Provability, Computability and Reflection, Volume 50

CONTRIBUTIONS TO MATHEMATICAL LOGIC PROCEEDINGS OF THE LOGIC COLLOQUIUM, HANNOVER 1 9 6 6

Edited by

H. A R N O L D S C H M I D T K. SCHfiTTE H.-J. T H I E L E

1968

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM

0 North-Holland Publishing Company

- Amsterdam - 1968

No part of this book may be reproduced in any form by print, photoprint, m i c r o f i h or any other means without written permission from the publisher

Library of Congress Catalog Card Number: 68-24434

P R I N T E D I N THE NETHERLANDS

PREFACE An international logic colloquium was held at Hannover, Germany, in August 1966. Partly due to its favourable location just before the International Congress of Mathematicians at Moscow, international attendance was remarkable. One-hour addresses were delivered by R. 0. Gandy, J. Suranyi, and A. Tarski, 42 half-hour papers were presented, and there were more than 80 participants, half of them from foreign countries. The colloquium was sponsored and substantially supported by the Division of Logic, Methodology and Philosophy of Science of the International Union for the History and Philosophy of Science. Additional support was given by the German Federal Government and the Technische Hochschule Hannover. The colloquium was organized by the Deutsche Vereinigung fur mathematische Logik und fur Grundlagenforschung der exakten Wissenschaften (DVMLG) as a European meeting of the Association for Symbolic Logic in cooperation with the British Logic Colloquium. Topics dealt with at the colloquium ranged from mathematical logic, recursion theory, and intuitionistic mathematics to philosophy and history of mathematics and foundations and philosophy of physics. The present volume contains only a selection of the papers presented at Hannover, concentrating mainly on the more strictly logical and foundational subjects. Many of the papers published here have been revised or extended since their presentation at the colloquium. The publication of the colloquium proceedings is overshadowed by the death of H. Arnold Schmidt who presided at the Hannover Colloquium and also suggested the edition of the present volume. The German logicians mourn for H. Arnold Schmidt who as founder and long-time president of the DVMLG deserved well of the promotion of mathematical logic in Germany. May 1968

K. SCHUTTE

SATURATED INTUITIONISTIC THEORIES *

P. H. G. ACZEL St. Peter’s College, Oxford Introduction According to the intuitionistic interpretation of the logical connectives, any verification of the sentence a v p must involve a verification of either u or of 0.Also any verification of Vxci(x) must involve a verification of .(a) for some a. This suggests that the collection of first order sentences that are verifiable in any intuitionistic mathematical theory form what we call below a saturated theory. The main result of this paper is Theorem 1. The proof of this theorem is almost identical to the Henkin proof of completeness of classical logic as presented for example by Lyndon [4]. A particular case of this theorem is that every intuitionistically consistent set of sentences can be extended to a saturated theory. Using this theorem we show that there is a close connection between saturated theories and the interpretation of intuitionistic logic given in Kripke. In fact the family of all saturated theories partially ordered by inclusion form what we call a Kripke structure. In section 4 we give new proofs that a) the pure theory of intuitionistic predicate logic with a non-empty set of individual constants and b) the theory of Heyting Arithmetic are saturated. To do this we introduce a relation Tlku between a set of sentences r and a sentence a. This relation is very similar t o the relation r1.s introduced in Kleene [2]. We then give a characterisation of this relation in terms of certain Kripke structures. In the final section we suggest how the methods of classical model theory may be extended to apply to intuitionistic logic. Note that we make free use of set-theoretic methods in this paper. We have attempted to give as smooth a generalisation of the semantics of classical logic as possible. Hence, of

* This work was carried out while the author received a grant from the Science Research Council. Most of the results are contained in a part of the author’s thesis submitted for D. Phil. 1

2

P. H. G . ACZEL

course, our results have no direct bearing on an intuitionistic semantics of formal intuitionistic logic. 1. Preliminaries

