n t methods. 1 5 . THEOREX. morphism R o : RNOS

-*

For each s p a c e S t h e r e i s a u n i q u e f r a m e ROES s u c h t h a t

NOS

OFS commutes.

-

-I

?.l!OS

R' KOFS

I t o r e o v e r Eo i s a n isomorphism.

1 6 . COROLLARY. For e a c h TD s p a c e S t h e b o o l e a n a l g e b r a HIJOS i s c a n o r i i c a l l y i s o m o r p h i c .to t h e power s e t o f S.

L e t u s look now a t t h e s t r u c t u r e t h e o r y o f frames.

Our e x p e r i e n c e w i t h t o p o l o g i e s t e a c h e s u s t h e i m p o r t a n c e of s e p a r a t i o n p r o p e r t i e s , and t h a t a a n y s u c h p r o p e r t i e s c a n be s t a t e d Thus we c a n e x p e c t p a r t l y o r wholely i n l a t t i c e t h e c r e t i c terms. t h a t such p r o p e r t i e s w i l l tic? r e l e v a n t h e r e . h!e c o n s i d e r four of these properties.

A FRAMEWORK FOR TOPOLOGY

247

A frame i l i s ( i ) b.g. (boolean g e n e r a t e d ) , or ( i i ) r e g u l a r , or ( i i i ) f i t , or ( i v ) c o n j u n c t i v e , if f o r e a c h a , b c fi w i t h a $ h , t h e r e a r e x , y ( i ) a v x = 1, h v x C 1, x c B!!, 1 6 . DEr"INITI0tJ.

or ( i i ) a v x = 1, y $ b , x or ( i i i ) a v x = 1, y $ b , X or ( i v ) a v x = 1, b v x # 1, respectively.

t!

E

with

~ =y0 , A Y 5

h,

-

I t i s e a s y t o check t h a t h . c . * r e g u l a r , r e E u l a r * f i t , fit c o n j u n c t i v e , and t h a t t h e s e i m p l i c a t i o n s a r e n o t r e v e r s i h l e (even f o r t o p o l o g i e s ) . !Je c o n s i d e r e a c h o f t h e s e p r o p e r t i e s i n turn. The f o l l o w i n g lemma i s v e r i f i e d e a s i l y . 1 7 . LEKXA. are equivalent.

F o r e a c h e l e m e n t a of t h e f r a m e H t h e f o l l o w i n g

( i ) For e a c h b a v x = 1 , b v x C 1.

c

H w i t h a $ b , t h e r e i s some x e BH w i t h

(ii) a = V { y c BH: y s a). T h i s lemma e x p l a i n s t h e c h o i c e of t h e term ' b o o l e a n E e n e r a t e d ' and shows t h a t a t o p o l o g y i s b.6. clopen sets.

10. COROLLARY.

i f and o n l y i f i t h a s a b a s e of

The a s s e m b l y o f e a c h f r a m e i s b . ~ .

R e g u l a r i t y fcr f r a m e s i s t h e o b v i o u s c e n e r a i i z a t i o n of regularity f o r spaces. I n p a r t i c u l a r a space S i s recrular ( i n t h e t o p o l o g i c a l s e n s e ) i f and o n l y i f OS i s r e g u l a r ( i n t h e above S e n s e ) . A t tP.e coment i t i s n o t c l e a r what the r e c u l a r i t y o f a frame e n t a i l s .

It i s i r x l u d e d h e r e s i m p l y as

3n

i n t e r m e d i a t e s t e p between b.g.

and

fitness. F i t n e s s was i n t r o d u c e d hy Isbell i n C 7 1 , however t h e i m p o r t a n c e o f f i t n e s s was d i s c o v e r e d by r'acr!ah.

Tc e x p l a i n t h i s

we

need a

s l i g h t detour. Y.e ' s a y a n u c l e u s j on a f r a m e ii a d n i t s a n e l e m e n t x of E i f j ( x ) = 1, and r.ore g e n e r a l l y j a d n i t s

subset

x

o f !! i f j C X 1 =

H. SIMMONS

248

The f i l t e r

i s c a l l e d t h e a d m i s s i b l e f i l t e r of j .

