Intuitionistic Logic, Model Theory and Forcing (Studies in Logic and the Foundations of Mathematics)

STUDIES I N LOGIC AND THE FOUNDATIONS O F MATHEMATICS

Editors

A. HEYTING, Amsterdam A.MOSTOWSK1, Warszawa A. ROBINSON, New Haven P. SUPPES, Stanford

Advisory Editorial Board Y . BAR-HILLEL, Jerusalem K. L. D E BOUVBRE, Sanra Clara H. HERMES, Freiburg i/Br. J. HINTIKKA, Helsinki J. C. SHEPHERDSON, Eristol E. P. SPECKER, Zurich

N O R T H - H O L L A N D P U B L I S H I N G COMPANY AMSTERDAM-LONDON

INTUITIONISTIC LOGIC MODEL THEORY AND FORCING

MELVIN CHRIS FITTING Herbert H. Lehman College The City University of New York

1969

N O R T H - H O L L A N D P U B L I S H I N G COMPANY AMSTERDAM-LONDON

Q

NORTH-HOLLAND PUBLISHING COMPANY, AMSTERDAM, 1969.

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner.

@$g#.g$ Library of Congress Catalogue Card Number: 79-102718 SBN 7204 2256 6

PUBLISHERS

NORTH HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH HOLLAND PUBLISHING COMPANY LTD - LONDON

PRINTED I N THE NETHERLANDS

ACKNOWLEDGMENTS

I would like to express my thanks to Professor Raymond Smullyan for his guidance and encouragement during the preparation of this thesis. I would also like to thank the library staff at the Belfer Graduate School of Science for much friendly unconventional assistance. This work was supported by National Aeronautics and Space Administration Training Grant NSG(T)144, and by Air Force Office of Scientific Research Grants AFOSR68-1375 and AFOSR433-65. With the exception of a few minor corrections and changes, this work duplicates the author’s Doctoral Dissertation, written at the Belfer Graduate School of Science, Yeshiva University, and submitted June 1968.

to my parents

INTRODUCTION

In 1963 P. Cohen established various fundamentd independence results in set theory using a new technique which he called forcing. Since then there has been a deluge of new results of various kinds in set theory, proved using forcing techniques. It is a powerful method. It is, however, a method which is not as easy to interpret intuitively as the corresponding method of Godel which establishes consistency results. Godel defines an intuitively meaningful transfinite sequence of (domains of) classical models Ma, defines the class L to be the union of the M a over all ordinals a, and shows L is a classical model for set theory [4; see also 31. He then shows the axiom of constructability, the generalized continuum hypothesis, and the axiom of choice are true over L, establishing consistency. In this book we define transfinite sequences of S. Kripke’s intuitionistic models [13]in a manner exactly analogous to that of Godel in the classical case (in fact, the M a sequence is a particular example). In a reasonable way we define a “class” model for each sequence, which is to be a limit model over all ordinals. We show all the axioms of set theory are intuitionistically valid in the class models. Finally we show there are particular such sequences which provide: a class model in which the negation of the axiom of choice is intuitionisticallyvalid; a class model in which the axiom of choice and the negation of the continuum hypothesis are intuitionisticallyvalid; a class model in which the axiom of choice, the generalized continuum hypothesis, and the negation of the axiom of

14

INTRODUCTION

constructability are intuitionistically valid. From this the classical independence results are shown to follow. The definition of the sequences of intuitionistic models will be seen to be essentially the same as the definition of forcing in [3]. The difference is in the point of view. In Cohen’s book one begins with a set M which is a countable model for set theory and, using forcing, one constructs a second countable model N “on top of” M. Forcing enables one to “discuss” N in M even though N is not a sub-model of M. Various such N are constructed for the different independence results. Cohen points out [3, pp. 147, 1481 that actually the proofs can be carried out without the need for a countable model, and without constructing any classical models; this is the point of view we take. It is the forcing relation itself that the center of attention [see 3, page 1471, though now it has an intuitive interpretation. A similar program has been carried out by Voptnka and others. [See the series of papers 22, 23, 24, 27, 6, 25, 7, 8, 26, 281. The primary difference is that these use topological intuitionistic model theory while we use Kripke’s, which is much closer in form to forcing. Also the VoptSnka series uses Godel-Bernays set theory and generalizes the F, sequence, while we use Zermelo-Fraenkelset theory and generalize theM, sequence. The Voptnka treatment involves substantial topological considerations which we replace by more “logical” ones. This book is divided into two parts. In part I we present a thorough treatment of the Kripke intuitionistic model theory. Part I1consists of the set theory work discussed above. Most of the material in Part I is not original but it is collected together and unified for the h s t time. The treatment is self-contained. Kripke models are defined (in notation different from that of Kripke). Tableau proof systems are d e h e d using signed formulas (due to R. Smullyan), a device which simplifies the treatment. Three completeness proofs are presented (one for an axiom system, two for tableau systems), one due to Kripke [13], one due independently to R. Thomason [21] and the author, and one due to the author. We present proofs of compactness and Lowenheim-Skolemtheorems. Adapting a method of Cohen, we establish a few connections between classical and intuitionistic logic. In the propositional case we give the relationship between Kripke models and algebraic ones [16] (which provides a fourth completeness proof in the

INTRODUCTION

15

propositional case). Finally we present Schiitte’s proof of the intuitionistic Craig interpolation lemma [17], adapted to Kleene’s tableau system G3 as modified by the use of signed formulas. No attempt is made to use methods of proof acceptable to intuitionists. Chapter 7 begins part 11. In it we define the notion of an infuifionistic Zermelo-Fraenkel (ZF) model, and the intuitionistic generalization of the Godel M a sequence. Most of the chapter is devoted to showing the class models resulting from the sequences of intuitionistic models are intuitionistic ZF models. This result is demonstrated in rather complete detail, especially sections 8 through 13, not because the work is particularly difficult, but because such models are comparatively unfamiliar. In chapter 8 the independence of the axiom of choice is shown. In chapter 9 we show how ordinals and cardinals may be represented in the intuitionistic models, and establish when such representatives exist. Chapter 10 establishes the independence of the continuum hypothesis. In Chapter 11 we give a way to represent constructable sets in the intuitionistic models, and establish when such representatives exist. Chapter 12 establishes theindependence of the axion of constructability. Chapter 13 is a collection of various results. We establish a connection between the sequences of intuitionistic models and the classical Ma sequence. We give some conditions under which the axiom of choice and the generalized continuum hypothesis will be valid in the intuitionistic class models (thus completing chapters 10 and 12). Finally we present Vop6nka’s method for producing classical non-standard set theory models from the intuitionistic class models without countability requirements [261. The set theory work to this point is self-contained, given a knowledge of the Godel consistency proof ([4], in more detail [3]). In chapter 14 we present Scott and Solovay’s notion of boolean valued models for set theory [19]. We define an intuitionistic (or forcing) generalizationof the Ra sequence (sets with rank) analogous to the Cohen generalization of the M a sequence, and we establish some connections between intuitionistic and boolean valued models for set theory.

CHAPTER 1

PROPOSITIONAL INTUITIONISTIC LOGIC SEMANTICS

0 1. Formulas We begin with a denumberable set of propositional variables A , B, C, ..., three binary connectives A ,v ,3 , and one unary connective -, together with left and right parentheses (, ). We shall informally use square and curly brackets [, 3, {, for parentheses, to make reading simpler. The notion of well formed formula, or simplyformula, is given recursively by the following rules: FO. If A is a propositional variable, A is a formula. F1. If Xis a formula, so is X. F2, 3,4. If X and Yare formulas, so are (XAY), (Xv Y ) , ( X I Y ) . N

Remark 1.1: A propositional variable will sometimes be called an atomic formula.

It can be shown that the formation of a formula is unique. That is, for any given formula X , one and only one of the following can hold: (1). Xis A for some propositional variable A. (2). There is a unique formula Y such that X is Y. (3). There is a unique pair of formulas Y and 2 and a unique binary connective b ( A , v or 3 ) such that Xis (YbZ). We make use of this uniqueness of decomposition but do not prove it here. We shall omit writing outer parentheses in a formula when no con-

-

20

PROPOSITIONAL INTUITIONISTIC LOGIC, SEMANTICS

CH. 1 6 2

fusion can result. Until otherwise stated, we shall use A , B and C for propositional variables, and X , Y and Z to represent any formula. The notion of immediate subformula is given by the following rules: 10. A has no immediate subformula. 11. X has exactly one immediate subformula : X . 12, 3,4. (XA Y), (Xv Y ) , ( X I Y ) each has exactly two immediate subformulas: X and Y. The notion of subformula is defined as follows: SO. X i s a subformula of X. S1. If X is an immediate subformula of Y , then X is a subformula of Y. S2. If X is a subformula of Y and Y is a subformula of 2,then X is a subformula of Z . By the degree of a formula is meant the number of occurrences of logical connectives ( , A ,v ,I) in the formula.

-

N

0 2.

Models and validity

By a (propositional intuitionistic) model we mean an ordered triple (9, 9, k), where B is a non-empty set, W is a transitive, reflexive relation on 9, and k (conveniently read “forces”) is a relation between elements of 9 and formulas, satisfying the following conditions: For any re9 PO. if T I=A and TWA then A k A (recall A is atomic). PI. Tk(Xr\ Y) iff r k X a n d r k Y. P2. Tk(Xv Y ) iff TkXor l‘k Y. P3. T k - X iff for all A E such ~ that r W A , A F X . P4. r k ( X x Y ) iff for all A E 9 such that r W A , if A k X , then A k Y.

Remark 2.1: For T e 9 , by r* we shall mean any A E such ~ that TWA. Thus “for all r*, q(T*)” shall mean “for all A e 9 such that I‘WA, cp (A)” ;and “there is a T* such that cp (r*)” shall mean “there is a A E 3 ’ such that TWA and cp (A)”. Thus P3 and P4 can be written more simply as: P3. Tk-Xiff for all T*, r * ) c X P4. r k ( X = Y ) iff for all r*,if r*I= X , then r*k Y.

A particular formula X is called valid in the model (9, W,k) if for all r E 3, T k X . X is called valid if X is valid in all models. We will show

CH. 1 fj§ 3,4

MOTIVATION. SOME PROPERTIES OF MODELS

21

later that the collection of all valid formulas coincides with the usual collection of propositional intuitionistic logic theorems. When it is necessary to distinguish between validity in this sense and the more usual notion, we shall refer to the validity defmed above as intuitionistic validity, and the usual notion an classical validity. This notion of an intuitionistic model is due to Saul Kripke, and is presented, in different notation, in [13]. See also [18]. Examples of models will be found in section 5, chapter 2.

53. Motivation Let (9, 9, k) be a model. Y is intended to be a collection of possible universes, or more properly, states of knowledge. Thus a particular r in 9 may be considered as a collection of (physical) facts known at a particular time. The relation W represents (possible) time succession. That is, given two states of knowledge r and A of '3, to say r W A is to say: if we now know r, it is possible that later we will know A . Finally, to say r I. Xis to say: knowing r,we know X , or: from the collection of facts r, we may deduce the truth of X. Under this interpretation condition P3 of the last section, for example, may be interpreted as follows: from the facts r we may conclude - X if and only if from no possible additional facts can we conclude X . We might remark that under this interpretation it would seem reasonable that if r k X and r@Athen A i= X,that is, if from a certain amount of information we can deduce X,given additional information, we still can deduce X,or if at some time we know Xis true, at any later time we still know Xis true. We haverequired that this holds only for the case that Xis atomic, but the other cases follow. For other interpretations of this modeling, see the original paper [13]. For a different but closely related model theory in terms of forcing see [5].

0 4. Some properties of models Lemma 4.1: Let (9, W ,k) and <S,9, I=') be two models such that for any atomic formula A and any r e Y, r C A iff r k' A. Then k and C' are identical.

22

PROPOSITIONAL INTUITIONISTIC LOGIC, SEMANTICS

-

CH. 1 5 4

Proofi We must show that for any formula X ,

r x r vx. This is done by induction on the degree of X and is straightforward. We present one case as an example. Suppose X i s Y and the result is known for all formulas of degree less than that of X (in particular for Y). We show it for X : r P X c> r P Y (by definition) e (W*) (r*pC Y) (by hypothesis) 9 (W*) (r* pc’ Y ) (by definition)

-

-

erv-y

OrP’x.

Lemma4.2: Let B be a non-empty set and 9 be a transitive, reflexive relation on 9. Suppose k is a relation between elements of B and atomic formulas. Then k can be extended to a relation I=’ between elements of B and all formulas in such a way that ( B , W , C’) is a model. Proofi We define I=‘ as follows: (0). if 1 A then r*V A , (1). I’C’ ( X A Y) if r C‘X and r k’ Y , (2). r C ’ ( X v Y ) i f r ~ ’ X o r r ~ ‘ Y , (3). r C’-X if for all r*, r*pC’X, (4). r C’ ( X I Y)if for all r*,if r*C’X, then r*k’ Y. This is an inductive definition, the induction being on the degree of the formula. It is straightforward to show that (9, 9,C’) is a model. From lemmas 4.1 and 4.2 we immediately have Theorem 4.3: Let 9 be a non-empty set and 9 be a transitive, reflexive relation on 9. Suppose k is a relation between elements of B and atomic formulas. Then i= can be extended in one and only one way to a relation, also denoted by b y between elements of 9 and formulas, such that (9, 9, k } is a model, Theorem 4.4: Let (9, 9, I=} be a model, X a formula and r,AEB. If r k X and T 9 A , then A I= X. Proofi A straightforward induction on the degree of X (it is known already for X atomic). For example, suppose the result is known for X , and I‘C-X. By definition, for all r*, r * y X . But TWA and 9 is transitive so any 9-successor of A is an 9-successor of r. Hence for all A*, A* pC X , so A P -X. The other cases are similar.

CH.

1@ 5 , 6

0 5.

23

ALGEBRAIC MODELS. ALOEBRAIC AND KRIPKE VALIDITY

Algebraic models

In addition to the Kripke intuitionistic semantics presented above, there is an older algebraic semantics: that of pseudo-boolean algebras. In this section we state the algebraic semantics, and in the next we prove its equivalence with Kripke’s semantics. A thorough treatment of pseudoboolean algebras may be found in [16].

Definition 5.1 : A pseudo-boolean algebra (PBA) is a pair (9?,Y) ST,T X , F Y

S, T X

T TI>

IS,

F=Y

I

In rules F - and F z above, ST means { TX T X E S ] .

CH.211

BETH TABLEAUS

29

Remark 1.1 : S is a set, and hence (S, TX} is the same as {S, TX, T X } . Thus duplication and elimination rules are not necessary.

If U is a set of signed formulas, we say one of the above rules, cal1 it rule R, applies to U if by appropriate choice of S, X and Y the collection of signed formulas above the line in rule R becomes U. By an application of rule R to the set U we mean the replacement of U by U,(or by U,and U, if R is F A , T v or T D ) where U is the set of formulas above the line in rule R (after suitable substitution for S, X and Y ) and U,(or U,,U,)is the set of formulas below. This assumes R applies to U.Otherwise the result is again U.For example, by applying rule F Z Ito the set { TX, FY, F ( Z 3 W ) }we may get the set ( TX, TZ, FW}. By applying rule T v to the set { TX, FY, T ( Z v W)} we may get the two sets ( T X , FY, T Z } and (TX, FY, T W } . By a conjiguration we mean a finite collection {S,, S,, ..., S,,} of sets of signed formulas. By an application of the rule R to the configuration {S,, S2,..., S,} we mean the replacement of this configuration with a new one which is like the first except for containing instead of some Si the result (or results) of applying rule R to S,. By a tableau we mean a finite sequence of configurations V,, V,, ..., Vn in which each configuration except the first is the result of applying one of the above rules to the preceding configuration. A set S of signed formulas is closed if it contains both TX and FX for some formula X.A configuration (Sl, S,,..., S,,} is closed if each Si in it is closed. A tableau V,,V2,...,V,, is closed if some Vi in it is closed. By a tableau for a set S of signed formulas we mean a tableau V,, V2, ...,W,,in which V, is ( S } . A finite set of signed formulas S is inconsistent if some tableau for S is closed. Otherwise S is consistent. X is a theorem if (FX)is inconsistent, and a closed tableau for { F X } is called a proof of X . If X is a theorem we write t, X. We will show in the next few sections the correctness and completeness of the above system relative to the semantics of ch. 1. Examples of proofs in this system may be found in 0 5. The corresponding classical tableau system is like the above, but in rules F- and F I , S, is replaced by S (see [20]). The interpretations of the classical and intuitionistic systems are different.

30

PROPOSITIONAL INTUITIONISTIC LOGIC, PROOF THEORY

CH.202

In the classical system TX and FX mean X is true and X is false respectively. The rules may be read: if the situation above the line is the case, the situation below the line is also (or one of them is, if the rule is disjunctive: F A , T v , Tz).Thus TX means the same as X , and FX means -X. Classically the signs T and F are dispensable. Proof is a refutation procedure. Suppose X is not true (begin a tableau with FX). Conclude that some formula must be both true and not true (a closed configuration is reached). Since this can not happen, Xis true. In the intuitionistic case TX is to mean X is known to be true (Xis proven). FX is to mean X is not known to be true (X has not been proved). The rules are to be read: if the situation above the line is the case, then the situation below the line is possible, i.e. compatible with our present knowledge (if the rule is disjunctive, one of the situations below the line must be possible). For example consider rule F x . If we have not proved X= Y, it is possible to prove X without proving Y,for if this were not possible, a proof of Y would be ‘inherent’ in a proof of X, and this fact would constitute a proof of XI Y. But we have S, below the line in this rule and not S because in proving X we might inadvertently verify some additional previously unproven formula (some FZES might become TZ). Similarly for F - . The proof procedure is again by refutation. Suppose X is not proven (begin a tableau with FX).Conclude that it is possible that some formula is both proven and not proven. Since this is impossible, X is proven. We have presented this system in a very formal fashion because it makes talking about it easier. In practice there are many simplifications which will become obvious in any attempt to use the method. Also, proofs may be written in a tree form. We find the resulting simplified system the easiest to use of all the intuitionistic proof systems, except in some cases, the system resulting by the same simplifications from the closely related one presented in ch. 6 9 4. A full treatment of the corresponding classical tableau system, with practical simplifications, may be found in [20].