We shall use a first order language L. This has a countable set Pred of predicate symbols, each one being n-ary for some integer n. In particular there is an 0-ary predicate symbol ‘El’ denoting absurdity. The atomic formulae have the form p (sl, ...,s), where p is an n-ary predicate symbol and each si is either one of a countable set of variables, or an individual constant taken from some arbitrary set. The formulae are built up from the atomic formulae in the usual way, using the connectives v , A , -+ and quantifiers Vx, Ax. A sentence is a formula with no free variables. If a is a sentence Ind(a) is the set of individual constants occurring in a, and if r is a set of sentences Ind(T)= U(Ind(a)l a ~ f )If. A is a set, St, is the set of sentences a such that Ind (a) E A . If is a sentence we shall write ka(t-,a) when a is a theorem of intuitionistic (classical) first order logic. We shall write rka(rk,a) if there are pl, ...,~ , E T such that l-(t,) P 1 ~ ( 2-f...(Bn-a)...). P Let = {a1 Ind (a) c Ind (r) and r t- a } Cn and Cn,(r) = {a1 Ind(a) E Ind(T) and I‘I,@}. DEFINITIONS 1) r is a theory (classical theory) if C n ( r ) = r (Cn, (r)=r); 2) r is consistent (a-consistent) if o $ C n ( r ) (a$Cn(T) and Ind(a)c Ind (r)) ; 3) r is complete (a-complete) if r is consistent and pECn(r) or f l - f l J ~ C n ( r ) for all p such that Ind(P)sInd(T) (F is a-consistent and p ~ C n ( r or ) /3-+aeCn(r) for all p such that Ind(P)GInd(r)); 4) r is prime if a v pECn(r) implies a ECn(r ) or pECn(T); 5) r is existential if v x a ( x ) ~ C n ( r )implies a ( a ) ~ C n ( r )for some individual constant a ; 6) r is saturated (a-saturated) if it is a prime, consistent (a-consistent) existential theory. If I n d ( r ) # 8 and r is consistent let Ur be the (classical) relational system Ur = (Ind (r),{ & - } p s p r e d ) where p r = { (al ...a,)] p (al... a,)ECn(T)} for each n-ary predicate symbol p . If U’ = ( A ’ , ( p ’ } p E p r e d ) is a relational system, let Val(U’) be the set of those sentences of St,, that are valid in U’. The following lemma will be useful below.

(r)

SATURATED INTWITIONISTIC THEORIES

3

LEMMA 1 1) Every a-complete theory is prime. 2) The union of a chain of a-consistent theories (a-complete theories) is an a-consistent theory (a-complete theory). 3) r is a saturated classical theory if and only if r=Val(U,). Proof. The proofs are all straightforward. We shall just prove 1). Let A be an a-complete theory. Let p v Y E A . Then p v y+a$A as A is a-consistent. Hence P-a$A or y+a$A as t(B+a)r\(y+a)-t(Bvy+a). But A is a-complete, so P E A or Y E A i.e. A is prime. 2. Completeness proof

The main result of this section is Theorem 1. The proof of this theorem follows closely the Henkin completeness proof of classical logic as given for example in Lyndon [4]. Lemma 2 is a generalisation of Lindenbaum's Lemma. LEMMA 2. Every a-consistent r can be extended to an a-complete theory A . Proof. Let X be the set of a-consistent theories A extending r such that Ind(A)=Ind(T). Then X is non-empty, as Cn(r)EX. Also X is closed under unions of chains by 2) Lemma 1. Hence by Zorn's Lemma X has a maximal element A say. A is an a-consistent theory. To show that A is acomplete assume p+a$A and pEStInd(d).Then A'=Cn(A u { p } ) is clearly a-consistent and hence is in X . Also A E A'. But A is maximal in X . Hence A = A ' and so P E A . Thus A is a-complete. If r is a set of sentences, a theory A is a rich extension of r if T c A and V X / ~ ( X ) implies E~ P(u)EA for some a. LEMMA 3. Every a-consistent r has an a-complete rich extension A. Proof. To each vxp(x)ECn(T) associate a new individual constant a,. Then A , = C n ( r u (p(aP)lvxp(x)ECn(r)}) is clearly a rich extension of r. We show that A , is a-consistent. Assume a ~ d , Then . there are Vx,pl(xl) ... Vx,&(x,)ECn(r) such that ru{Pl(as,),..., P,(a,,>}ka. Hence ru{p,(a,J, ..., P,- (a," - ,I} t- B, (a,,) a and so r { P l (a, 1 1 7 ...7 Pfl-1 ,)I tVx,P,(x,)+a as a,, occurs only in P,(a,,). But rkVxn/3,(x,). Hence r ~ { ~ ~ ( a..., , , P,-,(aP,_,)}t-a. ), Repeating the above we show eventually that Tt-a which contradicts the initial hypothesis. Hence A , is a-consistent and by Lemma 2 can be extended to an a-complete theory which clearly satisfies the lemma. +