!de s a y t w o n u c l e i j , k o n H

a r e c o m p a n i o n s , a n d w r i t e j - k , i f t h e y a d m i t e x a c t l y t h e same

e l e m e n t s o f ii.

T h i s p i v e s a n e q u i v a l e n c e r e l a t i o n o n NH a n d

r?

b l o c k i s , b y d e f i n i t i o n , a n e q u i v a l e n c e c l a s s of t h i s r e l a t i o n . C l e a r l y e a c h b l o c k i s c l o s e d u n d e r A , and so e a c h h l o c k h a s a minimum n em h er . 1 9 . TEEORCM.

S u c h minimum n u c l e i c a n b e d e s c r i b e d e x p l i c i t l y .

Let A h e a s u b s e t of t h e frame )I a n d l e t

f = A {ua: a

c

A),

c

= V {va: a

A).

Then g i s t h e minimum n u c l e u s w h i c h a d m i t s A a n d f

g*.

The f o l l o w i n g d e f i n i t i o n r e f o r n u l a t e s I s b e l l ' s n o t i o r ! of a fitted part. 20. D C F I I I I T I O M . A n u c l e u s j o n a frame if i s f i t t e d i f t h e w i s some s u b s e t A o f ii s u c h t h a t j = V { v a : a 6 A ) .

P u t t i n g t o g e t h e r lc), 20 we s e e t h a t a n u c l e u s i s f i t t e d i f a n d o n l y i f it i s t h e minimum mem!ler of i t s 1)locL. I n g e n e r c i l t h e o r d e r i n q of n u c l e i i s n o t d c t s r m i n e d by t h e o r d e r i n g of t h e i r a d m i s s i b l e f i l t e r s . Eowever for f i t t e d n u c l e i t h i s i s so. 21.

LE!;Mtl.

Let j ,k b e n u c l e i on t h e f r a n e H w i t h j f i t t e d .

Then

w he r e VCj), V(k) a r e t h e c o r r e s p o n d i n g a d m i s s i b l e f i l t e r s . T h i s l e a d s u s ,to t h e c h a r a c t e r i z a t i o n of f i t f r a m e s . a l l we h a v e a l o c a l v e r s i c n . 2 2 . LEMMA. are e c u i v a l e r i t .

F i r s t of

F o r e a c h e l e n e n t a of t h e f r a n e H t h e f o l l o w i n g

( i ) The n u c l e u s u i s f i t t e d . ( i i ) For e a c h b c ii t c i t h $ Li t h e r e i s some x u a ( x ) = 1, W b ( X ) f 1.

c 1:

with

A FRAMEWORK

( i i i ) For e a c h b X A y i b.

c Ii

FOR TOPOLOGY

w i t h a $ b t h e r e are x , y

249

E

ti w i t h a v x

= 1,

y $ b,

The f o l l o w i n g g l o b a l v e r s i o n i s d u e t o Macnab.

For e a c h frame H t h e f o l l o w i n g a r e e q u i v a l e n t . (i) H is f i t . ( i i ) Each n u c l e u s o n H i s a l o n e i n i t s b l o c k (or f i t t e d ) . ( i i i ) Each u - n u c l e u s o n H i s a l o n e i n i t s b l o c k (or f i t t e d ) . 23. THEOREi1.

T h i s t h e o r e m shows t h e i m p o r t a n c e of f i t n e s s , f o r i f a f r a m e H

i s f i t t h e n e a c h n u c l e u s o n 11 is d e t e r m i n e d s o l e l y by i t s a d m i s s i b l e f i l t e r a n d so NFi i s i s o m o r p h i c ( a s a p a r t i a l l y o r d e r e d s e t ) t o t h e s e t o f a d n i s s i b l e f i l t e r s o f H. T h u s vie c a n e x p e c t t h a t t h e s t r u c t u r e t h e o r y f o r f i t frames w i l l b e c o n s i d e r a b l y s i m p l e r t h a n t h e s t r u c t u r e t h e o r y for frames i n general. W e h a v e j u s t s e e n t h a t f i t n e s s i s c o n c e r n e d w i t h minimum members o f b l o c k s .

a b o u t t h e e x i s t e n c e an d p r o p e r t i e s of

!:'hat

maximum members of b l o c k s ?