0 2. Correctness of Beth tableaus Dejinition 2.1 : We call a set of signed formulas

{TXI, ..., TX,,FYi,

..., FY,}

m.2$3

31

HINTMKA COLLECTIONS

realizable if there is some model (9, W ,1) and some r e g such that C X I , .., k X,,,r pC Y,,..., pC Y,. We say that realizes the set. If {Sl, S,, ..., Sn} is a configuration, we call it realizable if some Si in it is realizable.

.

Theorem 2.2: Let gl,V2,...,V,, be a tableau. If V iis realizable, so is Vi+l. Pro08 We have eight cases, depending on the rule whose application from gi. produced gi+l Case(1): V i is { ..., { S , T ( X v Y ) } ,...I and Vi+, is { ..., {S,TX), {S,TY),...}. Since Vi is realizable, some element of it is realizable. If that element is not (S,T(Xv Y)}, the same element of Vi+ is realizable. If that element is (S,T(Xv Y)}, then for some model (Q,@, k) and some r e g , r realizes {S,T(Xv Y)}. That is, r realizes Sand rC(X v Y). Then r k X or I ' k Y, so either r realizes {S,TX} or {S,TY). In either case Vi+,is realizable. Case (2): Ci is (..., ( S , F ( - X ) } ,...} and %i+l is { ..., {ST,TX},...}. V iis realizable, and it suffices to consider the case that {S,F( X ) } is the realizable element. Then there is a model (9, 9,I=) and a re9 such that r realizes S and r pC - X . Since r pC - X , for some r * E 9 , r*k X . But clearly, if r realizes S, I'* realizes S, (by theorem 1.4.4). Hence r* realizes { S J X } and V i + lis realizable. The other six cases are similar. N

CoroZZury2.3: The system of Beth tableaus is correct, that is, if FIX,

X is valid. Proofi We show the contrapositive. Suppose X i s not valid. Then there is a model (9, W,C) and a r E Q such that r y X . In other words ( F X } is realizable. But a proof of X would be a closed tableau V,,V2,..., Vn in which is ( ( F X } } . But Vl is realizable, hence each Vi is realizable. But obviously a realizable configuration cannot be closed. Hence y IX.

8 3. Hintikka collections In classical logic a set S of signed formulas is sometimes called downward saturated, or a Hintikka set, if

T X AYES F X v YES

e- T X E S

*

and TYES, F X E S and F Y E S ,

32

PROPOSITIONAL INTTUITIONISTIC UxiIC, PROOF THEORY

TXVYES* FXAYES * T - X E S => T X ~ Y E => S F-XES * FXDYES

~~-1.263

T X E S or T Y E S , F X E S or F Y E S , FXES, F X E S or T Y E S , TXES, T X E S and F Y E S .

Remark 3.1 : The names Hintikka set and downward saturated set were given by Smullyan [20]. Hintikka, their originator, called them model sets.

Hintikka showed that any consistent downward saturated set could be included in a set for which the above properties hold with * replaced by c>. From this follows the completeness of certain classical tableau systems. This approach is thoroughly developed by Smullyan in [20]. We now introduce a corresponding notion in intuitionistic logic, which we call a Hintikka collection. While its intuitive appeal may not be as immediate as in the classical case, its usefulness is as great. DeJinition 3.2: Let 9 be a collection of consistent sets of signed formulas. 9 We call 9 a Hintikka collection if for any T X AYEr T X E r and T Y E T , F X v Y E r => F X E r and F Y E r , T X v Y E r * T X E r or T Y E r , F X A Y E ~ F X E r or F Y E r , T-XEr =+ F X E r , T X = Y E r =- F X E r or T Y E r , F-XET * forsome d E 9 , f T C A and T X E A , F X 3 Y E r => for some A E ~rT , c A , T X E A ,F Y E A . Definition 3.3: Let 9 be a Hintikka collection. We call (9, 9, C) a model for 99 if (1). (9, W ,1) is a model, (2). r T EA *TBA, (3). TxEr a r u ,

FxEr =>rpcx.

Theorem 3.4: There is a model for any Hintikka collection. Pro08 Let 9 be a Hintikka collection. Define W by: r W A if r T s A .

1211.284

33

COMPLETENESS OF BETH TABLEAUS

If A is atomic, let r I = A if T A E ~and , extend k to produce a model (9, W,I=>. To show property (3) is a straightforward induction on the degree of X. We give one case as illustration. Suppose Xis Y and the result is known for Y. Then

-

T

-

YEr

=>

( V ' 4 ~ 9 (I", ) 5 '4 * T

-

YEA)

* (V'4€9)(r,cd+FY€A) * (VA E 9)(TWd + '4 pc Y ) =>

rk-Y,

and F - Y E ~e. ( 3 A ~ 9 ) ( r , ~ d a n d T Y ~ d ) => (34 E 9)(r9'4 and A P Y ) =>

rpc

-

Y.

It follows from this theorem that to show the completeness of Beth tableaus we need only show the following: If y ,X, then there is a Hintikka collection 9 such that for some r e g , F X e r .

5 4.

Completeness of Beth tableaus

Let S be a set of signed formulas. By Y ( S ) we mean the collection of all signed subformulas of formulas in S. If S is finite, Y ( S ) is finite. Let S be a finite, consistent set of signed formulas. We define a reduced set for S (there may be many) as follows: Let Sobe S. Having defined S,,,a finite consistent set of signed formulas, suppose one of the following Beth reduction rules applies to S,,: T A, F A , T v , F v , T - or T D . Choose one which applies, say F A . Then S,, is { U,FXA Y } . This is consistent, so clearly either { U,FXA Y, FX} or {U,F X A Y, FY} is consistent. Let S,,,, be { U,FXA Y, F X } if consistent, otherwise let S,,,, be {U,F X A Y, FY}. Similarly if T A applies and was chosen, then S,, is { U, T X A Y } . Since this is consistent, { U , T X A Y, TX, T Y } is consistent. Let this be S,,,,. In this way we define a sequence So,S,, S,, ... . This sequence has the property S,,CS,,~. Further, each S,, is finite and consistent. Since each S , , s Y ( S ) ,there are only a finite number of different possible S,,. Consequently there must be a member of the sequence, say S,,, such that the application of any one of the rules (except F- or F D ) produces S,, again. Call such an S,, a reduced set of S, and denote it by S'. Clearly any finite, consistent set of

34

PROPOSITIONAL I"IST1C

LOGIC, PROOF THEORY

c1i.284

signed formulas has a finite, consistent reduced set. Moreover, if S' is a reduced set, it has the following suggestive properties:

T X A YES' * FX v Y E S ' 3 T X v YES' 3 F X A Y E S ' T-XES' * TXZYES' * S' is consistent.

TXES' FXES' TXES' ~F X E S ' FXES', FXES'

and T Y E S ' , and F Y f S ' , or T Y f S ' , or F Y E S ' , or

TYES',

Now, given any finite, consistent set of signed formulas S, we form the collection of associated sets as follows : If F- XES , {ST,TX}is an associated set. If F X 3 Y E S , {S,,TX,FY} is an associated set. Let d ( S ) be the collection of all associated sets of S. d ( S ) is finite, since U E ~ ( Simplies ) U C Y ( S ) and Y ( S ) is finite. d ( S ) has the following properties: if S is consistent, any associated set is consistent and F -XES FXD YES

=- for some =>

U E ~ ( S S,s ) U, T X E U , forsome UECCP(S) S , E U, T X E U , F Y E U .

Now we proceed with the proof of completeness. Suppose y ,X. Then { F X } is consistent. Extend it to its reduced set So. Form &(So). Let the elements of &(So) be U,,U,,..., U,.Let S, be the reduced set of U,, ..., S, be the reduced set of U,.Thus, we have the sequence So, S,, S,, ..., S,. Next form d ( S , ) . Call its elements Un+,,Un+,,..., Urn.Let S,+, be the reduced set of U,,,, and so on. Thus, we have the sequence So,S,, ..., S,,, Sn+,,..., S,. Now we repeat the process with S,, and so on. In this way we form a sequence So, S,, S,, .... Since each S i s 9 ( S ) , there are only finitely many possible different S,. Thus we must reach a point S, of the sequence such that any continuation repeats on earlier member. Let Q be the collection {So,S,, ..., Sk}.It is easy to see that 9 is a Hintikka collection. But FXESoE 9. Thus we have shown:

Theorem 4.1 : Beth tableaus are complete.

CH.285

35

EXAMPLES

Remark 4.2: This proof also establishes that propositional intuitionistic logic is decidable. For, if we follow the above procedure beginning with FX,after a finite number of steps we will have either a closed tableau for {FX}or a counter-model for X. Moreover, the number of steps may be bounded in terms of the degree of X . The completeness proof presented here is in essence the original proof of Kripke [13]. For a different tableau completeness proof see ch. 5 9 6, where it is given for first order logic. For a completeness proof of an axiom system see ch. 5 5 10, where it also is given for a iirst order system. The work in ch. 1 Q 6 provides an algebraic completeness proof, since the Lindenbaum algebra of intuitionistic logic is easily shown to be a pseudo-boolean algebra. See [16].

9 5. Examples In this section, so that the reader may gain familiarity with the foregoing, we present a few theorems and non-theorems of intuitionistic propositional logic, together with their proofs or counter-models. We show (1). Y I A V - A , (2). kr--(Av -A), (3). Y p - A 3 A , (4). I-,(AvB)r>-(--A/\ -I?), (5). 1 f 1 - - ( A V B ) + - A V -4). For the general principle connecting (1) and (2) see ch. 4 5 8. (1). Y I A V - A . A counter example for this is the following: g = {r,A } rwr, r a ,

AWA.

A !=Ais the != relation for atomic formulas, and 1 is extended to all formulas as usual. We may schematically represent this model by

r

I

A!=A

36

PROPOSITIONAL INTUITIONISTIC LOGIC, PROOF THEORY

czz.285

We claim r pC A v - A . Suppose not. If rt A v - A , either rC A or r C - A . But r pC A . If rC - A then since I ' g A , A pC A . But A 1A, hence rpCAv - A . (2). I-,--(AV -A). A tableau proof for this is the following, where the reasons for the steps are obvious: {{I; - - N ( A v - A ) } } , {{T ( A v A)}} { { T - ( A v - A ) , F ( A -A))), { { T - ( A v - A ) , FA, F A } } , {{T ( A v A), T A ) ) , { { F ( A v A), T A } } , {{FA, F A , T A } ) .

-

9

-

- -

-

N

(3). y1- - A 3 A . The model of example (1) has the property that TI= - A but (4). k , ( A v B ) = , - ( - J A A -B). The following is a proof:

-

r pC A.

-- -- --

y 1-

-

{ { F ( ( A v B ) 2 ( A A B))}} , {{ T ( A v B), P ( A A B ) ) } , { { T ( Av B), T ( - A A -q}}, { { T ( A v B), T - A , T - B } } , { { T ( A v B), FA, T - B } } , { { T ( Av B), FA, F B } } , { { TA, FA, FB}, { TB, FA, FB}}.

- -

( A v B) 3 ( - A v -B)# A counter example is the following: (5).

s={r,A,Q}, ~ A QWQ, , TWA, rBQ

rwr, A

A t= A , SL C B is the C relation for atomic formulas, and C is extended as usual. We may schematically represent this model by

r

ACA

n

SLkB

~11.295

EXAMPLES

37

Now A 1 A, so A ! = A vB. Likewise Dk A v B. It follows t h a t r k - - ( A v B ) But if r k - - A v - - - B , either r ! = - - A or r ! = - - - B . If T k - - A , it would follow that D k A. I f r I= -3, it would follow that A != B. Thus TY-NAV-NB.

-

CHAPTER 3

RELATED SYSTEMS OF LOGIC

0 1. fiprimitive intuitionistic logic, semantics This is an alternative formulation of intuitionistic logic in which a symbol , which is then re-introduced as a formal abbreviation, - X for X 3 f . For presentations of this type, see [15] or [17]. Specifically, we change the definition of formula by adding f to our list of propositional variables and removing from the set of connectives. is re-introduced as a metamathematical symbol as above. Our definition of subformula is also changed accordingly. The definition of model is changed as follows: replace P3 (ch. 1 5 2) by P3‘: r f f. This leads to a new definition of validity, which we may calif-validity.

f is taken as primitive, instead of

N

-

-

Theorem 1.1: Let X be a formula (in the usual sense) and let X’be the corresponding formula with written in terms off. Then X is valid if and only if X’ is $valid. Pro08 We show that in any model (3, W ,k) N

rcx

iff

rcx’

(where we use two different senses of I). The proof is by induction on the degree of X (which is the same as the degree of X’).Actually all cases

~ ~ 3 1 2

f-pRIMITIvE

nmxnomnc LOQXC,PROOF THEORY

39

are easy except that of N itself. So suppose the result is known for all formulas of degree less than that of X,and Xis Y. Then N

rCx+rCNy -vr* r*pcy

-vr* r * y y f , but clearly this is equivalent to r k Y' x f since r*f f.Hence equivalently rw. 0 2. f-primitive intuitionistic logic, proof theory In this section we still retain the altered definition of formula in the last section with f primitive. We give a tableau system for this. The new system is the same as that of ch. 2 0 1 in all but two respects. First the rules T - and F - are removed. Second a set S of signed formulas is called closed if it contains TX and FX for some formula X , or if it contains This leads to a new definition of theorem, which we may call f-theorem.

u.

Theorem 2.1: Let X be a formula (in the usual sense) and let X' be the corresponding formula with written in terms off. Then Xis a theorem if and only if X' is anf-theorem. N

This follows immediately from the following:

-

Lemma 2.2: Let S be a set of signed formulas (in the usual sense) and let S' be the corresponding set of signed formulas with replaced in terms off. Then S is inconsistent if and only if S' isf-inconsistent. Proojl We show this in two halves. First suppose S is inconsistent. We show the result by induction on the length of the closed tableau for S. There are only two significant cases. Suppose first that the tableau for S i s %fly Q,,..., Qn; Ql is { { U ,F - X } } and Q, is {(UT, TX}}.Then by the induction hypothesis {U;,TX') is $inconsistent. Hence so is { U ,F X 3f }, i.e. S'. The other case is if V1 is { { U,T - X } } and V , is ({ U,F X ) } . Then by the induction hypothesis { U', FX'} isf-inconsistent. Hence so is { U ,T X ' 3f }, i.e. S'. The converse is shown by induction on the Iength of the closed $-tableau for S'. If this f-tableau is of length 2, either S' contains TX and

40

RELATED SYSTEM OF IxxjIC

CH.3 g 3 , 4

FX for some formula X,and we are done, or S' contains Tf,which is not possible since we supposed S' arose from standard set S. The induction steps are similar to those above. The results of this and the last sections, together with our earlier results give: X' isf-valid if and only if X ' is anf-theorem. This is not the complete generality one would like since it holds only for those formulas X' which correspond to standard formulas X . The more complete result is however true, as the reader may show by methods similar to those of the last chapter.

fi 3.

Minimal Iogic

Minimal logic is a sublogic of intuitionistic logic in which a false statement need not imply everything. The original paper on minimal logic is Johannson's [9]. Prawitz establishes several results concerning it in [15], and it is treated algebraically by Rasiowa and Sikorski [16]. Semantically, we use the f-models defined in 0 1, with the change that we no longer require P3', that is, that r y f. Proof theoretically, we use thef-tableaus defined in $2, with the change that we no longer have closure of a set because it contains Tf.We leave it to the reader to show that X is provable in this tableau system if and only if X is valid in this model sense, using the methods of ch. 2. Certainly every minimal logic theorem is an intuitionistic logic theorem, but the converse is not true. For example ( A A - A ) = I B is a theorem of intuitionistic logic, but the following is a minimal counter-model for it, or rather for ( A A ( A3f))3 B :

s={r}, r m, r u , rcf, and C is extended as usual. It is easily seen that I ' k A A ( A ~ f )but , r)CB.

fi 4.

Classical logic

-

Beginning with this section, we return to the usual notions of formula, tableau and model, that is, with and not f as primitive.

CH.355

MODAL LOGIC,

s4;

SEMANTICS

41

Some authors call a set Y of unsigned formulas a (classical) truth set if

X AYEY X v Y E Y -X€Y X z Y e Y

o o o o

X X X X

E Y and E Y or #Y, $ Y or

YEY, YEY, YEY.

It is a standard result of classical logic that Xis a classical theorem if and only if X is in every truth set. There is a proof of this in [20].

Theorem 4.1: Any intuitionistic theorem is a classical theorem. Proofi Suppose X is not a classical theorem. Then there is a truth set Y such that X q Y . We define a very simple intuitionistic counter-model for X , ( 9 , W , t.}, as follows: 9=p.),

YWY, Y t .A - A E Y , for A atomic, and I:is extended as usual. It is easily shown by induction on the degree of Y that 9 bY e Y€Y. Hence 9y X , and X is not an intuitionistic theorem. That the converse is not true follows since we showed in ch. 2 9 5 that y I A v - A . Thus we have: minimal logic is a proper sub-logic of intuitionistic logic which is a proper sub-logic of classical logic.