1. Every a-consistent THEOREM

r can be extended

to an a-saturated A .

4

P. H. G. ACZEL

Proof. By Lemma 2, I' can be extended t o an cc-complete theory A,. By repeated use of Lemma 3 there is a sequence A, c A, 5 . .. of a-complete theories such that A,+l is a rich extension of A , for each n. By 2) Lemma 1 A = U,<mA,is an a-complete theory. A is also existential, for if VX/I(X)EA then VX/~(X)EA, for some n and hence /I(u)EA,+,GA as A , , , is a rich extension of A,. Thus A is an a-saturated extension of r. Let T k a iff T s A implies a ~ for d every saturated A such that Ind(a)c Ind (A). We may reformulate Theorem 1 as

THEOREM 2 (INTUITIONISTIC ANALOGUE OF THE COMPLETENESS THEOREM FOR CLASSICAL LOGIC). TI-a iff r k a . We shall need the following corollary of Theorem 1 THEOREM 3. 1) Every @+/3-consistent r can be extended to a /3-saturated theory A such that a c d . 2) Every Axa(x)-consistent r can be extended to an a(n)-saturated theory A, for some a. Proof. 1) If r is @+/I-consistent then Tu {cc} is /I-consistent and by Theorem 1 has a P-saturated extension A. 2) If T is Axcc(x)-consistent then T u (cc(u)-+x(a)) is @(a)-consistent for a$Ind(T). For I n d ( a ( a ) ) s I n d ( r u { ~ ( u ) - + a ( u ) } and ) if T u (a(a)-+a(u))ta(u) then f k a ( a ) and so I't-Axa(x). Hence by Theorem 1, r c r u (a(u)+ a(a)} has an a(a)-saturated extension. Let card (A) be the cardinality of the set A . Let A G c B if A c B and B - A and B have the same infinite cardinality. Then A s c_ B implies there is a Csuch that A ~ s C E G B . Theorem 4 below is a strengthening of Theorem 1. It is a generalisation of the Lowenheim-Skolem Theorem of classical logic. THEOREM 4. Let A be infinite and Ind(T)z c A . If r is cc-consistent then r has an cr-saturated extension A such that Ind(A)S EA.

Proof. By examining the proof of Theorem 1 it is clear that at most new constants are introduced to define A = Un and say o$G, H) < o$G', H'). Then by the assumed isomorphism and the rule

-

-

pair e 3 -1 of U G * , H , , for some word W on 3G,H, not W"kG,.l for o-(G'H') n = 1,2, ... or oz(G',H')- 1but W 3 k,,,l.Since not o j ( G f , H ' ) l o 5 ( G , H ) this contradicts Theorem 11". As explained earlier this gives Result 5. Proof of Result 4. For given recursively enumerable degree of unsolvability D, let nDbe as described in Lemma 1. We assume some one recursive ordering of the words of IID and let o,(w) be the number of the word w in this ordering, plus l.I4 We assume e is not a generator of ll, and define o-(G', H')

ny = ( e :

eoD(,)