Here t h e s i t u a t i o n i s n o t so well

u n d e r s t o o d b u t we do h a v e t h e f o l l o w i n g two r e s u l t s ( b o t h d u e t o Macnab)

.

24. LLNM.4.

25. 'rEEORE;.I.

Each w - n u c l e u s o n a f r a e e i s naximum i n i t s b l o c k .

L e t F! b e a frame w i t h ACC.

Th en e a c h b l o c k of

H has a maximun member.

A p p a r e n t l y w e h a v e d e p a r t e d from o u r p l a n of l o o k i n g a t t h e s e p a r a t i o n p r o p e r t i e s of 1 6 , i n p a r t i c u l a r w e h a v e n o t y e t Eonsidered conjunctivity.

Iiowever, all i s

i d e l l ,

s i.nce i t t u r n s

cut' t h a t c o n j u c c t i v i t y i s c o n c e r n e i l w i t h (.jnor.:r o t h e r t h i n F s ) t h e maximality c f c e r t a i n nuclei. I t is a n e a s y e x e r c i s e t o s??ow t h a t A s p a c e S h a s a c o n j u n c t i v e t o p o l o g y i f a n d o n l y if f o r cLich 2 t S an\l L' E 0 s w i t h p

F

L', t h e r e i s some q

space.:

-.

5 w i t h q - 5 ! I n ;-

~n p a r t i c i l l a r for It c o n j a n c t i v i t y i s w e a k e r tiiar; TI an:! i n d e y n d u r i t o f T!.,. E

c a n .be shoi.in t h a t f c r n a n y t o p o l o e i c a l c u r y o s c s t h e T1 ? r c p e r t y c a n b e re?.Lace;; h:: t > , t > i:eake!- mr, t c o n j u r . c t i v 0 ~ V : ; : P ? - % : ' . din ah.ussior. of t h i s car. b e f o u r d i n Cl01 a r d C1?1.

4 fiil1t.r

For frarr.es c o n j u n c t i v i t y ( w k i c ? . i s termed s u b f i t n e s s t:y I s h e l l ) i s closely r:on:rncted wi!h b l o c k s z r u c t u r c , a s t h e .fcllcw?-p.q r e s u l t of ::acna:-. ' s r;:?ows.

n.

250 ^ ^

TtiF0P.EI.I.

SIMMONS

F o r e a c h f r a m e ii t h e f o l l o w i n g are e q u i v a l e n t .

H is conjunctive. The z e r o rkucleus i s alone :? i t s b l o c k . F o r e a c h two n u c l e i j ,l: e n F, j-k

* j**

L,* *

.

Each r e u u l a r ( o r b o o l e a n ) n u c l e u s o n II i s n a x i x u n i n i t s Each u - n u c l e u s o n t? i s maximum i n i t s b l o c k.. Each v - n u c l e u s o n I: is a l n n e

(01%

naximurr.) i r , i t s t1Qc)c.

L o o k i n g d t ?3, 2 4 , 26 and n o t i n c t!iat e a c h v - n u c l e u s i s f i t t e d s u g g e s t s t h a t we s h o u l d c h a r a c t e r i z e t h o s e f r a m e s f o r w h i c h e a c h This characterization w - nu cl eu s i s a l o n e i n i t s t l o c k ( o r f i t t e d ) . i s a g a i n ' p r o v i d e d b y >!acnab, a n d a q a i n it i s a s e p a r a t i o n p r o p e r t y . T h i s s e p a r a t i o n p r o p e r t y i s w e a k e r t h a n f i t n e s s an d i n c o m p a r a b l e with conjunctivity. F o r e a c h frame I! t h e f o l l o w i n g are e q u i v a l e n t . ( i ) F o r e a c h a , b c H w i t h a 1: b t h e r e i s some x c H w i t h w a (x) = 1, W b ( X ) f 1. ( i i ) Each 14-nucleus o n t: i s a l o n e i n i t s b l o c k ( o r f i t t e d ) . 27. TBEORI'W.