5 5. Modal logic, S4; semantics In this section we define the set of (propositional) S4 theorems semantically using a model due to Kripke [12] (see also [18]). S4 was originated by Lewis [14], and an algebraic treatment may be found in [16]. A natural deduction treatment is in [15]. The definition of formula is changed by adding 0 to the set of unary connectives. Thus for example ( A v 0 - A ) is a formula. 0 is read “necessarily”. 0 is sometimes taken as an abbreviation for 0 and is read “possibly”. (In [14] 0 was primitive.) The S4 model is defined as follows: It is an ordered triple (9, 9, b> where $9’ is a non-empty set, 9 is a transitive, reflexive relation on 9,

--

--

42

01.386

RELATED SYSTEMS OF LOGIC

and C is a relation between elements of 9 and formulas, satisfying the following conditions : M1. I ' C X A Y iff I'CX and f C Y , M2. f k X v Y iff f C X or f C Y , M3. f C - X iff r F X , M 4 . I ' C X x Y iff fpCXor I'CY, M5. r C U X iff forallf*, I'*CX. Xis S4 valid in (9, W,C) if for all f E B , I'bX. Xis S4 valid if Xis S4 valid in all S4 models. The intuitive idea behind this modeling is the following: 3 ' is the collection of all possible worlds. f W A means A is a world possible relative to r. r CX means X is true in the world f. Thus M5 may be interpreted: Xis necessarily true in f if and only if Xis true in any world possible relative to f. This interpretation is given in [12].

0 6. Modal logic, S4; proof theory We define a tableau system for S4 as follows: Everything in the definition of Beth tableaus in ch. 2 0 1 remains the same except the reduction rules themselves. These are replaced by MTA

S,TXAY S, T X , T Y

MF

MTv

S, T X v Y S, T X S, TY

MFv

S,FXvY S , F X , FY

MT-

S,T-X S, FX

MF-

S,F-X S, T X

MTD

S,TX=,Y s, F X TY

MFx

S-~ ,FX3Y S, T X , FY

MTO

S,TOX

MFO

S,FOX

I

IS,

A

S,FXA Y S , F X S , FY

I

so, FX

S, T X

I

where in rule M F O So is ( T U X T U X E S } . Again the methods of ch. 2 can be adapted to S4 to establish the identity of the set of S4 theorems and the set of S4 valid formulas. This is left to the reader. The original

CH.357

s4 AND

INTUITIONISTIC u)(iIC

43

proof is in [12]. We are more interested in the relation between S4 and intuitionistic logic.

0 7.

S4 and intuitionisticlogic

A map from the set of intuitionistic formulas to the set of S4 formulas is defined by (see [ 181) = U A for A atomic, M(A) M ( X v Y)=M ( X ) v M(Y), M ( X A Y) = M ( X ) A M ( Y ) , M(-X) =U-M(X), M ( X 3 Y )= 0 ( M ( X )3 M ( Y ) ) . We wish to show Theorem 7.1: If Xis an intuitionistic formula, X is intuitionistically valid if and only if M ( X ) is S4-valid. This follows from the next three lemmas.

Lemma7.2: Let (3,92, PI) be an intuitionistic model and (Y, 92, ks4) be an S4 model, such that for any r e 8 and any atomic A

r c, A o r c , , M ( A ) . Then for any formula X

r C,X o

C,,M(X).

Pro08 A straightforward induction on the degree of X.

Lemma 7.3: Given an intuitionistic counter-model for X,there is an S4 counter-model for M ( X ) . Pro08 We have (8,92, C,), an intuitionistic model such that for some r E 9 r pC ,X.We take for our S4 model (3, 9,Cs4) where , C, is defined by A k S 4 A if A k , A for A atomic and any A in 9, and kS4 is extended to all formulas. If A is atomic A Cs4M(A) 0 A cs4 C lA e (VA*) A* Cs4 A o (Vd*) A* I=, A csAk,A and the result follows by lemma 7.2.

44

RELATED SYSTEMS OF LOGIC

~ ~ 1 . 3 8 1

Lemma 7.4: Given an S4 counter-model for M ( X),there is an intuitionistic counter-model for X . Pro08 We have <S, 9,ks4>, an S4 model such that for some r )1s4 M ( X ) . We take for our intuitionistic model (9, W , kI) where bI is defined by A CIA if A kS4M(A) for A atomic and any A in 9, and kI is extended to all formulas. Now the result follows by lemma 7.2.

CHAPTER 4

FIRST ORDER INTUITIONISTIC LOGIC SEMANTICS

0 1. Formulas We begin with the following: (1). denumerably many individual variables x,y, z, w, ... (2). denumerably many individual parameters a, b, c, d,... (3). for each positive integer n, a denumerable list of n-ary predicates A", B", C",D", ... (4). connectives, quantifiers, parentheses, A , v, 3 , -, 3, V, (, ). An atomic formula is an n-ary predicate symbol A" followed by an n-tuple of individual symbols (variables or parameters), thus A"(tll, ...,.),lt A formula is anything resulting from the following recursive rules : FO. Any atomic formula is a formula. F1. If X is a formula, so is X . F2,3,4. If X and Yare formulas, so are (XA Y ) , ( X v Y ) , (Xr>Y). F5,6. If X is a formula and x is a variable, (Vx)X and ( 3 x ) X are formulas. Subformulas and the degree of a formula are defined as usual. The property of uniqueness of composition of a formula still holds. We note the usual properties of substitution, and we use the following notation : If X is a formula and tl and j3 are individual symbols, by X(,") we mean the result of substituting j3 for every occurrence of tl in X (every free occurrence in case tl is a variable). We usually denote this informally as follows: we write X as X ( a ) and X ( i ) as X ( p ) . It will be clear from the

-

46

FIRST ORDER 1 " I S T I C

LOGIC, SEMANTICS

CH.4 4 2,3

context what is meant. We again use parentheses in an informal manner and we omit superscripts on predicates. Although the definition of formula as stated allows unbound occurrences of variables in formulas, we shall assume, unless otherwise stated, that all variables in a formula are bound. Notation like X(x) however, indicates that x may have free occurrences in X . $ 2. Models and validity In this section we define the notion of a first order intuitionistic model, and first order intuitionistic validity, referred to respectively as model and validity. This modeling structure is due to Kripke and may be found, in different notation, in [13] (see also [18]). The notions of ch. 1, if needed, will be referred to as propositional notions to distinguish them. If B is a map from Y to sets of parameters, by 4 (r)we mean the set of all formulas which may be constructed using only parameters of B(r). By a (first order intuitionistic) model we mean an ordered quadruple (3,W,k, B), where Y is a non-empty set, W is a transitive, reflexive relation on 3,C is a relation between elements of 9 and formulas, and B is a map from Y to non-empty sets of parameters, satisfying the following conditions : for any r E Y QO. q r ) 5 p(r*), Q1. T C A *A E @(I') for A atomic, 42. r k A r*k A for A atomic, 4 3 . rk(XA Y ) e T k X a n d r k Y, 44. r k ( X v Y ) - ( X v Y ) E @ ( r )and r k X o r r k Y, Q5. T k - X e - X E @ ( r ) and for all r* r * F X , 46. rC(Xr> Y ) o ( X 3 Y ) E @ ( and ~ ) for all r*,if r*kX, I ' * k Y, 4 7 . I ' k (3x)X(x)-=for some a d ( T ) r k X ( a ) , QS. r k (Vx)X(x)efor every r* and for every a @ ( r * ) r*CX(a). We call a particular formula X valid in the model (9, W,k, 9 )if for all r E Y such that X E@ (r) r C X. X is called valid if X is valid in all models.

-

$ 3 . Motivation

The intuitive interpretation given in ch. 1 5 3 for the propositional case may be extended to this first order situation.

CH.484

47

SOME PROPERTIES OF MODELS

In one's usual mathematical work, parameters may be introduced as one proceeds, but having introduced a parameter, of course it remains introduced. This is what the map B is intended to represent. That is, for re9 r is a state of knowledge, and B(r)is the set of all parameters introduced to reach r. (Or in a stricter intuitive sense, B (r)is the set of all mathematical entities constructed by time r.)Since parameters, once introduced, do not disappear, we have QO. 42-6 are as in the propositional case. 47 should be obvious. Q8 may be explained: to know (Vx) X ( x ) at r, it is not enough merely to know X(u) for every parameter a introduced so far (i.e. for all u E B ( r ) ) . Rather one must know X ( u ) for all parameters which can ever be introduced (i.e. for all a& (r*)

r*c x(a)).

The restrictions Q1, and in 44, Q5 and 4 6 are simply to the effect that it makes no sense to say we know the truth of a formula X if X uses parameters we have not yet introduced. It would of course make sense to add corresponding restrictions to 4 3 , 47 and QS, but it is not necessary. The original explanation of Kripke may be found in [13]. For a different but related model theory in terms of forcing see [ S ] .

5 4.

Some properties of models

Theorem 4.1: In any model (9, W ,C, B), for any

r E 9 ,

xdyr).

if I'k-X,

Proofl A straightforward induction on the degree of X, Theorem 4.2: In any model (9, W , C, Y), for any formula X , if

r u , r*!=x.

Proofi Also a straightforward induction on the degree of X. Theorem4.3: Let 9 be a non-empty set, W be a transitive reflexive relation on '3, and B be a map from 92 to non-empty sets of'parameters such that B(r)sP(r*)for all reg. Suppose C is a relation between elements of 9 and atomic formulas such that r I= A A E @ (r).Then k can be extended in one and only one way to a relation, also denoted by k, between 9 and formulas, such that (9, 92, !=, 9)is a model. Proofi A straightforward extension of the corresponding propositional proof. Definition 4.4: Let (9, W,C, 9)be a model and suppose a is some

48

FIRST ORDER 1 ” I S T l C

LOQIC, SEhfANTlCS

CH.455

parameter such that a# UreSS((r). By (9, W ,I:,9)(3 we mean the model <S, W ,C’, P’) defined as follows: Y ( r )is the same as S(r)except for containing a in place of b if B (r)contains b. For A atomic rkA=>rk‘A(:), and I:’ is extended to all formulas. Lemrna4.5: Let(g, 9, k,8>bearnodel,a$Ur,.S(r),(9, W , I : ‘ , Y ) be (9,W ,I:,9’) (:). Then for any formula X not containing a

r k x O F c ’x )(: . Proof: By an easy induction on the degree of X. Definition 4.6: Let (9, 9, k, 9)be a model and suppose a is some paraBy (9, W , b y@ ) b E a we mean the model meter such that a$ UregB(r). (9, W ,I=’, P’> defined as follows: 9‘ (r)is the same as S (r)except for containing a as well as b whenever S(r)contains b. For A atomic I‘ I:A => r C’ A’, where A’ is like A except for containing a at zero or more places where A contains b, and I:’ is extended to all formulas.

Lemma 4.7: Let ( 9 , W , k, 9) be a model a$ U r s S 9 ( r ) ,and let ( 9 ,W,k’, S’) be <S, 9, k, 8 ) b = , . Then if X is any formula not containing a, and if X ’ is like X except for containing a at zero or more places where X contains b

r cx 0r ~ x ’ .

Proof: Again an easy induction on the degree of X.

0 5.

Examples

We show that two theorems of classical logic are not intuitionistically valid: (1). k c (VX) ( A (4 v - A (41, but the following is an intuitionistic counter-model for it. We take the natural numbers as parameters. Let 9 = {ri i = 0,i , 2 , ...}, T i W F j iff i < j S(rJ = {1,2,..., i, i 11

I

+

m.465

49

EXAMPLES

r,CA(i) iff i < n and k is extended to all formulas. We may give this model schematically by

I We claim no rik

--

etc.

(Vx) ( A ( x ) v --A (x)). Suppose instead that

rikNN(vx)(A(x)v -A(x)).

Then for some j > i

r jc (VX) ( A( x ) v

-

A (x)) .

Butj+lE@(Tj), so rjkA(j+l)vwA(j+l). But rjpl A ( j + 1)sincej+ 1>j, and if rjk - A rj+lpl A ( j + l ) , a contradiction.

( j + l),then since I'j9Wj+l,

(2). k c (V.1 ( A v N x ) ) = ( A v ( w q x ) ) , but an intuitionistic counter-model is the following, where parameters are again integers: 3 = {Tt,

r,> 9

r m Z ,w r l , r,wr, , @(b) = {l),@(r2) = {1,2)

Y

r1wi),r,c~(i), ~ , c A ,

and C is extended to all formulas. Schematically, this is

::g

:::,A

To show this is a counter-model, first we claim

rl C ( V X ) ( Av B ( x ) ) .

50

FIRST ORDER INTUITIONISTIC LOGIC, SEMANTICS

cnI.4Q6,7

This follows because rl t B ( 1 ) . Hence r,CAvB(l)

and r , C A , s o r,CAvB(l)

and r , k A v B ( 2 ) .

rl )cA and moreover r,)c( V x ) B ( x ) rl y A v (VX) B(x). But

since r , v B ( 2 ) . Thus

Q 6. Truth and almost-truth sets In classical first order logic, a set 9 ' of formulas is sometimes called a truth set if o X E Y and Y e Y , (1). XA Y E Y (2). X v Y e 9 e X E or ~Y E Y , (3). N X E Y -3 X4Y, (4). X = Y E Y o X # Y or Y E Y , (5). (3x) X(X)ESP o X ( a ) e Y for some parameter a, (6). (Vx) X ( X ) E Y e-X ( a ) e Y for every parameter a, where there is some k e d set of parameters, X and Y are formulas involving only these parameters, and (5) and (6) refer to this set of parameters. We now call Y an almost-truth set if it satisfies (1)-(5) above and (6a). (Vx) X ( X ) E Y * X ( U ) E Y for every parameter a. It is one form of the classical completeness theorem that for any pure (i.e. with no parameters) formula X, Xis a classical theorem if and only if X is in every truth set. We leave the reader to show Theorem 6.1: If X is pure and contains no occurrence of the universal quantifier, X is in every truth set if and only if Xis in every almost-truth set.

Q 7. Complete sequences The method used in this section was adapted from forcing techniques, and is due to Cohen [3].

m.468

51

A CONNECTION WITH CLASSICAL. LOOIC

Definition 7.1 : In the model (59,g,I.,

@)¶

r,AEQ

=>

we call QE B an W-chain if

r W A or A W r .

If V is an W-chain, by V we mean { X I for some r E V ,

-

rC X I .

If V is an W-chain, V is called complete if for every formula X with parameters used in v' X v XEW.

Lemma 7.2: Let +? be a complete W-chain in the model (9, W,k, 9'). Then V' is an almost-truth set. Pro08 This is a straightforward verification of the cases. We give case (4) as an illustration. Suppose (XDY)EW. Then for some r& rCX3 Y. Now either X $ V or XEW.If XEV, then for some A EV A I.X. Let i2 be the 3-last of r and A . Then 51k Xand Q C X x Y,so 51k Yand YEW'.Thus X#U' or YE%?.

Conversely suppose (XI Y)#V'. Then -X#W, since Q' is closed under modus ponens and contains X D (X3Y) as is easily shown. But X v N X E V , hence XEW. Further Y # V , since again Y I ( X = Y ) E Q ' . N

Lemma 7.3: Let 3 { { T (3.) C X ( 4 = Y(X)l, T Y ( a ) , T X ( a ) l } W 3 x ) [X(X)= Y ( x ) l ,T Y ( 4 , T X ( a ) } } { { F [ X ( a ) 3 Y(a11, T Y ( a ) , T X ( a ) } ), { { T [ X ( a )= Y ( a ) l , T Y(a), T X ( a ) ) ) , { { F X ( a ) , T Y ( a ) , T X ( a ) } ,{ T Y ( a ) , T Y ( a ) , T X ( a ) ) ) , W X ( a ) , T Y ( a ) , T X ( a ) } ,{ T Y ( a ) ,m a ) , T X ( a ) } } .

-

-- ---

5 2.

-

-

-

-

x(~)l}~,

9

9

9

-

Correctness of Beth tableaus

Dejinition 2.1: Let S = { T X , ,..., TX,, FY, ,..., FY,} be a set of signed

~~1.552

CORRECTNESS OF BETH TABLEAUS

55

formulas, ($9, W , I=, 9)a model, and r E 9 . We say I' realizes S if XjE@(I'), yj~@(r), and r C X , , rgc Y, (i=1 ...n , j = l ...m). A set S is realizable if something realizes it. A configuration Q is realizable if one of its elements is realizable.

Lemma 2.2: Let Q stand for either the sign T or the sign F. If S,QX(b) is realizable and if a is a parameter which does not occur in S or in X (so a # b ) then S,QX(a) is realizable. Proo$ Suppose in the model (9,W , 'F, 9 ) I' realizes S,QX(b). Choose a new parameter c # ( J r e g 9 ( I ' ) ( we can always construct a new parameter). Let (9, W , k', 9') be (9, 99,C, 9)(:) (see ch. 4 9 4). Since a does not occur in S or X , by lemma 4.4.5, in this new model r realizes S,QX(b). But now a$UrEgPi'(r), so we may define a third model ($9, W,k", 9"') as (8, 9, C', Pi'),,=a, By lemma 4.4.7 in this third model r realizes S,QX(a). Lemma 2.3: If S,T(3x)X ( x ) is realizable, and if a does not occur in S or X ( x ) , then S,TX(a) is realizable. Proofi Suppose in the model ($9, 9,I=, 9) r realizes S,T(3x) X(x). Then I ' b ( h ) X ( x ) , so for some b&'(r) r b X ( b ) . Thus r realizes S,TX(b). If a=b we are done. If not, by lemma 2.2 we are done. Lemma 2.4: If S,F(3x) X ( x ) is realizable and if a is any parameter, S,FX(a) is realizable. Prm$ Suppose in the model (9, W,'F, 9) r realizes S,F(3x) X ( x ) . Then I' pC ( 3 x ) X ( x ) . If a O ( r ) , r pC X ( a ) and we are done. If a#9(r), a cannot occur in S or X by the definition of realizability. But 9(r)#B so there is a b@(I') with b # a and r pC X(b). Thus S,FX(b) is realizable Now use lemma 2.2. Lemma2.5: If S,T(Vx)X(x) is realizable and if a is any parameter, S,TX(a) is realizable. Proofi Similar to that of lemma 2.4. Lemma 2.6: If S,F(Vx) X ( x ) is realizable and if a is any parameter which does not occur in S or X ( x ) , then S,,FX(a) is realizable. Proofi Suppose in the model (9, W,C, 9) r realizes S,F(Vx) X ( x ) . Then r )I (Vx) X ( x ) . But X ( X ) E @ ( ~so ) , there is a r*such that I'* pC X ( b ) for some b E B ( r * ) . Of course r*realizes S,. If b=a we are done. If not, since S,,X(b) is realizable, by lemma 2.2 we are done.