=

1) -

Further, we define Ilw to be the finite presentation whose generators are the generators of n D and of n;, and whose defining relations are the defining relations of JT, and of I l y . Once more we refer to Rabin's overall plan for his Theorem 1.1, p. 176 of Rabin [40] as stated on p. 177, the first seven paragraphs of 9: 1.3. We now define Il, to be as specified by Rabin when our Il, is identified with his no, our Zi'Y with his 112 - and consequently our I l w with his n. We shaIl eventually show that since Il, is torsionfree, i.e., has no elements of finite order (as also shown below), the family 8 = {Il,},, w ranging over the words of the R of Lemma 1, is the desired C(D)of Result 4. THEOREM 111 (Rabin). For any word w of R , XI, is trivial i f and only if 1 in I l D .

W=

The "plus 1" of this definition is not necessary but simplifies the argument. (Cf. the proof of Result 5 . )

l4

26

W. W. BOONE

For that special case of Rabin’s Theorem 1.1 in which the Markov property P is the property of being trivial, his 17, may be taken as the group presentation having no generators and no relations, so that his n ( w ) is his 17,. For each w,our 17: cannot be embedded in the trivial group. Thus Theorem I11 is simply (**) of our proof of Result 1, with P taken to be the property of being trivial. Throughout Rabin’s proof of (*) of our proof of Result 1, his w is arbitrary but fixed; but note that nothing in his argument precludes his IZ2 varying with w as we require. 2 denotes isomorphism (between groups presented). is torsion free. Let w and w’ be any two distinct THEOREM I v . Suppose , ~ only ifboth w = 1 and w’=1 in IT,. words of R. Then I l W ~ 1 7ifand From Theorems I11 and IV, and a verification that n, as given by Lemma 1 is torsion free, Result 4 is immediate. Certainly 1 = 1 in IZ, so that by Theorem 111, for any w of R,w = 1 in 17, if and only if IZ,EIZ,. Thus (?w,w ~ R ) w =1 in IZ, reduces to the isomorphism problem on 5. By Theoif and only ifeither rem IV, for any two distinct words w,w’of R, IZ,r17,, (a) both w = l and w ’ = l in 17, or (b) w is w’.Thus the isomorphism problem l 5on 5reduces - indeed by bounded truth-tables - to (?w,W E R)w = 1 in If,. Thus to show Result 4 it remains only to show Theorem IV and to verify the hypothesis of that theorem. Theorem IV follows at once from Theorems I11 and IV‘. THEOREM IV‘. Suppose n, is torsion free. Where w and w‘ are two distinct words of n,, suppose w # 1 and w‘# 1 in IZ,. Then not l7, z IZ,,. As we explain later, Theorem IV” implies Theorem IV’. With m and n positive integers, mln means m divides n. By the order of the element g of a group we mean the least positive integer m such that g”’= 1 in the group. A group G is torsion free except f o r N , N a positive integer, if (1) at least one element of G has order N ; ( 2 ) for every positive integer m, if m is the order of some element of G , then mlN. (Note that torsion free except for 1 is equivalent to torsion free in the usual sense.) THEOREM IV”. Suppose 17, is torsion free. Where w is a word of 17, suppose w# 1 in 17,. Then 17, is torsion free except f o r O,,(W). Same as footnote 13, but with “Result 4” instead of “Result 5”. Or Rotman [43], the “Britton’s Lemma” section, pp. 265-271. Lemmas 3 and 4 of Britton [lo] are, for our purposes, stated as Lemmas 12.3 and 12.4 respectively in Rotman [431. 15