A t t h i s p o i n t I m u s t e n d my t a l e , b u t o f c o u r s e it i s n o t t h e e nd of t h e s t o r y . Many t o p i c s h a v e n o t S e e n d i s c u s s e d h em ?, a n d

t h e r e a r e many p r o b l e m s s t i l l t o b e s o l v e d .

For f u r t h e r d B t a i l s

of t h e s u b j e c t t h e r e a d e r s h o u l d c o n s u l t C13, C 3 3 , C 4 3 , C51, [ E l , C 9 1 a n d i n p a r t i c u l a r C71 w h i c h c o n t a i n s a w e a l t h of m a t e r i a l . The r e a d e r s h o u l d b e w a r n e d t h a t , b e c a u s e of d i f f e r e n c e s i n n o t a t i o n , t e r m i n o l o g y a n d a p p r o a c h , it i s sometimes v e r y d i f f i c u l t t o c o m p ar e t h e work of d i f f e r e n t a u t h o r s ( a n d r e a l i z e t h e y a r e t a l k i n F a b o u t t h e sane t h i n g ) . F o r t h i s r e a s o n I hope t h a t a f u l l a nd c o h e r e n t a c c o u n t of t h e s u b j e c t w i l l b e a v a i l a b l e i n t h e n e a r future.

A FRAMEWORK FOR TOPOLOGY

25 I

REFERENCES

c11 R. Beazer a n d D.S.

Kacnab; Modal e x t e n s i o n s o f h e y t i n g a l g e b r a s , t o a p p e a r i n C o l l o q u i u m Yath.

C21

J . Benabou; C.

C 31

T r e i l l i s lo c a u x e t p a r a t o p o l o g i e s , SBminaire

Ehresmann, 1 9 5 7 / 5 8 I'ac.

C.I!.

Dowker and D.

d e s S c i e n c e s d e P a r i s , 1959.

Papert;

Q u o t i e n t f r a m e s and s u b s p a c e s ,

P r o c . London Math. Soc. ( 3 ) 1 E ( 1 9 6 6 ) 275-296.

C43

C.H.

Dowker and D.

f r a m e s , pp.

Papert Strauss;

S e p a r a t i o n axioms f o r

223-240 of T o p i c s i n T o p o l o g y Ed. by

A

Cs&z&,

North Holland (1972). C51

C.ti.

Dowker and D. S t r a u s s ; P r o d u c t s a n d sums i n t h e c a t e g o r y of f r a m e s , pp. 208-219 o f CateRorical T o p o l o g y , S p r i n g e r L e c t u r e n o t e s i n P l a t h e m a t i c s v o l . 540.

C61

C. Ehresmann; G a t t u n g e n von l o k a l e n S t r u k t u r e n , J b e r . D e u t s c h . Math.-Verein. 6 0 ( 1 9 5 7 / 5 8 ) 59-77.

C71

J.R. I s b e l l ; 5-32.

C81

D.S. Macnab; An a l g e b r a i c s t u d y o f modal o p e r a t o r s on h e y t i n l ! a l g e b r a s w i t h a p p l i c a t i o n s t o t o p o l o g y and s h e a f i f i c a t b o n ,

Atomless p a r t s of s p a c e s , Math.

Scand. 3 1 ( 1 9 7 2 )

T h e s i s , Aberdeen, (1976). ; Modal o p e r a t o r s on h e y t i n g a l g e b r a s , S u b m i t t e d

f9l

for p u b l i c a t i o n . c 1 0 1 I). S. Macnab and H. Simmons ; submitted f o r publication.

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c111 S. P a p e r t ;

An a b s t r a c t t h e o r y o f t o p o l o g i c a l s u b s p a c e s , ? r o c . Camb. P h i l . SOC. 60 (19611) 1 0 7 - 2 0 3 .

c 1 2 1 H. Simmons; The l a t t i c e t h e o r e t i c p a r t of t o p o l o g i c a l . s e p a r a t i o n p r o p e r t i e s , t o a p p e a r i n Sroc. W i n b u r g h >lath. SOC.