56

FIRST ORDER INTUITIONISTIC LOGIC, PROOF THEORY

cH.5$3

Theorem 2.7: Let Vl, V,, ..., V nbe a tableau. If V i is realizable, so is Vf+lProo$ We pass from V i to V,+, by the application of some reduction rule. All the propositional rules were dealt with in ch. 2. The four new (first order) rules are handled by lemmas 2.3-2.6.

X is provable, X is valid. Proofi Exactly as in the propositional situation.

Corollary 2.8: If

0 3. Hintikka collections This section generalizes the definitions of ch. 2 0 3 to the first order setting. Recall that a finite set of signed formulas is consistent if no tableau for it is closed. We say an infinite set is consistent if every finite subset is. 9,by 9(r)we Let B be a collection of sets of signed formulas. If mean the collection of all parameters occurring in formulas in r. If r , A e B , by TWA we mean P ( r ) G P ( A )and T,GA.

Definition 3.1: We call B a (first order) Hintikka collection if, for any rE9, r is consistent and

TXAYET * T X e r and T Y E r , FXvYeT = . F X e r and F Y E r , TXvYeT = s T X e r or T Y E T , or F Y E r , FXAYer *FXer *FXer, T NxEr TXIYET - F X E ~ or T Y e r , F -X& *forsome A e Q r W A and T X E A , F X ~ Y E *forsome ~ A E Q , r W A and T X e A , F Y e A , T ( V X ) X ( X ) E =r F T X ( a ) e T for all aeP(r), F ( 3 x ) X ( x ) E f =S F X ( a ) E r for all aeP(r), T ( 3 x )X ( X ) E=5~ T X ( a ) € f for some a e 9 ( r ) , F (Vx) X ( x ) E r =5 for some A E 9 TWA and forsome U E B ( A )T X ( a ) f A . Definition 3.2: If B is a Hintikka collection, we call (9, W,C, 9)a model for B if (1). (9,W ,t=, 9)is a model,

CH.564

HINTIKKA ELEMENTS

51

(2). B and 9 are as above, (3). for all rE9 T X E r a r t X a n d FXEr-TpCX. Theorem 3.3: There is a model for any Hintikka collection. Pro08 Suppose we have a Hintikka collection 9. B and W are as defined above. If A is atomic, let T t A if TAET, and extend =l to all formulas. The result <S,9, k, B ) is a model. We claim it is a model for S. We show property (3) by induction on the degree of X. The propositional cases were done in ch. 2 Q 3. Of the four new cases we only do two as an illustration. Suppose the result known for all subformulas of the formula in question. Then T ( V x ) X ( x ) E ra ( V d ~ 9 ) ( r 9 =A T(VX)X(X)EA) (since rTE A if r W A ) ( V A E ' ~ ) ( ~ W A ( ( V a ~ p ( ( d )T) X ( a ) € A ) ) + (VA E q ( r m => (paE B ( A ) ) A c x (a)))

-

.=,

r c ( v ~x )(XI

Conversely

F (Vx) X (x) E r =5 (34 E 9)(TWA and (3a E B(A)) ( F X (a). A)) * ( 3 4 ~ 9 ) ) F W and(3aeB(A))(AyX(a))) d r y (vX> x ( x ) . Thus, as in the propositional case, to establish the completeness of Beth tableaus we need only show that if Xis not provable, there is a Hintikka collection 9 and a re9 such that F X E r .

0 4. Hintikka elements DeJinition 4.1 : Let r be a set of signed formulas and P a set of parameters. We call a Hintikka element with respect to P if r is consistent and T X A Y E ~ * T X E ~and T Y E r , FXvYer ~ F X and E ~F Y E r , T X V Y E T = > T X E ~or T Y E I ' , FXhYBr *FXEr or FYEL", T-XEr => F X E r , TX=YEr *FXEr or T Y E r ,

r

58

F'IRST ORDER INTUITIONISTIC LOOK, PROOF THEORY

CH.554

T(Vx) X ( x ) d =-T X ( a ) E r for each U E P , F ( 3 x ) X ( x ) ~ =r FX(a)ET for each U E P , T ( 3 x )X ( x ) E r TX ( a ) E r for some U E P . Theorem4.2: Let r be an at most countable, consistent set of signed formulas. Let S be the set of all parameters occurring in formulas in r. Let a,, u2, a3,... be a countable list of parameters not in S. Let P=Su {u,, u2,a3,...}. Then r can be extended to a Hintikka element with respect to P. Proofi Order the (countable) set of all subformulas of formulas in r, using only parameters of P : X,, X,, X,, .... We define a (double) sequence of sets of signed formulas: Let r0=r. Suppose we have defined r,,which is a consistent extension of T o , using only hitely many of a,, a,, a3,.... Let A,' =r,,.We define A,,,..., 2 A:+' and let r,,,=A",'.We do this as follows: Suppose we have defined A: for some k (I GkG'n). Consider the formula x,. ~t

most one of

neither is, let A:+

TX,. FX, can be in A'C, (since it is consistent). If If one is in A:, we have several cases.

=A :.

Case (la). x k iS Y v Z and TxkEd:. Then one of At,TY or At,TZ is consistent. Let A:+ be Af:,TYif consistent, and A:,TZ otherwise. Case (lb). xk iS Y v Z and F&€d:. Then A:,FY,FZ is consistent. Let this be A:+'. The cases (2a). T X A Y , (2b). FXA Y, (3). T - J X , (4). T X 3 Y, are all treated in a similar manner. Case (5a). xk is (3x) X(x) and T&€&. Since Af: uses only finitely many of a,, u,, u3,..., let ui be the first one unused. Let A:+' be Ak,,TX(a,). Since a, is new, this must also be consistent. Case (5b). xk is (3x) X ( x ) and FX,EA:. Let At+' be A: together with FX(a) for each aES, and each @=aiwhich has been used so far. Then A:+ is again consistent. Case (6). T(Vx)X ( x ) , is treated as we did case (5b). Case (7). If the signed formula does not come under one of the above cases let A:+' =A:.

'

CH.555

COMPLETENESS OF BETH TABLEAUS

59

Thus we have defined a sequence r0,rl,r2, .... Let n=Urn.We claim Kl is a Hintikka collection with respect to P . The verification of the properties is straightforward.

0 5. Completenessof Beth tableaus Supposing X to be not provable, we give a procedure for constructing a sequence of Hintikka elements. First we order our countable collection of parameters as follows:

S,:

1

1

1

a,, a z , a 3 ,

...

s2: a : , a3; , a 3: , ... s3: a.: , a.z , a.3 , ... .. .. ..

where we have placed all the parameters of X in S,, and let Pn= =s, v s, v ... v s,. For this section only, by an F-formula we mean a signed formula of the form F - X , F X 3 Y or F ( V x ) X . We many assume once and for all an ordering of all formulas. Now we proceed: Step(0). X is not provable, so { F X } is consistent. Extend it to a Hintikka element with respect to P,. Call the result rl. Step (1). Take the first F-formula of rl. If this is F - X , consider rlT,TX.This is consistent. Extend is to a Hintikka element with respect to P z , call it rz.If the first F-formula is F X 3 Y, extend rlT,TX,FYto a Hintikka element with respect to P J Z . If the first F-formula is F ( V x ) X ( x ) , extend Tl,,FX(a:) to a Hintikka element with respect to P J Z . In any event rzis a consistent Hintikka element with respect to P2. Now call the first F-element of rl “used”. The result of step (1) is

{rl,rz1.

Supposeat theend of step (n)we have the sequence {rl,rz,r3, ..., rzn} where each Tr is a Hintikka element with respect to P,. Step (n+ 1). Take the first “unused” F-formula of T i , proceed as in step (I) depending on whether the formula is F - X , FX= Y or F ( V x ) X. Produce from rlT,TX or rlT,TX,FY or rlT,FX(a;“+’)a Hintikka element with respect to Pzn+l, call it rz,,+i, and call the formula in question “used”. Repeat the same procedure with the first “unused” F-formula of rz,producing a Hintikka element with respect to Pzn+z,

60

FIRST ORDER INlTJITIONISTIC LOGIC, PROOF THEORY

c~.5§6

call it r2A+2. Continue to rzn, producing a Hintikka element with respect call it rzn+l. The result of the n+lst step is thus to PZn+,,

Let '3 be the collection of all r,,generated in the above process. We claim 4 is a Hintikka collection. Each r n E % is a Hintikka element with respect to P,,, so B(T,,) is P,,. Since r,,is a Hintikka element with respect to s(I',,), to show B is a Hintikka collection we have only three properties to show. Suppose for some r,,€'3,F(Vx) X(x)Er,,. By the above construction there must be some r&'3 such that r , , T E r k , B(r")=g(rk)and Fx(a)Erk for some parameter a. Thus (3fke%)r,wrkand F X ( u ) M k for some a&(rk). The cases F- and F= are similar. Thus '3 is a Hintikka collection and F X E ~ , Eso~ our , completeness theorem is established. We note that in the Hintikka collection '3 resulting, every formula is a subformula of X . We remark also that the construction of 54 and of this section could be combined into a single sequence of steps. This proof is a modification of the original proof of Kripke [13].

0 6.

Second completeness proof for Beth tableaus

The following is a Henkin type proof and serves as a transition to the completeness of the axiom system presented in the next few sections. A proof along the same lines but using unsigned formulas was discovered independently by Thomason [21] and by Aczel [l]. The similarity to the aIgebraic work of ch. 1 $ 6 is also noted. Recall that a finite set of signed formulas r is consistent if no tableau for it is closed. An infinite set is consistent if every finite subset is, Defhition 6.1. : Let P be a set ofparameters and f a set of signed formulas. We call r maximal consistent with respect to P if (1). every signed formula in r uses only parameters of P, (2). r is consistent, (3). for every formula X with all its parameters from P,either T X E ~ or F X E r or both r , T X and r,FX are inconsistent.

SECOND COMPLETENESS PROOF FOR BETH TABLEAUS

c ~ . 5 § 6

61

Lemma6.2: Let I‘ be a consistent set of signed formulas, and P be a non-empty set of parameters containing at least every parameter used in r. Then r can be extended to a set A which is maximal consistent with respect to P. Proofi P is countable, so we may enumerate all formulas with parameters from P: XI, X2, X3, ... Let A o = r . Having defined A,,,consider X,,,,. If A,,,TX,,+, is consistent, let it be A,+l. If not, but if A,,FX,,+, is consistent, let it be A,,+l. If neither holds, let A,,+, be A,. Let A = A,,. The conclusion of the lemma is now obvious.

u

Definition 6.3 : Let r be a set of signed formulas and P a set of parameters. We call r good with respect to P if (1). r is maximal consistent with respect to P, (2). T(3x)X ( x ) ~ r * T X ( a ) € rfor some UEP.

Lemma 6.4: Let r be a consistent set of signed formulas, and S be the set of parameters occurring in r. Let {u,, a2,a3,...} be a countable set of distinct parameters not in S, and let P = S u {al, a,, u3,...}. Then r can be extended to a set A which is good with respect to P. Proof: P is countable, order the set of formulas with parameters from P: X,, X,, X3, . We proceed as follows: (1). Let A0=T. (2). Extend A, to a set A , maximal consistent with respect to S. (3). Take the fist X , (in the above ordering) of the form T(3x)X ( x ) suchthatT(3x)X(x)EAl butfornoaESisTX(u)EA,. LetA2=A,,TX(al). Since a, is “new”, A z is consistent. (4). Extend A 2 to a set A3 maximal consistent with respect to S u (al}. (5). Take the first Xi of the form T(3x) X(x) such that T ( 3 x ) X(x)eA3 but for no a ~ S {al} u is TX(ar)€A3. Let A4=A3,TX(a2). Again A4 is consistent. (6). Extend A4 to a set A s maximal consistent with respect to S u {ai,az} And so on. Let A = A,. We claim A is good with respect to P. First A is consistent since each A,, is consistent. If X has all its parameters in P,then for some n all the parameters of X are in S u {ul, u2,..., a,,}.But in step (2n) we extend Azn to AZnfl, a set maximal consistent with respect to S u {al, a2,..., a,,}. Thus TX or

u

62

FIRST ORDER INTUITIONISTIC LOGIC, PROOF THEORY

c1i.556

FX is in A,,,, and hence in A , or neither can be added consistently. Thus A is maximal consistent with respect to P. Finally suppose T(3x)X ( x ) ELI. We note that the formula dealt with in step ( 5 ) is different from the one dealt with in step (3), and the one dealt with in step (7) is different again. Thus we must eventually reach T(3x)X ( x ) , and so for some ~ E TPX ( ~ ) E AHence . A is good with respect to P.

Now let us order our countably many parameters as follows: S,:

1

1

1

a2, a$,

... ...

a:, a ; ,

...

a,, a 2 , a 3 ,

s,: a : , s ~ :a : ,

...

2

. ..

. ..

and let P,=S, u S, u ... u S,. Let 93 be the collection of all sets of signed formulas which are good with respect to some P,.We claim B is a Hintikka collection. Suppose r E 9 . Then r is good with respect to some Pi, say P,. Then P(r)(the collection of all parameters of r)is P,. Suppose T X A YET but TX$r. If I",TXA Y is consistent, so is r,TXr\ Y,TX, and so I' is not maximal. Thus TXEI'. Similarly TYEr. Hence T X A YEr-TXEr and T Y E r . Similarly we may show

Moreover

T ( 3 x ) X ( x ) ~ r+ T X ( a ) E r for some a € P ( r ) , since r is good with respect to P,.

CH.5g7

AN AXIOM SYSTEM, d i

63

Suppose F - X E r . Since is consistent, rT,TX is consistent. Extend it to a set A which is good with respect to P,,+l.Then B ( T ) = 9 ( A ) and rTS A , so r W A and TXEA . Similarly, if F X 3 YEr, there is a A E such ~ that r9A, TXEA and FYEA . Finally, if F(Vx) X ( x ) E r , since a;+ does not occur in r, T T , F X ( d +’) is consistent. Extend it to a set A which is good with respect to P,,+l. Again r9A and FX(a:+’)EA for 4” E ~ ( A ) . Thus 9 is a Hintikka collection. To complete the proof, suppose X is not provable. Then { F X } is consistent. Since it has only finitely many parameters, they must all lie in some P,,. Extend { F X } to a set r good with respect to P,,.Then re9 and FXd‘.This establishes completeness. Remark 6.5: The model resulting from this Hintikka collection is a “universal” model in that it is a counter-model for every non-theorem. This is not the case for the model of 5 5. We will show later that, in a sense, this Hintikka collection is the analog of a classical truth set.

0 7. An axiom system, d , The following system was chosen to give a fairly quick completeness proof. It is very close to the system of [lo] p. 82. Axiom schema: 1. X ” ( Y ” X ) , 2. ( X 3 Y ) 3 ((X” ( Y 3 2))2 ( X I . Z ) ) , 3. ((x”z)A(Y3z))~((xV Y)s>z), 4. ( X h Y ) ” X , 5. ( X A Y ) 3 Y, 6. X I ( Y D ( X A Y)),

7. X”(XV Y), 8, Y ” ( X v Y), 9.

(XA

-” -

-x)3Y,

10. (X” X)3 X, 11. X ( a ) ( 3 x ) X ( x ) , 12. (Vx) X ( x ) 3 X ( a ) .

64

FIRST ORDER INTUITIONISTIC LOOIC, PROOF THEORY

CH.557

Rules: 13.

X(u)3 Y (3x1 X ( x ) y Y 3 X(u) = l

14.

Y

y = (VX) X ( x ) X, X 3 Y 15. Y . In rules 13 and 14 the parameter u must not occur in Y. In a deduction from premises the parameter u must not occur in the premises either. We use the usual notation, if X can be deduced from a finite subset of S, we write S k X . We use k X for 0 !-X . In the next three sections we establish the correctness and completeness of d,. We introduce a second system d 2 ,equivalent to d,, to aid in showing correctness. For use in showing completeness we need the following three lemmas : Y

Lemma 7.1: The deduction theorem holds for d l . Proof: The standard one (e.g. [lo] $5 21, 22). Lemma 7.2: i- ( W A Y ) 2 X ,

Proof: (1). ( W A Y ) = X (2). ( W A Z ) X X (3). W = ( Y v Z ) (4). w (5). Y v z (6). W D ( Y = ( W AY ) ) (7). Y 3 ( W A Y ) (8). W x ( Z D( W A2)) (9). z 3 ( W A 2) (10). Y X X (11). Z 3 X (12). ( Y V Z ) X X (13). X (14). W 2 X

by hypothesis, theorem , by hypothesis, theorem, by hypothesis, theorem , premise, by (31, (41, rule 15 , axiom 6 , by (41, (61, rule 15, axiom 6 , by (41,(81, rule 15, via (11, (3, via (21, (9) , via (10, (1l), axiom 3, by (51, (121, rule 15, deduction theorem cancelling premise (4).

CH.558

A SECOND AXIOM SYSTEM, d a

65

Lemma 7.3: If a does not occur in W, Y ( x ) or X , k ( w A Y ( Q ) ) = X , k W D ( 3 X ) Y(X) kW3X

Proofl (1). ( W A Y ( U ) ) D x (2). w D ( 3 x ) Y ( x )

I

by hypothesis, theorems ,

premise , by (21, (31, rule 15> axiom 6 , by (31, (9,rule 15, via (11, (61, by (7), rule 13, by (4), (81, rule 15, deduction theorem cancelling premise (3).

5 8. A second axiom system, d 2 We introduce a second, very similar, axiom system, and prove equivalence. -01, has the same axioms as d,, as well as rules 13 and 14. It does not have rule 15. Instead it has rules 14a.

X(a) (VX)

X(X)

15a. (Vx,) ...(Vx,,) X, (3xl) ...(gx,,)X x Y Y provided all parameters of (Vxl) ...(Vx,,) X are also in Y (n may be 0). To show the two systems are equivalent, it suffices to show 14a and 15a are derived rules of d,, and 15 is a derived rule of -01,. To show 14a is a derived rule of d,, suppose in -01, we have X(a). Let T be any theorem of -01, with no parameters. By axiom 1, X ( a ) ~ ( T = X ( a ) )so , by rule 15, T 3 X ( a ) . Since a is not in T,by rule 14, T x ( V x ) X ( x ) . But also T, so by rule 15, (Vx) X ( x ) . To show 15a is a derived rule of d l , suppose in -01, we have (Vxl)... (Vx,,) X ( x , , ..., x,,) and (3x,) ( j x , , ) X ( x l , ..., x , , ) Y, ~ and all

...

66

FIRST ORDER INTUITIONISTIC LOGIC, PROOF THEORY

CH.599

parameters of (Vx,) ...(Vx,,) X(x,, ...,x,,) are in Y. From (Vx,). ..(VxJ X(x,, ..., x,,), by axiom 12, X(a,, ..., a,,). From axiom 11, X(a,, ...a,,) 2 (3x,) ...(3xJ X ( x , , ..., x,,), so by rule 15, ( j x , ) (3x,,) X(x,, ..., x,,) and by rule 15 again, Y. Finally to show rule 15 is a derived rule of d2,suppose we have X and X I Y in d2. Let a,, u2 ..., a, be those parameters of X not in Y. Since we have X(a,, ..., a,,), by rule 14a, (Vxl) ...(Vx,,)X ( x , , ..., x.). Similarly, since X(a,, ..., an)3Y and a,, ..., a,, do not occur in Y, by rule 13, (3x,) ...(3xn) X ( x , ,..., x,,)= Y. Now by rule 16a, Y. Thus d,and d2are equivalent. For use in the next section we state the straightforward

...

Lemma 8.1: If in d 2we can prove X(a), there is a proof of the same length of X(b) for any parameter b. (note: a does not occur in X ( b ) =

x (a)(;)). 5 9.

Correctness of the system d 2

Theorem 9.1: If X is provable in d,, X is valid. Pro08 By induction on the length of the proof for X.If the proof is of length 1, X is an axiom and we leave the reader to show validity of the axioms. Suppose the result is known for all formulas with proofs of length less than n steps, and Xis provable in n steps. We investigate the steps involved in the proof of X . Axioms have been treated. Suppose X ( u ) x Y in rule 13 is provable in less than n steps where a is not in Y. Then X(a)=, Y is valid. Then ( 3 x ) X ( x ) 3 Y is provable. We I=, 9 ) and any r& and wish to show it is valid. Take any model where ak is the last parameter used in A:. Let A = r,, then A has the following properties : A uses exactly the parameters of Q. X$A since X $ r , for any n. A is deductively complete with respect to Q . A has the Or-property. For if Y v Z EA, say Y v Z = Z,, then Y v ZEA , for some m. We can take m>n. Then Y v Z = Z , e A C , so either Y or Z is in A;,%A. Similarly, A has the 3-property.

'.

u

CH.

5 8 10

COMPLETENES OF THE SYSTEM d i

69

Lemma 10.4: If r is nice with respect to P: (1). X A YE^ (2). X v YE^ (3). - X E r

+XErand YEr, +XEI'or YE^, *X#r, (4). A '= YE^ =.X$ror YE^, (5). (3.) X ( x ) E r CJ X ( a ) E r for some u E P , (6). (Vx) X ( x ) E r X(a) E r for every a E P.

Proofi (1). By axioms 4, 5 and 6 , since r is deductively complete with respect to P. (2). X v Y d j X E r or YEr, since r has the Or-property. The converse holds by axioms 7 and 8. (3). If - X E ~ ,X $ r since r is consistent (using axiom 9). (4). If 1 2 YE^, either X # r or YE^ since r is deductively complete with respect to P . (5). If ( 3 x ) X ( x ) ~ I ' X , ( a ) ~ rfor some a 6 P since r has the 3property. The converse is by axiom 11. (6). By axiom 12.

Lemma 10.5: Suppose r is nice.with respect to P , and {ulyu2,u3...} is a set of distinct parameters not in P . Let Q = P u {al, u2, u3...I. Then (1). If X has all its parameters in P but - X # r , I' can be extended to a set A nice with respect to Q such that X EA. (2). If X= Yhas all its parameters in P but Xx Y#T,I'can be extended to a set A nice with respect to Q such that XEA and Y#A. (3). If X ( x ) has all its parameters in P but (Vx) X ( x ) # r , r can be extended to a set A nice with respect to Q such that for some ~ E Q , X(a)#A. Proofi (1). Since - X # r , {r,X } is consistent, for otherwise T,XF-X. So by the deduction theorem C-X= - X , and by axiom. 10 rF - X , so - X E ~ Since . ( r , X } is consistent, there is some Y such that r , X y Y. Now use lemma 10.3. (2). r , X y Y for otherwise, by the deduction theorem r l - X = Y, so XI>YEr. Since r , X y Y,use lemma 10.3. (3). a, # P . We claim r y X(ul). Suppose r l - X ( a l ) .For the conjunction, call it W, of some finite subset of r, 1 FVzX(al). But a, does not

I0

FIRST ORDER INTUITIONISTIC LOGIC, PROOF THEORY

CH. 5 8 10

occur in W.By rule 14 t- W D (Vx) X(x), so C r (Vx) X(x), (Vx) X ( x ) E r . Since r y X(al), use lemma 10.3.

Now we proceed to show completeness. We arrange the parameters as follows : S,: a il , a,,l a l3 , ...

s,:

a : , a:, a : ,

s3: a.: , a., , a.3 , .. .. .. 3

3

... ...

and let P,=Sl uS,u...uS,, . Let 3 be the collection of all nice sets Let with respect to any Pi. If r e g , r is nice with respect to, say, P,,. 9’(r)=Pn. Let r W A if B(T)EB(A) and rc A. For any X, let r C X iff X d ‘ . By lemmas 10.4 and 10.5 ( 9 , W , I=, 9’) is a model. Finally, suppose y X. All the parameters are in, say, P,. Since 0 y X, by lemma 10.3we can extend 0 to a set r, nice with respect to P, such that X#r.Thus r e g , X E @(r)and r y X. Remark 10.6: This is a “universal” model in the sense of Q 6.

In ch. 6 5 4 we will show that the set of all theorems using only parameters of P, is itself a nice set with respect to P,. This would make the final use of lemma 10.3 above unnecessary.

CHAPTER 6

ADDITIONAL FIRST ORDER RESULTS

0 1. Compactness We call an infinite set S of signed formulas realizable if there is a model (9, W ,C, 9')and a re9 such that for any formula X

T X E S *X E @( T ) and T t X , F X E S X E @ ( r ) and r y X . There is a similar concept for sets of unsigned formulas U. We say Uis satisfiable if there is a model (9, 92, C, 9') and a I ' E g such that for any formula X X E U + X X E @ ( ~and ) rkX.

Lemma 1.1: Let U be a set of unsigned formulas and define a set S of signed formulas to be { TX I XEU}.Then (1). U is satisfiable if and only if S is realizable (2). U is consistent if and only if S is consistent. Pro08 Part (1) is obvious. To show part (2), suppose U is not consistent. Then some finite subset {ul, ..., us}is not consistent, so from it we can deduce any formula. Let A be an atomic formula having no predicate symbols or parameters in common with {ul, ..., u,}. Then kl(U1 A * * * A U n ) 3 ~ .

72

ADDlTIONAL FIRST ORDER RESULTS

~~1.661

Hence there is a closed tableau for

so there is a closed tableau for

By the way we have chosen A , there must be a closed tableau for { T(u, A A u,)} and hence for { Tu,, ...,Tu,}. Thus S is not consistent. The converse is trivial.

-..

Because we have this lemma, we will only discuss realizability and consistency of sets of signed formulas. Lemma 1.2: Let S be a set of signed formulas. If S is realizable, S is consistent. Pro08 If S is not consistent, some finite subset Q is not consistent. That is, there is a closed tableau Vl, V, ..., V, in which gl is {Q}. If Q were realizable, by theorem 5.2.7 every gi would be, but a closed configuration is not realizable. Lemma 1.3: Let S be afinite set of signed formulas. If S is consistent, S is realizable. Pro08 Let S be { TX,, ..., TX,, FYI,..., FY,}. S is consistent if and only if (F(X1 A

**.

A

x,) 13 (Y1 V * * * V Y,)}

is consistent. If this is consistent, (XIA AX,)^ (& v v Y,) is a . non-theorem, so by the completeness theorem, there is a model (9, 2,k, 9)and a r E 3 such that X j € @ ( r ) ,$ ~ @ (and r)

rpc(xlA...Ax,)D(Y~ V - v Y,). But then for some r* r*tx,A - A x,, r*)cY, v . - v Y,, so r*realizes S. This method does not work if S is infinite, but the lemma remains true, at least for sets with no parameters. The result can be extended to sets with some parameters, but we will not do so.

m.651

73

COMPACTNESS

Lemma 1.4: Let S be an infinite set of signed formulas with no parameters. If S is consistent, S is realizable. Proofl The proof can be based on either of the two tableau completeness proofs. If we use the first proof, that of ch. 5 0 5, change step 0 to: “Sis consistent. Extend it to a Hintikka element with respect to PI. Call the result r1”.Continue the proof as written. The lemma is then obvious. If we use the proof of ch. 5 0 6 the result is even easier. S is consistent, so by lemma 5.6.4, we can extend S to a set I‘ which is good with respect to P,.The result follows immediately. Theorem 1.5: If S is any set of signed formulas with no parameters, S is consistent if and only if S is realizable. Corollary 1.6: If every finite subset of S is realizable, so is S. Corollary 1.7: If U is any set of unsigned formulas with no parameters, U is consistent if and only if U is satisfiable.

Remark 1.8: The last corollary could have been established directly by adapting the completeness proof of ch. 5 Q 10. Definition 1.9: For a set of formulas U, by X E u.

r b U we mean I‘ C X for all

Corollary 1.10 (strong completeness): Let U be any set of unsigned formulas with no parameters. Then UtIX if and only if in any model <S,W , C, P),for any I ‘ E S , if l-k- U, I‘k-X. Proofi Ut, X if and only if { TY Y EU >u { F X } is inconsistent,

I

Corollary 1.11: (cut elimination, Gentzen’s Hauptsatz): If S is a set of signed formulas with no constants and {S, T X } and {S, F X } are inconsistent, so is {S}.

Remark 1.12: This may be extended to sets S with some parameters. To be precise, to any set S which leaves unused a countable collection of parameters. It follows that in the completeness proof of ch. 5 Q 6 a set A maximal consistent with respect to P actually contains TX or FX for each X with parameters from P.

74

ADDITIONAL FTRST ORDER RESULTS

CH. 682

$2. Concerning the excluded middle law If S is a set of unsigned formulas, by S t , X and S k , X we mean classical and intuitionistic derivability respectively. Let X(a,, ..., a,) be a formula having exactly the parameters al, ..., a,. By the closure of X we mean the formula (VxiJ

(Vxi.) X(xil, ' " 9 xi,)

(where x i j does not occur in X ( a l , ..., a,,)). Let M be the collection of the closures of all formulas of the form X v -X. We wish to show:

Theorem 2.1: If X has no parameters,

t C Xe M F I X . We first show:

Lemma 2.2: Let (9,92,k, 9')be a model, r E 9 , and suppose Y E M rk-Y. Then r can be included in a complete .%chain V such that %" is a truth set (see ch. 4 Q 6). Pro08 Enumerate all formulas beginning with a universal quantifier:

XI,x2,x,,....

Let T o=r. Having defined r,,,consider X , + l . If X,+ $ @ (r:)for any T:, let I'n+l=I',,. Otherwise there is some r,*such that X,+,E@(~,*). Say X,,, is ( V x ) X ( x ) . We have two cases: (1). If r,*b-(Vx)X(x), let

r,+,=r:.(2). If r,*pC (Vx) X ( x ) , there is a r,**and an a ~ P ( r , *such *) that r,** y X(a). Let r,+,be this r,**. Let the %chain V be {r0,rl,r2,...I. Since YEM=-Tb Yand T = T 0 ,

V is a complete %chain by the definition of M, and so V' is an almosttruth set. Thus we have only one more fact to show:

Y (ct) E V' for every parameter CI of

Q'

+ (Vx) Y (x) E V' .

Suppose (Vx) Y (x, al, ..., a,,) $ V' (where al, ..., a, are all the parameters of Y ) . If some ai is not a parameter of W, we are done. So suppose each at occurs in V'. Then for some r,,€V, all ai@(I',,) and r. j! (Vx) Y(x, al, ..., an). But by the construction of Q, there is a rm(m2n) such that r, pC Y(b,al, ..., a,) for some b~P(r,,,).But

r t (vxl) ...(vx,)

(VX) [Y (x, x ~..,., x,) v

Y (x, x ~..., , x,,)]

~1-1.683

and r9Wm, so

75

SKOLEM-LOWENHEM

r,t

Y ( b , cxl ,... a,) v

N

Y ( b , 011, ..., a n ) ,

thus rmk-Y(b, a,, ..., a,). -Y(b, a1,..., cx,)~%‘, so Y(b,a,..., cxn)$V’ for a parameter b of V’. Now to prove the theorem itself: If M t , X then for some finite subset {ml, ..., m,} of M

t,(ml A A mn)3 X . By theorem 4.8.2 (and the completeness theorems) kc(ml A * . * A m,)

3

x.

But kcml A .--Am,, hence tCX. Conversely, if MY ,X,let S be the set of signed formulas

I

{ F X ) u {TY Y E M } . Since MY ,X, S is consistent. Then by the results of the last section, S is realizable. Thus there is a model (’3, W ,k, 9’) and a r E B such that Y E M TC ~ Y, X e @ ( T ) and ry X. But X has no parameters, so X v X E M .Thus k X v X , so C X. Now by lemma 2.2 there is a truth set containing -X. Hence Y .X. N

-

-

0 3. Skolem-Ldwenheim By the domain of a model (9, 9, k, 9’) we mean ursgY(r). So far we have only considered models in which the domain was at most countable. Suppose now we have an uncountable number of parameters and we change the definitions of formula, model and validity accordingly, but not the definition of proof.

Theorem 3.1: X is valid in all models if and only if X is valid in all models with countable domains. Pro08 One half is trivial. Suppose there is a model (9, W , k, 9’) with an uncountable domain in which X is not valid. The correctness proof of ch. 5 # 2 or 9 is still applicable. Thus Xis not provable. Since Xis not provable, if we reduce the collection of parameters to a countable number (including those of X), X still will not be provable. Then any of the completeness proofs will furnish a counter-model for X with a countable domain.

76

c1i.694

ADDITIONAL FIRST ORDER RESULTS

This method may be combined with that of Q 1 to show Theorem 3.2: If S is any countable set of signed formulas with no parameters, S is consistent if and only if S is realizable in a model with a countable domain. Theorem 3.3: If U is any countable set of unsigned formulas with no parameters, U is consistent if and only if U is satisfiable in a model with a countable domain.

Remark 3.4: In part 11, we will be using models with domains of arbitrarily high cardinality.

5 4.

Kleene tableaus

The system of this section is based on the intuitionistic system G3 of [lo]. The modifications are due to Smullyan. The resulting system is like that of Beth except that sets of signed formulas never contain more than one F-signed formula. Explicitely, everything is as it was in ch. 2 Q 1 and ch. 5 Q 1 except that the reduction rules are replaced by the following, where S is a set of signed formulas with at most one F-signed formula.

KT v

S,TXvY S, T X ]S, T Y .

KFV

ST,FXVY ST,

FX

ST,F X v Y KTA

S,TXAY S, T X , T Y

KTKT

3

KT 3 KT V

S,T-X ST, FF S,TX3Y ST,F X I s, T Y S, T ( h )X ( X ) S, TX(a> S, T ( V X )X (x) S, T X ( a )

where in KT3 and W V the parameter a does not occur in S or X ( X ) .

c~.6§4

77

KLEENE TABLEAUS

There are several ways of showing this is actually a proof system for intuitionistic logic. We choose to show it is directly equivalent to the Beth tableau system, that is, we give a proof translation procedure. We leave it to the reader to show the almost obvious fact that anything provable by Kleene tableaus is provable by Beth tableaus. To show the converse, we need

Lemma 4.1: If a Beth tableau for { TX,, ..., TX,, FYI,..., FY,) closes, then there is a closed Kleene tableau for

{TX,, ..., TX,, F ( Y , v ..*v Y,)}. Proofi The proof is by induction on the length of the closed Beth tableau. If the tableau is of length 1, the result is obvious. Now suppose we know the result for all closed Beth tableaus of length less than n, and a closed tableau for the set in question is of length n. We have several cases depending on the first step of the tableau. If the first step is an application of rule F A , the Beth tableau begins

{{% FXi, .-.,FXn, FY A z}}, {{ST,FXI, * - * , FXn, F Y } {ST, F X I , * * * , FXn, F Z } } 2

3

and proceeds to closure. Now by the induction hypothesis there are closed Kleene tableaus for {&, F(X1v v Xn v Y ) } and {ST,F ( X , v v Xn v Z ) } . We have two possibilities: (1). If Y is not “used” in the first tableau, or if Z is not “used” in the second tableau, a Kleene tableau beginning

---

{{ST, F(X1 {{ST, F ( X 1

x,

v *’* v v (Y v *..v Xn))}

A

Z))}}

9

9

must close. (2). If both Y and Z are “used”, a Kleene tableau beginning {{snF(Xi v * * . v Xn v ( Y {{ST,

F(‘Y

A

A

z))}],

Z)>>

9

{{ST, F Y I , {ST,FZ}} Y

must close. The other cases are similar and are left to the reader.