Logic ColLoquium '7: A . ;la c in ty r e , L . Pacholski, J. Paris Icds. I 0 A'orth-Iio1hif1 Picblishing Company, 1978

BETH'S THEOREM AKD SELF-REFERENTIAL SENTENCES

C. Smorygski

It is well-known t h a t GBdel's s e l f - r e f e r e n t i a l sentence i s Henkin's s e l f - r e f e r e n t i a l sentence is provable, hence equivalent t o any theorem. When Macintyre and Simmons 1973 gave a s i m i l a r e x p l i c i t d e f i n i t i o n of LBb's s e l f - r e f e r e n t i a l sentence, i t became c l e a r t h a t 0.

Introduction,

equivalent t o consistency,

something g e n e r a l was happening,

Two groups, one Dutch and one I t a l i a n , The purpose

conjectured as much and eventually proved a theorem t o that e f f e c t . of t h e p r e s e n t note is t o give a new proof of t h i s theorem.

The Theorem is most conveniently stated i n t h e n o t a t i o n of modal logic and

is, perhaps, b e s t viewed as a theorem about modal l o g i c . consider is c a l l e d language of

&

Propositions:

Truth

E?ES.

& (

c o n s i s t s of

aigms. P, Ti L

9, r r

n.

M O ~ operator. ~L

Vi 3 ,

.,pn),

Sentences are denoted by c a p i t a l Roman l e t t e r s A , B , C: o r , A(pl,..

The axioms and rules of

w e wish t o emphasize some atoms occurring i n A.

A_xigs.

Al.

The

...

g p ~ ~ i c1, ~ At .

Prcpositioml

The modal system w e

f o r LBb's Theorem ) and is defined as followst

Khes.

Tautologies

UA A O ( A 3 B ) + D B A3. DA3nOA A2.

Rl.

A , A--)B

R2.

A

/

aA

/

when

& are

B

o(nA+A)+aA.

A4.

The choice of axioms and r u l e s is best explained by t h e i r breakdown i n t o categories:

t h e l o g i c a l ones ( A l , Rl ), t h e d e r i v a b i l i t y conditions ( A2, A3,

A2 ) , and t h e formalized Lbb's Theorem ( A 4 ), be a c o n s i s t e n t r.e. extension of Let

'68@( = p r i m i t i v e

a r i t h m e t i c formulated with q u a n t i f i e r s ), l e t 'Q' sentence

q , and

l e t P r p ( . ) be i t s "canonical" proof-predicate.

i n t e r p r e t a t i o n s of

qnt o atoms

.

p

&

in

..,,p, 1'

7t

recursive

be t h e code of a n arithmetical

W e can d e f i n e

ql,..., vl,. ..,q )of any sentence

Each assignment of arithmetical sentences

y i e l d s a n i n t e r p r e t a t i o n A(

A(pl,, , , p ) constructed from t h e s e atoms i n t h e obvious way:

i.

A(yl,..., q n ) = 'pi. T ( A ) , then A( yf,...,

If A(P l , . . . , ~ n ) = pi, then

ii-iii.

iv-viii.

I f A(pll...,pn)

=

P r o p o s i t i o n a l OpeAtOKS are preserved.

253

vn)

is 0=0 ( 0=1 ) .

C. SMORYNSKI

254

Hence anything we prove about a modal sentence i n

&

w i l l i n t e r p r e t as a

metatheorem about a l l of its i n s t a n c e s i n r . Self-reference1

The c l a s s of s e l f - r e f e r e n t i a l sentences we consider i n t h i s

paper are those t h a t correspond t o fixed-points of a p p r o p r i a t e modal functions. To explain t h i s , note first that t h e sentence

'4

constructed by d i a g o n a l i z a t i o n

to satisfy,

q- q('Q' ),

occurs only as a

code i n

and n o t a5 a subformula.

The corresponding condition

on a fixed-point equation, Pt)A(P) s is t h a t p occur only i n s i d e t h e scopes of boxes (

) i n A(p).

It is t h e s e

fixed-points p t h a t we shall be i n t e r e s t e d i n . We can now s t a t e t h e Theorem we wish t o prove: THWREM.

L e t A(p,ql,,..,qn)

be a modal sentence with only t h e atoms shown and

such that a l l occurrences of p l i e w i t h i n t h e scopes of boxes. sentence B(ql,. i.

ti.