78

ADDITIONAL FIRST ORDER RESULTS

CH.685

Thus the two tableau systems are equivalent. Now we verify a remark made at the end of ch. 5 0 10. Lemma 4.2: (Godel, McKinsey and Tarski): t,X v Y iff tlX or tl Y. Proofi Immediate from the Kleene tableau formulation. Lemma 4.3: (Rasiowa and Sikorski): If tl(3x) X ( x , al, ..., a,) where a, ,..., a, are all the parameters of X , then klX(b, a,, ..., a,) where b is one of the ai.If X has no parameters, b is arbitrary and kl(Vx) X(x). Pro08 A Kleene tableau proof of (3x) X ( x , a,, ..., a,) begins

{ { W x ) X(X, a,, a,)}}, {{FX(b, a,, --.a,)}}, a*.,

Y

and proceeds to closure. If b is some ai,we are done. If not, we actually have a proof, except for a different first line, of (VX)

8 5.

X ( X , 01,. * * Y a,).

Craig interpolation lemma

Theorem 5.1: If FIX= Y and X and Y have a predicate symbol in common, then there is a formula 2 involving only predicates and parameters common to X and Y such that k , X > Z and kIZ 3 Y; if X and Y have no common predicates, either tl X or t, Y.

-

The classical version of this theorem was first proved by Craig, hence the name. The intuitionistic version is due to Schiitte [17]. Essentially the same proof was given for a natural deduction system by Prawitz [15]. We give basically the same proof in the Kleene tableau system. For another proof in this system see [ll]. We find it convenient to temporarily introduce two symbols t and f into our collection of logical symbols, letting them be atomic formulas, and letting them combine according to the following rules.

x v t=

tv

x =t,

Xvf=f vx=x, X A t= t A X=X, x A f = f AX=f, -t=f, -f=t,

CH.645

CRAIG INTERPOLATION LEMMA

79

x 3 t = f 3 x = t, t3X=X X3f=-X, (3x)t = (Vx)t = t ,

( 3 x ) f = (Vx)f = f . By a block we mean a finite set of signed formulas containing at most one F-signed formula. When we call a block inconsistent, we mean there is a closed Kleene tableau for it. By an initialpart of a block we mean any subset of the T-signed formulas. We make the convention that if S is the finite set of unsigned formulas {XI,..., X,,)then TS is the set {TX,, ..., TX,,}. We further make the convention that for a set S of formulas, S, and S, represent subsets such that S, nS, =8 and S, u S, =S. By [S] we mean the set of predicates and parameters of formulas of S, together with t and$ Now we define an interpolation formula X for the block (TS,P Y ) (where S is a set of unsigned formulas and Y is a formula) with respect to the initial part TS,, which we denote by { TS, F Y } / {TS,}, as follows ( X may be t or f,but we assume t and f are not part of S or Y ) : Xis an ( T S , FY}I(TS,} if (1). [XI E CSll" CSZY y1, (2). { TS,, F X } is inconsistent, (3). (TX,TS,, FY} is inconsistent (we have temporarily added to the closure rules: closure of a set of signed formulas if it contains Tf or Ft). Lemma 5.2: An inconsistent block has an interpolation formula with respect to every initial part. Proofi We show this by induction on the length of the closed tableau for the block. If this is of length 1, the block must be of the form

{ TS,TX,FX}. We have two cases: Case (1). The initial part is ( T S , , T X } . Then X is an interpolation formula. Case (2). The initial part is { T S , ) . Then ( TS,, TX,FX} is inconsistent and t is an interpolation formula. Now suppose we have an inconsistent block, and the result is known for all inconsistent blocks with shorter closed tableaus. We have several cases depending on the k s t reduction rule used.

80

ADDITIONAL FIRST ORDER RESULTS

c~.6§5

K T v : The block is { T S , TX v Y, FZ}, and { T S , TX, F Z } and { T S , TY, F Z } are both inconsistent. Case (1). The initial part is { T S , , TX v Y } . Then by the induction hypothesis there are formulas U, and U, such that

U,is an {TS, T X , FZ}/{TS,,T X } , U2is an { T S , T Y , F Z } / { T S , , T Y } . Then U, v U, is an { TS, TX v Y, F Z } / {TS,, TX v Y } . Case (2). The initial part is { T S , } . Again, by hypothesis, there are U,, U2such that U,is an { TS, T X , F Z } / { T S , } , U,is an {TS, T Y , F Z ) / { T S , } . Then U, A U2is an {TS, TX v Y, F Z } / { TS,}. KF v : The block is { T S , FX v Y } , and { T S , F X ) or { T S , FY} is inconsistent. Suppose the first. Let the initial part be { T S , } . By hypothesis there is a U such that

U is an {TS, F X ) / { T S , ) . Then U is an { TS, FX v Y } / {T S , } . K T A : The block is { T S , TX A Y, F Z } , and { T S , TX, TY, F Z } is inconsistent. Case (1). The initial part is { TS,, TX A Y } . By hypothesis there is a U such that U is an {TS,T X , T Y , F Z } / { T S , , T X , T Y ) . Then U is an { T S , TX A Y, F Z } / {TS,, TX A Y ) . Case (2). The initial part is { T S , } . By hypothesis there is a U such that U is an { TS, T X , T Y , F Z } / { T S , } . Then U is an { T S , TX A Y, F Z } / {T S , } . KF A : The block is { T S , FX A Y } , and { T S , FX) and ( T S , FY} are both inconsistent. Suppose the initial part is { T S , } . By hypothesis there are U,, U2such that U, is an { TS, FX}/{T S , } , U, is an { T S , F Y } / { T S , } . Then U,A U, is an { T S , FX

A

Y}/(TS,}.

c~.685

CRAIQ INTERPOLATION LEMMA

81

KF- : The block is { TS, F - X ) , and { TS, T X } is inconsistent. Suppose the initial part is { TS,}. By hypothesis there is a U such that U is an ( T S , T X } / {T S , } . Then U is an {TS, F - X } / { T S , } . KT- : The block is {TS, T - X , FY), and { TS, F X } is inconsistent. Case (1). The initial part is { TS,}. By hypothesis there is a U such that

-

U is an { TS,FX}/{ TS,}.

Then U is an { TS, T X , FY}/{TS,}. Case (2). The initial part is {TS,, T - X ) . By hypothesis there is a U such that U is an { T S , F X } / { T S 2 } . We claim that U is an { T S , T X , F Y ) / {T S , } .

-

-

First we verify its predicates and parameters are correct. By hypothesis [UJc[S2]n[Sl,X], so immediately [-U]c[S,,-X]n[Sz, Y ] . We have the following two blocks are inconsistent:

{TSz,F U } {TS,, T U , F X ) . 3

It follows that the following two blocks are also inconsistent:

{TS,, T {TS2, T

- u, N

X,F U}, FY) N

3

and we are done. K F 3 : The block is { TS, F X 2 Y } , and { TS, T X , F Y } is inconsistent. Suppose the initial part is {TS,}.By hypothesis there is a U such that

u is an ( T S , TX,F Y ) / { T S , } . Then U is an { TS, FX =I Y ) / {TS,}. K T 2 : The block is { TS, TX 3 Y, F Z ) , and { TS, F X ) and { TS, TY, F Z } are both inconsistent. Case (1). The initial part is {TS,). By hypothesis there are U,, U2 such that U, is an { T S , FX}/{T S , } , U2 is an {TS, TY, F Z } / { T S , ) ,

82

ADDITIONAL FIRST ORDER RESULTS

c~.6$5

Then U,A U2 is an { TS, TX 3 Y, F Z } / {TS,}. Case (2). The initial part is {TS,, T X = Y}. By hypothesis there are U,, U2 such that U , is an { T S , F X } / { T S 2 ) , U2 is an { T S , T Y , F Z } / { T S , , T Y } . We claim U,= U2 is an { TS, TX 3 Y, F Z } / {TS,, TX 2 Y}. By hypothesis cull [S21 n [ S l y XI CU2l CS,, y] n [S2, 21 so [ul= U2]E [SlyX = Y] n [S2, Z] . 9

We have that the following four blocks are inconsistent: (1). w 2 , (2). {W,,TS,, (3). { TSI, TYY W), (4). { m2,TS2, and we must show the following two blocks are inconsistent:

fw, w, m,

The first follows from (2) and (3), and the second from (I) and (4). KF3 : The block is { TSyP(3x) X ( x ) } , and { TS, FX(a)}is inconsistent. Suppose the initial part is { TS,}. By hypothesis there is a U such that U is an { T S , FX (a)}/{ T S , } . Then [U]E[Sl]n[S2, X(a)]. Case (1). a# [U]. Then U is an { TS, F(3x) X ( x ) } / {TS1} Case (2). UE [ U ] , U E [SJ Again U is an { TS, F(3x) X ( x ) } / {TS,} Case (3). U E [U], a$ [S2]. Then ( 3 x ) U(z) is an {TSY F (3.1 x ( x ) } / { T S J* K T 3 : The block is { TS, T(3x)X(x), FZ}, and { TS, TX(a), FZ} is inconsistent, where a# [S, X ( x ) , Z ] .

m.685

CRAIG INTERPOLATION LEMMA

83

Case (1). The initial part is { TS,, T(3x) X ( x ) } . By hypothesis there is a U such that U is an {TS,T X ( a ) , F Z } / { T S , , T X ( a ) } . Then U is an { TS, T ( 3 x ) X ( x ) , F Z } / {TS,, T(3x) X ( x ) ) . Case (2). The initial part is { TS,). By hypothesis there is a U such that

u is an { TS,T X (a), F Z } / {TS1}. Then U is an { TS, T ( 3 x ) X ( x ) , F Z } / {TS,}. KFV: The block is { TS, F(Vx) X ( x ) ) , and { TS, FX(a)} is inconsistent, where &[S, X ( x ) ] . Suppose the initial part is {TS,}. By hypothesis there is a U such that U is an {TS, F X ( a ) ) / ( T S , ) . Then U is an { TS, F(Vx) X ( x ) } / {TS,}. KTV: The block is { TS, T ( V x ) X ( x ) , F Z } , and { TS, TX(a), F Z } is inconsistent. Case ( 1 ) . The initial part is { TS,, T ( V x ) X ( x ) } . By hypothesis there is a U such that U is an {TS,T X ( a ) , F Z } / { T S , , T X ( a ) } . Case (la). a # [ U ] . Then U is an { T S , T ( V x )X ( X ) , F Z ) / { T S , , T ( V x )X ( X ) } *

Case (lb). a E [ U ] , aEISl, X ( x ) ] . Again U is an {TS, T ( V x ) X ( x ) , F Z } / { T S , , T ( V x ) X ( x ) } . Case (Ic). a ~ [ v ] a#[S,, , X(x)]. Then (Vx) U(:) is an { TS, T ( V x ) X ( x ) , F Z } / {TS,, T ( V x ) X ( x ) } . Case (2). The initial part is { TS,}. By hypothesis there is a U such that

u is an { T S , T X ( U ) ,FZ)/{TSl}. Case (2a). a # [ U ] . Then U is an { TS, T ( V x ) X ( x ) , F Z } / { TS,). Case (2b). a€ [ U ] , a€ [Sz, X ( x ) , Z ] . Again U is an { TS, T ( V x ) X ( x ) , F Z } / {TS,}. Case (2c). a € [ U ] , a$ [Sz, X ( x ) , Z ] .

84

ADDITIONAL FIRST ORDER RESULTS

c~.656

Then (3x) U ( l )is an { TS, T ( V x ) X ( x ) , F Z } / {TS,). Now to prove the original theorem 5.1 : Suppose t I X 3 Y. Then { TX, F Y ) is inconsistent. By the lemma, there is a U such that U is an { TX,F Y } / {T X } . We have three cases: (1). U = t . Then since { Tt, F Y } is inconsistent, t, Y . (2). u=f. Then since (TX, F f ) is inconsistent, { F - X } is also inconsistent cf is not in X).Thus t-,-X. (3). U # t , u z f . Then U is a formula not involving t orf, all the parameters and predicates of U are in X and Y, and since { T X , FU} and (TU, F Y } are both inconsistent, tIX=, U and kI U 2 Y.

5 6.

Models with constant B function

In part I1 we will be concerned with finding countermodels for formulas with no universal quantifiers, and we will confine ourselves to models with a constant B function. To justify this restriction, we show in this section Theorem 6.1: If X is a formula with no universal quantifiers and y ,X, then there is a counter-model (’3, 9, k,P) for Xin which B is a constant function. Definition 6.2: For this section only, let a,, u2, a3,... be an enumeration of all parameters. We call a set r of signed formulas a Hintikka element if r is a Hintikka element with respect to some initial segment of a,, u2, a3,... (see ch. 5 Q 4). Lemma 6.3: If S is a finite, consistent set of signed formulas with no universal quantifiers, S can be extended to a Jinite Hintikka element. Proof: Suppose S is the set ( X , , X,, ..., Xn} where each X i is a signed formula. We d e h e the two sequences {pk},{&) as follows: Let

P, = 0, Qo = x,, ..., X”. Suppose we have defined Pkand Qkwhere Pk = Y,, ..., q , Q k = w,,..., w,,

85

MODELS WITH CONSTANT 9 FLINGTION

CH.686

and Pk u Qk (considered as a set) is consistent. To define Pk+, and Qk+l we have several cases depending on W, : Case atomic: If W, is a signed atomic formula, let pk+l

= Y,,

. * a ,

Y,, WI

Qk+l

= W2, ..*,Ws.

Case T v : I f W, is TX v Y, either TX or TY is consistent with Pk u Qk, say TX. Let Pk+i

= Y1, ...,

x, TX V

= w2, ...,

Y,

w,,T X .

Case F v : If W, is FX v Y then FX, FY is consistent with Pku Qk. Let Pk+l

= Y1, ..-,Y,, FX

V

Y,

Qk+l

= W2,

..., W,,F X , F Y .

CasesTA, F A , T - , T I aresimilar. Case T3 : If W,is T(3x) X(x), let a be the first in the sequence a,, a2,... not occurring in Pk or Q k . Then TX(a) is consistent with Pk u Qk. Let Case F 3 : If W, is F(3x) X(x), let {a,,, ..., sit} be the set of parameters occurring in P k u Q k such that no FX(a{,) occurs in PkuQk. Then {FX(ai,),...,FX(a,,)} is consistent with PkuQk. Let Pk+l

=pk>

Qk+l

=

W ~ , * *w,,FX(ai,),...,FX(ai,),F(3x)X(x), .Y

After finitely many steps there will be no T-signed formulas left in the Q-sequence because each rule T v , T A , T-, T 3, T3 reduces degree, and no rule F v , F A , F 3 introduces new T-signed formulas. When no T-signed formulas are left in the Q-sequence, no new parameters can be introduced since rule T3 no longer applies. After finitely many more steps we must reach an unusable Q-sequence. The corresponding P v Q-sequence is finite, consistent, and clearly a Hintikka element.

Remark 6.4: The above proof also shows the following which we will need later: Let R be a finite Hintikka element. Suppose we add (consistently) a finite set of F-signed formulas to R and extend the result to a finite Hintikka element S by the above method. Then

R, = s*.

86

ADDITIONAL FIRST ORDER RESULTS

c~.656

Since R s S , certainly R T s S T .That STGRT also holds follows by an inspection of the above proof; no new T-signed formulas will be added. Now we turn to the proof of the theorem itself. We have no universal quantifiers to consider, so we may use the definition of associated sets in ch. 2 9 4. Suppose X is a formula with no universal quantifiers, and y ,X. Then { F X } is consistent. Extend it to a finite Hintikka element S:. Let T,,,.., T,,be the associated sets of S:. Extend each to a finite Hintikka element, Sy, ..., S: respectively. Thus we have

s:, sy, ...)s:. For each parameter Q of some S: and each formula of the form P(3x) X ( x ) in St, adjoin FX(u) to S8 and extend the result to a Hintikka element S;. Do the same for S:, .... S,",producing S:, ..., S: respectively. Thus we have now

s;, s:, ..., s; .

Let T,,+iy...,T, be the associated sets of SAYS ; , . . . ,S:.Extend each to a Hintikka element, S:+,, ..., Sz respectively. Thus we have now

st, s:, ...)s,1,s,,,0 1, ..., sl),. For each parameter u used so far, and for each formula of the form F(3x) X ( x ) in S;, adjoin FX(u) to St and extend the result to a finite Hintikka element St. Do the same for each. Thus we have now

s;, s:, ...) s,",s;+ 1, ..., s;

.

Again take the associated sets, and extend to finite Hintikka elements, producing now 2 1 0 s;,~:,...r~n,Sn+l,...,S~,Srn+l ,...,s~.

Continue in this manner. Let 4)

W

s o = U S:, s l =U s:, k=O

k=O

By the remark above, for each n, S,T=S;T=S;T=.*..

Thus if S: has as an associated set S;, S,,, ES,.

etc.

c~.6$6

MODELS WITH CONSTANT

9 FUNCTION

87

It now follows that {So,S,, ..,} is a Hintikka collection. For example, suppose F- YES,..Let k be the least integer such that F- YES:. By the above construction, there is some set S," such that S: is an associated set of S: and TYES:. But then Sjk,sS,?, so by the above SjTcS,,and TYES,.The other properties are shown similarly. Moreover, 9(S,,)=9(Sm)for all m and a, as is easily seen. (Recall that 9's) is the collection of all parameters used in S.) Now as in ch. 5 0 3 there is a model for this Hintikka collection, and this model will have a constant B map, so the theorem is shown.

CHAPTER 7

INTUITIONISTIC M , GENERALIZATIONS

6 1. Introduction Here and in the rest of part I1 we restrict our considerations to the following language: a countable collection of bound variables x, y, z, ..., a collection of parameters (or constants) of arbitrarily high cardinality f, g, h, ..., one two-place predicate symbol E (we write ~ ( xy,) as (xE~)), and the usual connectives, quantifiers and parentheses. In all the models (9, W ,k, 9)which we will consider in part 11, the map B will be constant, and so we will simply write the range 9’ of 9 instead of 9, thus (9, W,by 9’),where a(r)= 9’for all I ’ E S . We call a model (9, PZ, b, 9’) an intuitionistic ZF model if classical equivalents of all the axioms of Zermelo-Fraenkel set theory, expressed without the use of the universal quant@er, are valid in it. As a special case, suppose {S, W ,C, 9’)is an intuitionistic ZF model and 9 has only one element r. Then this is (isomorphically) a classical model for ZF. If we define a truth function on all formulas over Y by u(X)=T u(X)=F

if if

rcx, ryx,

ZF map to T. Thus the notion of intuitionistic ZF model is a generalization of the classical notion. Suppose (9, 9, k, 9’) were an intuitionistic ZF model such that v will be a classical truth function, and all the axioms of

92

INTIIITIONlSl'K M a GENERALIZATIONS

~11.711

-AC was valid in it, where AC is some classically equivalent form of the axiom of choice expressed without use of the universal quantifier. It follows that the axiom of choice is classically unprovable from the axioms of ZF. For otherwise ZF kcAC , so for some finite subset A,, ..., A , of ZF A,,

..., A,k,AC.

We may suppose A,, ..., A , stated without the universal quantifier.

kc(A, A . * * A A n ) 2 A C . So by the results of ch. 4 4 8 kl--((Al

A-..AA,)=,AC),

equivalently, kI(Al

A

A

A,)

3

--

AC.

But <S,92,k, 9'>is an intuitionistic model in which A,, ..., A,, -AC are valid, a contradiction. Thus to show the classical independence of the axiom of choice it suffices to construct an intuitionistic ZF model in which AC is valid. Similar results hold for the independence of the continuum hypothesis and of the axiom of constructability. In this chapter we will define intuitionistic generalizationsof the classical Ma sequence of Godel [4], which provide intuitionistic generalizations of L , the class of constructable sets. We will show these generalizations are intuitionistic ZF models. In later chapters we will give specific intuitionistic generalizations of L establishing the independence of the axiom of choice, the continuum hypothesis and the axiom of constructability. The specific models constructed, and most of the general methods will be those of forcing, due to Cohen [3]. It is the point of view that is different. No classical models are constructed, complete sequences and countable ZF models are not used. In [ 5 ] , Gregorzyk noted the foundations of a connection between forcing and intuitionistic logic. In [13] Kripke discussed the relationship between forcing and his models.

-

Remark 1.1 : For the rest of part I1 we shall distinguish informally between constants, bound variables, and free variables. We shall use

c~.7$2

THE CLASSICAL Ma SEQUENCE

93

x, y, z, ... for both bound and free variables. This is an informal distinction. Formally, free variables and constants are both parameters in the sense of part I since free variables are simply place holders for arbitrary constants.

0 2. The classicalMu sequence Let V be a classical ZF model, In [4]Godel defined over V the sequence Ma of sets as follows. M,=B. Ma+, is the collection of all (first order) definable subsets of Mu. MA= Ua< A Ma for limit ordinals, 1. Let the class L be U a E V M UGodel . showed that L was a classical ZF model. As an introduction to the intuitionistic generalization, we re-state the Godel construction using characteristic functions instead of sets. Now of course “E” is to be considered as a formal predicate symbol, not as set membership. Let M be some collection and let v be a truth function on the set of formulas with constants from M. We say a (characteristic) function f is dejinable over ( M , v ) if domain ( f ) = M , range ( f ) E { T , F ) , and for some formula X ( x ) with one free variable and all constants from Myfor all aEM f(a>= lJ ( X (4)* Let M’ be the elements of M together with all functions definable over (MY

0).

We define a truth function v’ on the set of formulas with constants from M‘ by defining it for atomic formulas. I f f , gEM’ we have three cases : (1). f, gEM; let v ’ ( f E g ) = v ( f E g ) . (2). f e M , g E M ’ - M ; let v ‘ ( f E g ) = g ( f ) . (3). f E M ’ - M ; let X ( x ) be the formula which definesf over (My v). If there is an h e M such that ~ ( ( V X ) ( X E ~= X ( X ) ) ) = T ,

and U’(hEg) = T ,

94

INTUITIONISTIC M a GENERALIZATIONS

CH.763

let u’(fEg) = T .

Otherwise let U’(fES) =F .

(Case (3) reduces the situation to case (1) or case (2).) We call the pair ( M ‘ , v ‘ ) the derived model of ( M , v ) . Now let M, =0 and let uo be the obvious truth function. Thus we have (M,, v,). Let ( M a + l ,v a + l ) be the derived model of (Ma, v,). If A is a limit ordinal, let M A = U a c A M , . Let v , ( f E g ) = T if for some a < l , v , ( f ~ g ) = T .Otherwise let u , ( f ~ g ) = F . Thus we have (MA,vA). Let L=

u Ma. U € V

Let v(fEg)=Tif for some aeV, u , ( f e g ) = T . Otherwise let u ( f e g ) = F . Thus we have the “class” model (L,v ) . The reader may convince himself that this construction is essentially equivalent to Godel’s, so that if A is any axiom of ZF, v(A)=T. Thus ( L , u ) is a classical ZF model, though not a standard one. For a boolean generalization of this type of sequence see ch. 14 5 7.

0 3.

The intuitionisticMa sequence

Suppose we have a model (9, 9 , C, 9)(recall that Y is a set, the range of the B map, and that there is only one predicate symbol E). For convenience, let P be the collection of all W-closed subsets of 3. We say a functionf’is definable over (3,92, k, 9)if domain (f)= 9, range ( f ) c P , and for some formula X ( x ) with one free variable, all and no universal quantifiers, for any U E Y constants from 9,

f(4= {rI r C X ( 4 >‘ Let 9’ be the elements of Y together with all functions definable over (3,9, k, 9). We define a C’ relation by giving it for atomic formulas over 9‘. Iff, g E Y ’ we have three cases: (1). f, g E Y ; then let r k ’ ( f e g ) if r k ( f e g ) . (2). f e y , g E Y ‘ - Y : let r k ’ ( f e g ) i f r e g ( f ) . (3). f € Y ‘ - Y ; let X ( x ) be the formula which defines f over

THE INTUITIONISTIC Me SEQUENCE

CH.783

(9, W,t, 9). Let

and

95

rC’(f E g ) if there is an h ~ such 9 that r t (3x) (x E h = x (x))

- -

r t’(h€ 9).

(This reduces the situation to case (1) or case (2)) We call the model (9, 9, t’, Y >the derived model of (9, 9, k, 9). Now let V be a classical (first order) model for ZF. We define a sequence of intuitionistic models in V as follows: Let (9, W,to,Y o )be any intuitionistic model satisfying the following five conditions: (1). (9, W,co,

V

YO)€

(2). Y ois a collectionof functions suchthat, iff€ Yo, domain (f) E Y o and range (f) E P. (3). forf, g € y o , rC0(fEs)iffrw(f). (4) (extensionality). for f , g, h e Y o , if r t , - ( 3 x ) - ( x ~ f = x ~ gand ) r ko (f ~ hthen ) r to ( g ~ h ) . ( 5 ) (regularity). 9,is well-founded with respect to the relation xsdomain ( y ) .

-

N

Remark 3.1: Ifweconsider the symbols v , A , -, 3 , V, 3, (, ), E, x,, x2, x g ,... to be suitable “code” sets, formulas are sequences of sets, and hence sets. It is in this sense that (1) is meant. See also fj 14.

Next let ( 9 ,W, Y,,,) be the derived model of (9,9, C,, 9,). If A is a limit ordinal, let 9A=Uax€g).

-

-

Remark 4.5: The reader may show the two simple facts used above, and often later: Xis dominant implies -Xis dominant and t,(VX)(X(X)3"

Y(x)).

Y(x))r+x)-(X(x)3

Lemma 4.6: Iff, g E 9 , and r I = , ( f c g ) then rC,+,(fcg). Pro08 rC,(fcg). Suppose for some I'* and some hcY,+, r*k , + , ( h e f ) . If h e y a , by dominance r*t = , ( h ~ f )But . r*k,(fcg) so -(hEg). as above r*C-, -(htzg), and by dominance r* If h ~ Y , + , - 9 , , since f e Y , a n d r * k , + l ( h ~ f ) , it mustbethecasethat h is hx for some formula X over 9,, and there is some k E Y , such that

-

CH.755

97

A LI'ITLE ABOUT EQUALITY

r*k a + l ( k E f )and r*k , - ( 3 x ) - ( x ~ k f X ( x ) ) . Since both k, f EY,,by dominance r * k , ( k E f ) . Thus r*I=,- -(keg), and by dominance r* ( k E g ) . That is for any r**there is some r***such that r*** ( k E g ) . But also ~ * * * ~ , - ( ~ x ) - ( x E ~ E X ( kX e) y) , , so by

-

-

definition I'***ka+l(hxEg). Thus r * k a + l - - ( h x E g ) . Hence I'l=a+l(Vx)( x ~ f 2 - x E g ) , so r t U + , - ( 3 x ) - ( x € f ~ x € g ) . Lemma 4.7: Iff, g € Y a and r k a ( f s g ) , then r k ( f s g ) . Proofi First, by transfinite induction, for any >cc, I=, (fsg). The successor ordinal step is given by lemma 4.6. Suppose 1 is a limit ordinal, A >a, and the result is known for all fl such that cc o : (3, (3X) [ @ E X A (3Y) (YEX 3 Y ' E X ) ] .

- -

Proof.- If we show there is an fE 9,+1 - 9,such that o C f is valid in (9, W,t=o+l, c400+l>, the result will follow by dominance of o ~ x . Let X ( x ) be the formula -(3Y)

-

{[-(3Z) " ( Z E y

3 Z'EY) A

0 € Y ] 3 X€Y).

There is an fx E 9, + - 9,. We claim o Efx is valid in 9,+, This ).follows from the next four lemmas: (9, 92, Lemma12.2: I f I ' k a f = O ~ g = O t h e n r k , f = g . Proof.- r k a - ( 3 x ) ( x f ~) A -(3x) ( x e g ) so by intuitionistic logic r ~ , - ( ~ x ) - ( x E ~ = x E r~k ), f,= g . Lemma 12.3: I'k,+lOE fx. Proof.- By the results of 5 9 for some g e 9 , rk,g=0. Suppose for some r* r*C, -(&) - ( z E k 2 z ' E k ) A 0 ~ k . Then r*k,OEk, that is r * k o ( 3 w ) ( w = ~ A w E ~ )so , for some S E Y , r * k , S = @ ~ s E kBylemma . 12.2r*k,s=gY s o r * C , - - ( g ~ k ) . We have shown r k,(vx) {[- (32) ( Z E X 3 Z'EX) A EX] 3 EX))

-

--

or equivalently r t = , D ( ~ x ) - { [ - ( 3 z ) N ( z E ~ ~ Z ' E X ) ~ ~ E ~ 1 ~ g E x ) ,

r kco x (91, ka+lS E f X .

But I'ka+lg=O,

so by definition rt=,+lOefx.

Lemma 12.4: If g E 9 , , there is an h E 9 a + 1 - Y , such that h =g' is valid in ( 3 , W , k a + l , y a + l > . Proofi Let Y(x) be the formula ( x E g ) v ( x = g ) . There is an h y E 9a+l - 9,. We will show

rka+l- (

3 w ) - [ ~ ~ h y ~ ( ( ~ ~ g ~ ~ = g ) ] .

105

(2).

r*k,+

(s = g ) . Since trivially

r*

Ca+

( g E hy) ,

r*ka+l --(dy). Thus we have

rkaCl N ( 3 W ) N [ ( W E gw~ = g ) = , w ~ h ~ ] Lemma 12.5: If I‘b,+l

( g € f x ) ,then (g’e f x ) . Proof: r k o + l ( g e f X ) ,so there is an h g Y , such that r k , + l ( g = h ) r \ (h€Jx). Since h ~ 9 , ,for some tcl>.

Proofi All the elements of the class 9’are functions. We have assumed 9, is well-founded by the relation xedomain ( y ) . It then follows that 9 is also well-founded by xdornain ( y ) .

The formula Iv

-

{(YY) (Y EX)

3

(3Y) [Y

EX A

-

(3.) (z E X

A Zf

YIII

(*>

is equivalent to - { ( 3 y ) ( Y ~ x ) A - ( ~ Y ) [ Y E x A - ( 3 z ) ( ~ EZxE Y~1 1 1

which is obviously dominant. Then for somegE9’,, r k , ( g e f ) . SupposefE9,and r ! = , ( 3 y )(pf). We claim rka--(3y)[yEf

A-(IZ)(ZEf

AZEy)].

Suppose otherwise. Then there is some r*such that

r*ka

-

( 3 y ) [ y ~ fA

-

(32) (

Z E ~A Z E ~ ) ] .

We define a set W to be

1

(x x E 9,and for some

r** r**i=,(x~f)}.

W is not empty since g E W. The relation xEdornain (y) well-founds W. Let s be a “smallest” element of W. That is, S E W, but for no f E W is iEdornain(s). Since SE W, for some r** r**!=,(sE~). We claim r**k,-(3z)(ZEf

A ZES).

Suppose not. Then for some r***

r***k,(jZ) Thus for some r E 9,

( Z E ~A ZES).

r***k , ( r E f )

A

(re$).

Since r*** k,(res), there is some fEdomain(s)such that r*** k,(r=t)~ A (~Es).But then r*** = !-, ( t c f ) , so for some I‘****

-

r**** i=,(t~j),

CH. 7 9 14

107

DEPINAEILITY OF % MODELS

so t~ W , a contradiction. Thus r**ka-(3z) ( z ~ f ~ z But ~ s r** ) . k,(s€f) so r * * C a ( g y ) [ y g f A -(gz)(z€f A Z ~ Y ) ] , and this contradicts

r*C, Thus

-

(gy) [ y ~ fA

rka--(3y)[y€f

-

(3.1

( Z E ~A Z E Y ) ~ .

A -(32)(Z€f

A Z€v)].

But r was arbitrary. We have shown that for each f €9, the following : is valid in (g,W,C,, 9,) (3Y)(YEf)-+Y)[Yd

A+Z)(ZEf

AZEY)I.

The theorem now follows by the dominance of (*).

5 14.

Definability of the models

One of our initial assumptions was that (B, 92, ko, Y o ) € V. The definition of the sequence was an inductive definition. It should be clear that the definition can be carried out in V itself. That is, not only is (59, W,I,,Y",)E V for each CIEV,but moreover

Theorem 14.1: There is a formula F ( x , y ) over V which defines the sequence of (3, L%?, Pa, Ye).That is, for x , YE V F(x, y ) is true over V if and only if x is some ordinal CI and y is (3,92, k, 9,)(In . fact F ( x , y) can be absolute, as should be obvious.) Of course (3,92, kY9)is not in V, since, in particular, Y is not a set. But we do have

Theorem 14.2: Let X(xl, ..., xn) be any formula with no constants and no universal quantifiers. There is a (classical) formula R, (z, xl, ..., x,) with constants from V such that for any r e 9 and el, ..., c,EY, r i=X ( c , , ..., cn) if and only if R, (r,cl, ..., en) is true over V. Proofi By induction on the degree of X. Suppose X is atomic, ( x ~ y ) . Let R,(z, x, y ) be the formula Z € g A

(301)(0?'dhd(Ct)A

X € Y a A

Y€YaA

Zk,(XEy))

108

a.7815

INTUITIONISTIC M s GENERALIZATIONS

(where we have used the obvious abbreviations allowed by the above theorem). Suppose X is not atomic but the result is known for all formulas of lesser degree. IfX(x, ,..., xn) is Y(xl,..., x n ) v Z ( x l,..., x,,), by hypothesis there are formulas Ry(w, xi, ..., xn) and R,(w, x1,..., x,,). Let R,(w, xl, ..., x,) be the formula R,(w, xl, ..., x n ) v&(w, xl, ..., x,,). The case X is Y AZ is similar. Suppose X ( x l , ..., x,,) is Y(xl,..., x"). By hypothesis there is a formula RY(z,xi,..., xn). Let R,(z, xl, ..., x,,) be the formula

-

-(2W)(W€g

A Z B W A Ry(W,Xl,

...,&))a

The case X is YD Z is similar. Suppose X ( x , ,..., xn) is ( 3 y ) Y ( y , x1,...,x,,). By hypothesis there is a formula RY(w,y, xl, ...,x,,). Let R,(w, xl, ...,xn) be the formula (3y) (3a) [ o r d i n u l ( a ) ~ y € Y , A R,(w, y, xl,

0 15.

..., x,,)].