.., q

There is a

) with only t h e atoms shown and such t h a t

Lt- 0LP-(P,q1s...

,qn)l+Cl(p-)

&k Bt)A(B,q l,...,qn).

P a r t i is not t h e a u t h o r ' s f a v o r i t e phrasing of uniqueness, b u t i t has t h e v i r t u e of being statable without f u r t h e r notation. The Theorem was independently conjectured by G. Sambin and t h e a u t h o r , had a

1975 ) and t h e 1975, t o appear i n Smoryrfski A ), and was f i n a l l y proven independently by D.H.J. de Jongh ( announced i n SmoryLski 19121 and again i n Smoryiski 1972 ) and Sambin ( Sambin 1976 ).

s p e c i a l case s e t t l e d independently by C. Bernaxdi ( Bernardi author ( announced i n SmoryLski

In S e c t i o n 1 w e s h a l l prove Beth's Theorem f o r t h e I n t e r p o l a t i o n Theorem f o r usual argument,

&

&.

Actually, we s h a l l prove

and then o b t a i n Beth's Theorem from it by t h e

Our proof of t h e I n t e r p o l a t i o n Theorem is model-theoretic and is

based on R. Solovay's proof ( Solovay

with r e s p e c t t o Kripke models.

1976 ) of t h e Completeness Theorem f o r 2

This is followed i n S e c t i o n 2 by d e Jongh's proof

of t h e e x i s t e n c e and uniqueness r e s u l t , from which we quickly conclude t h e Theorem,

We hasten t o emphasize t h a t o u r proof is a pure e x i s t e n c e proof and does n o t r e s u l t i n a n a c t u a l e x p l i c i t d e f i n i t i o n of t h e fixed-point,

Moreover, u n l e s s we

discount t h e e f f o r t involved i n e s t a b l i s h i n g Beth's Theorem, o u r proof is no simpler than a d i r e c t proof which e x h i b i t s t h e fixed-points e x p l i c i t l y .

What o u r

proof does do is g i v e a t r i v i a l l o g i c a l explanation f o r t h e e x p l i c i t d e f i n a b i l i t y of fixed-points,

For information on t h e computation of fixed-points, s e e o u r survey Smorynski A .

1.

Beth's Theorem.

Definition.

The basic t o o l of modal l o g i c is t h e Kripke modeli

A Kripke model is a t r i p l e (X,.

3.1

THEOREM.

L

compact f o r

There i s a c.u.b.

.

E.L e t (Pi)irl

(111 =

v2. over J by t h e formula such t h a t D

iP

N o ) be an

yti

i s A--compact

6 Ja

f c(f,);

iP

6

9=

F(i,p).

yi

.

G ( v ) < k as 111e1(,

I$cK

and

F(i,p)s

Suppose X #

We s h a l l show t h a t i f ateC then J6

rf&. We

shall f i n d X o s X ,

thus p r o v i n g t h e d e s i r e d r e s u l t .

and parameter pbJ,

hence, sincea(QCO. X = D

C(

P

and, by

i s a l i m i t o r d i n a l and p C J d

) n

v n

E

m')

E

*Ris

f o r a l l non-

constant f ( x ) E M x l , then t h e sub-structure of * R w i t h domain I- Bf(B) + a : f ( x ) E M x l , n E E+ , a E MI, i s c l e a r l y a -ring containing M.

z

This motivates the following: Lemma 3. I

L e t M be a countable E - r i n g .

a

i;

a < a + I holds f o r no a E M.

M c * R ( & * C ) and a E *R be such t h a t

Then t h e r e i s B

f(B) i s i n f i n i t e f o r a l l non-constant f (x)

E

dxl.

E

* R , s.th.a

i;

B < a + I and

288

A.J.

WILKIE

Proof -

Let

a

5

B

i be

the a l g e b r a i c c l o s u r e of M i n

< a + I and 16

since

- 0)

i s countable.

where 0 # a.

E

ao, (6

c

*R s.th.

This i s c l e a r l y p o s s i b l e

i.

Then i f f ( x ) c Mcxl has degree n 2 I we have

-

el,...,

M, and

multiplicities (clearly some a l c M.

"61 , and pick B

i s i n f i n i t e s i m a l f o r "0 0 c

On E M a r e a l l r o o t s of f , counting

is a l g e b r a i c a l l y c l o s e d ) .

Note t h a t

Now i f f(B) were f i n i t e , (*) implies t h a t a l l of

- e l ) , ...,

(6

- en)

-

Since M i s a E - r i n g and "ao

E

I=

To.

In p a r t i c u l a r , a.

Z we may p i c k b

and deduce from t h i s i n e q u a l i t y t h a t 18 E

Bi

--

for

0'

would be f i n i t e s i n c e none of them a r e i n f i n i t e s i m a l

by the choice of B and t h e f a c t t h a t M n 3 N E w s . t h . X IB O i l 5 N. Thus i=1

t h a t there is some a

f

i=l

M s.th. a

5

- bl

E

M, r

a < a + I (since a

5

E

s.th.

P

is finite.

E

and

* naOb + r

But t h i s c l e a r l y implies

6 < a + 1)

c o n t r a d i c t i n g t h e lemma hypotheses, and t h e r e f o r e showing t h a t f ( 6 ) i s i n f i n i t e which proves the lemma.

0 Lemma 3.2 For any E - r i n g M, t h e r e is an M*

I.

s. th. M

cM*.

Proof Let M' G *R be any countable Z - r i n g and a f i e l d of f r a c t i o n s o f ) MI.

E

*R be a l g e b r a i c over (the

Then by lemma 3.1 and t h e comments immediately

p r e c e d i n g it we can f i n d a countable containing an element b s . t h . a

5

E -ring M" s. th. M'

b < a + 1.

C M"

b *R

Since we may suppose M countable

the lenuua now follows from lemma 2 . 2 and t h e f a c t t h a t *R i s r e a l closed using t h e obvious union of chains argument (noting t h a t t h e c l a s s of E -rings is closed under unions of ascending chains). Theorem 3.3 L e t F(xl

,...,x" )

some model of .I

E

n x ,

+

( i . e . F(x)

,...,xnl

(n

E

z').

Then F(Z)

-

L! 0 has a s o l u t i o n i n

0 is n o t provable from Io) i f f t h e r e is a prime

SOME RESULTS AND PROBLEMS ON WEAK SYSTEMS OF ARITHMETIC idea1,j

, of ZCGl

( a ) F(;) (b)

9,

E

zc;1

and

-

can be d i s c r e t e l y ordered

EE’, 3 k

\Jd

(c)

s. th. :

,,...,kn

E

i . e . can be expanded t o a model of To, and

s.th. g(k

P

,,...,kn)

0 (mod d) V g ( n

Proof -

E

3.

,...,

+

an) E M”. 3 : Suppose t h a t M + I o and M F F ( a t ,...,a ) = 0, a = ( a , -+ L e t 9 = {g(G) E E C ; ] : PlF g(a) = 0). Then i s c l e a r l y an i d e a l of E“‘x1

!

., 0, 1 , 2 , ...) t o t h e ...,an, and hence i s an i n t e g r a l donain - so 9

i s isomorphic (under just +,

and

generated by a t , ideal

-

and i s d i s c r e t e l y orderable.

...,b

Also F(;)

...,kn

f

Z.+ and

choose b l ,

1

5

i

This i s p o s s i b l e s i n c e M i s a E - r i n g .

5

n.

MI= g(dbl + r l ,

...,dbn

d i s c r e t e l y ordered.

E

M, k t ,

f

5) .

E

d

E

+ r ) = 0 so t h a t g ( r l , n

subring of M

i s a prime

To show (c) l e t

s . t h . a. = dbi + ri f o r Now i f g

...,rn) 5

E

9 , then

0 (mod d) since M i s

+ : Now suppose 4 i s a prime i d e a l of z c x l having p r o p e r t i e s ( a ) , (b) and ( c ) . Using (b), d i s c r e t e l y order the i n t e g r a l domain = M, say. Let K be the

+

ordered f i e l d of f r a c t i o n s of M.

We s h a l l c o n s t r u c t a

E

- r i n g Mo s . t h .

This i s c l e a r l y s u f f i c i e n t t o prove t h e theorem by lemma 3.2 and

M G M0 C-K . (a).