Power set axiom

We wish to show in this section that the power set axiom is valid in (9, B, c, 9). Then for some smallest ordinal uo Let co be afixed element of 9. C~EY,,. Thus uo is also fixed. We first want to show that for a fixed I'EY there is a Po such that for any CEY, if r C ( c ~ c , ) there , is some deYb, such that r i=(c= d). After showing this we will show that in fact there is one Po which will do for all r e g . For the above fixed co, a. and r,for cl, c2 E Y such that r k (cl c co) A (czcco), if for all r* and for all ~ E Y , , r*c((tEcl)

=(tEC2)),

then r I.(cl =c2). The proof is as follows: Suppose for some r* and some ~ E TY * k ( k c , ) . Since I'i=(clsco) r*t- -(hec0). Then for any r**there is a r***such that I"*** t ( k c , ) . But c O ~ Y a oso, there is some t E Y , , , such that I ' * * * k ( h = t ) ~ ( t e c , , ) . Since r***k ( k c , ) , r*** k ( t q ) . Now by hypothesis, since ~EY,,,r * * * i = - - ( t E ~ J ; SO r * * * k - - ( h ~ Thus ~ ~r)*. C - - ( k ~ ~ ) . We have shown r i= (Vx) (XEc1 3 XE c2) or r k (cl E c2). Similarly r k (c2 c c ~ ) .

--

--

CH. 7 15

109

POWER SET AXIOM

Thus (speaking intuitively) to decide if two subsets of co are equal at r we can confine ourselves to elements of Yaoprovided we look at all r*. Now let P be the collection of all elements c e Y such that TC (czc,). We define (intuitively) a function U on P by

I

U ( c ) = {(r*, t ) t e Y a 0 and r * C ( t E c ) } . By the above result, for cl,c ~ E Pif, U(cl)= U(cz),then r k (cl =cz). Let B be the range of Uon P. U:P+ B is a function but not one-to-one. So we cut down its domain to a new domain P' on which U is one-to-one. Thus for ueB, for U - l (u) choose some single element x from the class of all y e P such that U ( y ) = u . Let P'={U-'(u) U E B } .Let U' be U restricted to P'. Then U is an isomorphism between P' and B. Suppose we could show for some posV P ' E Y ~Then ~ . if CEYand I'!=(CEC,), CEP,so there is some deP' such that U(c)=U(d), so T t (c=d) and d e Y p 0 .Thus we would have the desired result. We now show P' z Y p o for some Po€ V.

I

Lemma 15.1: There is a formula F ( x ) over V such that X E Piff F(x) is true over V. Pro08 Let R, (z, x , y ) be the formula defining z k ( x s y ) as given in the last section.Let F ( x ) be R, (I',x , co). Lemma 15.2: There is a formula G(x, y ) over V such that y ~ U ( xiff ) G ( x , y ) is true over V. Proofi Let R, (2,x , y ) be the formula defining 2 k ( x ~ y )Let . G(x, y ) be F ( x ) A (3r,s) [ y = ( r , s ) A r , g

A

~

€A r.@r 9 A~ R E ( r~ , s ,x)].

Lemma 15.3: For any c e Y U ( c ) e P ( gx Y a o ) V. e ( P ( x ) is the power set of x in V.> V (and is defined by G(c, x)). Proofi U(c) is a subset of B x Lemma 15.4: B EV. Pro08 By lemma 15.4 ( U ( x ) ~€9') is a subset of P ( S x Y a o ) eV. (It is a definable subset, defined by

I

(3a) (ordinal ( a ) ~ ( 3 c( )c e Y a ~ G ( cx))) ,

).

Lemma 15.5: There is a formula H ( x , y ) such that X E ~ for , y a subset of 9, if and only if H ( x , y ) is true over V (that is, a choice function). Proof: That Y can be well ordered in .V is straightforward.

110

INTUITIONISTJC M a OENERALIZATIONS

CH. 74 15

Theorem 15.6: P' G Y,, for some Po E V. ProoJ The function U - l (u) can be defined by: U-' (u) is that x such that H ( x , y ) where y = { z e P I U(z)= U(u)}, which can be formalized. Now P' is the range of U-' (u) on B. By the axiom of substitution in V P ' E V. Hence P' G Y f lfor o some Po E V,since P' -c Y and Y is a class. Thus we have our first assertion. We have written it out fairly completely as illustration. From now we will only indicate the steps. Above, for b e d r we produced an appropriate But the procedure can itself be defined over V. Since Y E V ,by the axiom of substitution again, there is a maximum floeV which works for all 3.Thus we have shown : There is a j3,~ V such that for any CEY and any reg,if TI=( c s c , ) , then for some d~ 9,, r k (c =d). Now we can show the following, from which the power set axiom follows, since co was arbitrary:

a,.

Theorem 15.7: The following is valid in ('3, W ,b, 9'): (3y)

- (32)

[(ZEY)

=(z c coy '

Proofi Let X ( x ) be the formula ( x c c 0 ) , with c , E ~ ~ ,Let . Po be as above, and let y=max(ao, Po). Then Y E V. Considerf,EY,+, -9,. We claim ~ ( ~ Z ) - [ ( Z E ~ ~ ) = ( Z C C ~is) valid. ] Let Y and suppose r*pC ( h ~ f , ) Then . for some r** r**b (hefx), so there is some ~ E Ysuch , , that r * * k ( t = h ) ~ ( t E f , ) . By dominance T**ky+l(tEf x ) , r * * k Y X ( t ) , so r**k,(fcco), by permanence r * * b ( t ~ c , ) .Thus r**C-- ( h ~ c , ) ,so T*W-(hsc0). We have shown

-

r c(vx) [- ( h E c,> or equivalently,

rc

- r* - c (3x1

2

-

(h~j-~)]

. co). Then for some r** r** k (hcco).

[ ( h ~ j -=)~()h G c,)]

Conversely suppose pl (h There is some t E Y B o such that r * * k ( h = t ) . So r * * b ( ( t c c , ) stable.) By dominance r**k Y ( t c c,,),

r**c, x (t) ,

( x c y is

CH. 7 9 16

X-EQUIVALENCE

111

and the theorem follows.

Remark 15.8: Above we obtained Po by two applications of the axiom of substitution. These could have been combined into one step as in Cohen [3]. This proof was based on that one, which followed a suggestion of Solovay. We find this two step approach more intuitive, but the treatment in Cohen is more elegant.

0 16. X-equivalence Definition 16.1: Let X be a formula with no universal quantifiers and all constants in 9,.We call (3,92, Fa, 9,) X-equivalent to ( 9 , W , t, 9') if for every Y which is an instance of a subformula of X with all constants in 9,for , any re9

rc,y-rcy.

Theorem 16.2: Let X be as above, with all its constants in 9,There . is is X-equivalent to an ordinal PEV, a/3,). By theorem 16.2 there is some SEV, yG6,such that ('3, 9, C,, 9,) is cp-equivalent to (9, W ,t, 9>. Since cp is a formula over 9,, cp is also a formula over 9,. Thus it + - 9, We . claim defines a function f v E 9,

rk

- (3X)

[X E f

v

E

(3 W ) (W E Co

A

x ( W , X))] ,

which is what we wanted. We' now proceed with the proof. Suppose r*pc- ( c ~ f J . Then for some r** r**k(cEf,). Since f,E9,+i-9,, there is some d E 9 , such that r * * C ( c = d ) ~ ( d ~ By f~). dominance r**ka+l(d~f,).r**C,cp(d), But (9, W ,Cay 9,) is cpequivalent to ('3, 9, C, 9)hence

r**I= cp(& r**C""'p(C),

r*y -cp(c>, r*y - ( 3 w ) ( w e c ,

AX(W,~)).

Thus we have shown

r C (VX)

[ (3W ) (W € CO A

Conversely suppose

N

r*y - ( 3 w )

x(W, X))

3 N (X

E

f,)] .

( w ~ c , n X ( w c)). , Then for some

r**k(3W)(W€C,

r**

A x ( W , C)).

By the above lemma, there is some c ' E such ~ ~ that ~ r**k (c'=c). Hence r * * P - N ( g w ) ( w ~ c , ~ X ( wc')), , that is, r**C--cp(c'). But C ' E ~ ~ ~ C and Y ,('3, E ~ W ,,C,,~ Y 6 )is cp-equivalent to (S,W ,C, 9), hence

-- --

r**c, r**k , + l r**C

But

cp(c')

"

(C'EfQ?),

(C'Ef,).

r**k(c'=c),

y

m.7§17

so

AXIOM OF SUBSTITUTION

r**k

ru

(C E

&,) ,

r*gc 4 E f p ) . We have shown

The assertion now follows.

115

CHAPTER 8

INDEPENDENCEOFTHE AXIOM OF CHOICE

0 1. The specific model The model given here is adapted from the one of Cohen [3]. We have changed it from showing directly that there is an infinite set with no countable subset to showing directly that there is a set with no choice function. The change was made because the notion of countability requires much more machinery in these models. See [3, p. 1361 for a brief introduction to the model. Following ch. 7 9 3, a sequence of models and a class model are defined if the 0th model is fixed. We now define a specific ( ' 3 , W , k0, Yo). All the work is relative to a classical.mode1 V. Let e be some formal symbol. By aforcing condition we mean a finite consistent set r of statements of the form (nem) and -(nem) (n>O, m a 1). ((nem) can be some ordered triple in V, say ( n , 0, m ) . Anything convenient. Similarly -(nem) can be some other triple, say ( n , 1, m ) . We have written it like this for reading ease.) Let 99 be the collection of all forcing conditions, and let W be set inclusion E. Before defining Y o we , define the following partition of the integers:

I, = {1,3, 5,7, ...>, Il= {2,6, 10, 14,...}, I2 = (4,12,20, 28, ...}, in general

etc . In = {2"(2k

+ 1) I k = 0, 1,2, ...>.

m.8§1

117

THB SPECIFIC MODEL

This partition has the properties that each I,, is infinite, and if n E I,,, n >m. Now we define Y o .It consists of the functions A

A

A

0,1, 2, **.,

to, ti,

s1, 82, s3,

t , *.*, ~

whose definitions are the following: For each integer n the function A has domain k. Proo$- We show for a n y r E 9 ry o(sm=s,,). SupposeI'k,(sm=sn)for some Since r is a forcing condition, it is finite, so we may choose an integer k such that neither (kern), -(kern), (ken), -(ken) belong

118

INDEPENDENCE OF THE AXIOM OF CHOICE

c~.8§2

to r. Let A be r u {(kern), -(ken)}. Then A E S and I'WA. By definition A k0 (&EX,,,). Since A C, (3x)- (xEs,,,=xEs,,), by intuitionistic logic Aka- -(&Es,,). Then for some A* A*Fo(ft~s,,), which means (ken)EA*. But (ken)Ed*, a contradiction.

-

-

Thus all five conditions are met so the resulting class model <S,5% k, 9) is an intuitionistic ZF model.

0 2. Symmetries Let G be the collection of all permutations n of integers such that n permutes the elements of one I,, and is the identity on all I,,,for m f n . We may extend any R E G to 9 as follows: n(A) = A ,

(4

n = sn (n) n(tJ = t,,, n(T)= T .

9

Let X be the formula X ( x , cl, ..., cn) where n has been defined for cl, ..., c,,. Let n(X) be X ( x , ~ ( q ) , . .n(c,,)). ., If f x E 9 a + l - 9 a , let n( fx)bef,(,,. Thus n is extended to 9. We also extend n to B by N

We note that

( n e m)s r o (n e n (m))E n (r), (n e rn) B r o ( n e n (m))E II (r). N

r E S implies n (r)E 8.

Theorem 2.1: For any formula X with all constants in Yaywith no universal quantifiers, any r e 3,and any 'ICE:G

r c,x e n ( T )Ca7C(X), and

r c x o n (r)c 7z ( X ) .

Pro08 A straightforward induction on a and the degree of X. Dejinition 2.2: Let N be some collection of integers. By GNwe mean the subset of G leaving N invariant.

Lemma 2.3: Let f€9. There is a finite set N of integers such that if ZEGN, n(f)=f.

~~.8@3,4

119

FUNCTIONS.AXIOM OF CHOICE

ProoJ IffeYo, we have two cases. If f i s not some s,,, let N=0. If f i s s,,, let N = {n}. Suppose the result is known for all g E 9,LetfE . Y,, - Ya. Thenf is fx for some X ( x , el, ..., cn), where cl, ..., ~~€9,. By hypothesis there are , Let finite sets Nl, ...,N,, of integers such that if Z E G ~ n:~(ci)=c,. N = N , u-.-uN,. Then if m G N ,n ( f X ) =f n c X , = f x .

0 3. Functions We introduce the following formula abbreviations:

(x, ~ ) E Z ordpr(x) relution(x) function(x)

--- - -

for (3w) [wEx A w = {y, z} A x = {y, w > ] , for ( 3 w ) [ W E Z A w = (x, y ) ] , for (3y) [ y ~ 3 x (32) ( 3 w ) ( y = ( z , w ) ) ] , for (3y) EX 3 ordpr(y)], for relarion(x) A (3y) (32) ( 3 4 ) (3u) N

[(( y, 2 ) EX A (u, u ) E X A y = u) 3 z = u] ,

dornain(x)=y

for

-(32)(3w)-

- -

[(Z,W)EXIZE~] A

(32)

[ Z E Y I( 3 w ) ((2, W > E X ) ] .

Theorem 3.1: All the above formulas are dominant.

0 4. Axiom of choice Let AC (T) be the formula

- (W

(3x) {fmction(x) A dornain(x) = T A

[YET= ( 3 4 ( z vA

(v,Z)E41).

-

That is, AC (T) says that T has a choice function. In this section we show that AC (T)isvalid in (9, W ,I=, 9'). In fact, it is valid in (g,W,t,, 9,) for every a ; the same proof holds for each case. We first show a preliminary Lemma 4.1 : Iff E 9 'and r k (fE t,,) then for some m E I,,

r l a (f = sm). Pro08 r I = ( f e t , )so there is some bEdornain(t,,) such that

r q j =b)A(bEf,,).

120

INDEPENDENCE OF THE AXIOM OF CHOICE

~ ~ 1 . 8 8 4

Now suppose there is some r E 5 ? such that I'kAC(T). Then for some FE9,

r i=function ( F ) A domain ( F ) = T A

-

(3Y)"

C Y E T I

(32) ( Z E Y

A

(YY

Z)EF)I.

There is a finite set N of integers such that if Z E G ~ z(F)=F. , Let n=l+maxN.

-

ri=( 3 ~ ) [ Y E T

3

( 3 2 ) (W

A

(Y, Z>EF)I

and r k (t, E T ) hence

rk

(3.1 ( z E t,

A

(tny Z> E F ) .

Then for some I'* r*C(3Z)(ZEtn

A

(t,, z ) E F ) .

For some a E : Y r*C(mEt,)

A

(t,,a)EF.

By the above lemma, for some mEZn, I'*b (a=$,). Hence

r*

N N

( n = l +maxNy hence m#N. Choose an integer k > n such that k f m and neither ( p e k ) nor - ( p e k ) belongs to r**for any integer p , but kEI,. (T** is finite but I,, is infinite, so this is possible.) Let TL be the permutation 7c(m)=k, n(k)=m, on all other integers 7c is the identity. Since myk$N, xeGN.Now SO

7~

(r**) i= 7~( EF)

9

n(r**)C En(F),

n:(r**) c (t,, sk) E F . But A =r** u n: (r**) is itself a forcing condition. It is finite, and since r**and 7c (r**) must be the same except for statements involving m and k , and m is not (a second element of any statement) in .n(T**) and k is not in r**,z(T**) and are compatible. Thus A E and ~ r**WA and n(T**)WA. So A C function F , (since r t= function F ) A i= (tny S r n ) E F , A k {t,,, Sk) E F .

r**

c~i.894

AXIOM OF CHOICE

121

It then follows by intuitionistic logic that

a != or since (x =y ) is stable,

--

(s, = Sk),

A k (s, = s~). But m # k , contradicting theorem 1.1. Thus for all I ' E ~ so

As we showed in ch. 7 9 1 the axiom of choice is now classically independent.

CHAPTER 9

ORDINALS A N D CARDINALS

0 1. Definitions Continuing ch. 8 0 3 we introduce the following formula abbreviations: range(x)=y for - ( ~ z ) ( ~ w ) - [ ( W z , ) E X D W EA ~ ] (3w) [ W E Y 2 (3.) ,..., (0, a,}, {I,Ul}, a z } , . . . , (0, a,), (1, a,>, (2, a,>,

...,

{Z

w.

We leave it to the reader to show that this well-ordering can be expressed in the model. The only nontrivial part of the well-ordering is ao, a,, a,, ...) since the subscripts are not part ofthemodel. But Wi&tei?i provides this ordering. Thus the axiom of choice is valid in the model of ch. 10.

0 8.

Continuum hypothesis

In this section we show that the generalized continuum hypothesis is valid in the model of ch. 12. More generally we show the following: Theorem 8.1: Suppose ('3, 9?,C, 9) is ordinalized, (3,9?,I=, EL, and 9 and 9,are countable in L. Then the generalized continuum hypothesis is valid in (3,9, !=, 9').

We devote the rest of this section to the proof. We remarked in ch. 7 9 14 that the definition of the sequence of intuitionistic models is absolute. If L is the class of constructable sets of V, since ('3, 9, I,, EL, the construction of the sequence is the same over V or over L. Thus in this case we may assume in all the preceding work V was L. (We use the continuum hypothesis in L.) Trivially card(Y,+,)= EE0.card(9'J in L. Since ('3, 9, k, 9)is ordinalized and 9,is countable in L, it follows by the work of ch. 9 that for any ordinal a of L, if a d w and if B is the least ordinal such that & ~ 9then ' ~curd , (u)=card(Yg)in L. We use P ( x ) to denote the power set operation both in L and in