3

To t h i s end we f i r s t observe t h a t

i s Noetherian) so t h a t f o r each prime p c ?Z+ t h e r e a r e

...,

g(ff\’), ff:)) = 0 \lg E .y Op c o n s t r u c t i o n (see e.g. C I Ip.!+~ ) . Now f o r k

E + ,k

E

2 2 and 1 5

i

-

+

i s f i n i t e l y generated (since Z c x l

ip’, ..., a 2 ) c

0 s.th. P t h i s follows from (c) and a standard

Say a:‘’

ff

= (see

Section 2 ) .

denote t h e l e a s t non-negative

s o l u t i o n f o r x E Z of t h e simultaneous congruences: x E a

(PI)

k

i,kl-l

m

n

(mod p I

k p j j (kj E

z+,

1 S j 5 m) i s the prime f a c t o r i z a t i o n of k. j=t li s o l u t i o n e x i s t s by t h e Chinese Remainder Theorem. C l e a r l y r i ( p ) = a!’)

where k =

any prime power pa. from which i t follows t h a t

1.2-1

Such a for

29 0

A.J.

Jk

r n ( k ) ) E 0 (mod k)

g ( r l ( k ) #...,

WILKIE

Z+ and Vg

E

c o n s i s t e n t l y define the r e l a t i o n Div on

2x

z[z1 s . t h . 8

= f +

Div (k.8)

@ 3f

f(rl(k),

...,r n ( k ) ) E 0 %

We now l e t

c

.

3

E

We may t h e r e f o r e

M by

3

and

(mod k ) .

8 {E

have domain

:k

E

iZ+, 8

- we

E

M, Div (k,8)) with t h e ordering

%

induced by K.

Clearly M G % (as a s e t )

and

For t h i s , f i r s t observe t h a t our c o n s t r u c t i o n has ensured t h a t

(of K).

ri(kl)

kllk2

NOW suppose 6.B'

8 = f +

9 , B'

k' I f ' ( r l ( k ' )

f

E

= f' +

(mod k l ) f o r 1

rick2)

M, k,k'

E

5

so kk' I f ( r l ( k k ' )

Similarly

,..., r n ( k k ' ) )

I

kk' k'f ( r l (kk')

n.

5

,..., r n ( k k ' ) ) * f ' ( r l ( k k ' ) ,.. .,rn(kk'))

by the above observation, and hence Div (kk',68'). therefore closed under * .

i

i s closed under +

Z+,Div (k,B) and Div ( k ' , 8 ' ) . Say E EEGI. Then k I f ( r , ( k ) , . . . # r n ( k ) ) ,

4 , f(z),f'(z)

,. .., r n ( k ' ) )

must show

+ k f ' ( r l (k!'), +

El

Div (kk',k'B + k8') which implies

Thus 'f; 6 . Br' ; ~

% and %

is

...,rn(kk')) E Mo,

so

so t h a t i s closed under +.

%

r;8


and the monotone system

These sets can both be obtained in the form A

=

v ( U )

where the sentence unary predicate Let now

cp

I

X E P(A)

U

(fi8U)

L

.

and

8

contains besides the symbols of

(O

L U { U

,a sentence of

expansion of

I= Cp J

T

3 , be

a theory of

X

.

given

8

K

It can be easily derived from Chang-Makkai theorem the following are equivalent: a) There is a formula A rl-

=

rl

of

T

with

L

such that

A

CD-(U)

for all models

fi C

b) For all countable

5 k= T

IAl > 1

&(U)

29 7

.

is countable

L

the new

a countable

(see [CK]) that

M. ZIEGLER

29 8

Our main r e s u l t is

.

(1 1 ) Theorem

The f o l l o w i n g are e q u i v a l e n t :

a ) There is a formula

such t h a t

Ac

for all

&(u)

=

of L

q

& I= T

b) For a l l c o u n t a b l e

.

T

is a non-empty monotone

$(u)

system w i t h c o u n t a b l e base. ( 1 . 2 ) C o r o l l a r y (Beth's theorem f o r monotone systems) Let

be a formula of

4

a ) There is a formula

=



b) For a l l

Ail=