16

ALGEBRAIC SYSTEMS AS A WHOLE

21

The central idea t o verify the “torsion free” hypothesis is the following lemma. In its statement and proof we assume known 3 2 of Britton [lo116 and 6 2 of Boone [7]. LEMMA 5. Suppose Cond,,,,(E*;E;p,, U E V ) .Then E* has an element of jinite order N , if and only if E has an element of order N . The “if” part of the lemma is immediate by Lemma 3 of Britton [10]16, i.e., since E is imbedded in E*. Suppose (T) W N =1 in E* and W”# I in E*, n = 1, 2,. . ., N - 1. To obtain the “only if” part of the lemma, we show by induction on the number of occurrences of the p,,, U E V ,in W that (*) for a certain word U of E, U N =1 in E and Un#1 in E, n= 1,2, ..., N - 1. If W is p-free, we may take W itself to be U , by Lemma 3 of Britton [loll6, i.e., since E is imbedded in E*. Suppose W is not p-free. Clearly, by Lemma 4 of Britton [lo], either (a) Wis notp-reduced or (b) Wisp-reduced but W 2is not p-reduced. If (a), note by Lemma 0 of Boone 171, that (p [ W])N= 1 in E* and (p [ W])” # 1 in E*, n = 1,2, ...,N - 1, where p [ W] contains fewer occurrences ofp,,, U E V,than Wcontains. Suppose (b) and write Was A B where B A is notp-reduced. Since B(AB)”B-’=(BA)“, m=O, 1, 2, ..., in the free group, (7) holds with BA taken as new W. Note that BA falls under case (a) and contains just as many occurrences of p,,, U E V as , AB, i.e., the old W. This completes the inductive argument for (*), and hence shows the lemma. By a Britton Tower we mean a sequence of finite presentations of groups such that for any member n and succeeding member IT’ of the sequence Cond,,,(n’; n ; p , , U E V ) Here . the sequence may be finite, or infinite. Lemma 6 follows at once from Lemma 5 by finite induction.

6. For any Britton Tower, one member has an element offinite order LEMMA N if and only if every member has an element of order N . Thus if one member of a Britton Tower is torsion free, or torsion free except for N , so also is every member. Now, at this point, we must slightly modify our original argument for Lemma 1 so as to make the new Il, torsion free. Assume (1) of the proof of Lemma 1. By Lemma 39 of Boone [6],p. 566, for the Thue system Trqo, there defined, nESif and only if chs;+’q,hd= 1 in TIqO1. Writeq-, for theq, of qqolto avoid a notational confusion. Then, by Theorem X of Boone [5], p. 250, for the Thue system TrqOl* there defined, n e S if and only if q,chs~+’q-,hd=g, in Tcqol*. We now identify TLqO1* with the T on p. 22 of Britton [lo] by taking the present q1 to be his qo, and the remaining symbols of TLq0,* to be his sl, s2,..., sM. (Cf. (11) of Britton [lo], pp. 29, 30.) The crucial point is that for this choice of T, the group presentation G of Britton

28

W.

W. BOONE

[lo], pp. 22, 23, has N=O. Of course, our equivalence (2) of the proof of Lemma 1 still holds, although r ( n ) now has a slightly different form. Clearly, this G can be taken to be the 17, of Lemma 1. For this G, what with N=O, Britton’s argument for Lemma 6 of Britton [lo] admits of a certain modification, viz., changes (a), (b), (c), and (d) of pp. 59, 60 of Boone [7]. In a moment we must refer to certain material on p. 25 of Britton [lo] in this modified form, but the revised version is almost explicitly spelled out in the middle of p. 60 of Boone [7]. LEMMA 7. 17,, i.e., the presentation G of Britton [lo] pp. 22, 23, with N = O as just described, is torsion free. We assert that the sequence

is a Britton Tower. Here F(x, y ) is the usual presentation of the free group on x , y ; and the G’s are as described on pp. 22, 23 of Britton [lo]. For Cond ,LB(G4; F(x, y ) ; sb,b = 1, 2,. .., M ) is verified by Boone [7], Lemma 18, pp. 72, 73; CondJLB(G,;G,; li, ri, i= 1 , 2,..., P) by Boone [7], Lemma 4, p. 65; Cond,,,(G,; G3; qa,a=O, 1, ..., N ) by Britton [lo], p. 25, lines 10-20 modified as just explained; CondJLB(G,;G,, t ) by Britton [lo], p. 24, lines 13-15; and CondJLB(G;GI; k ) holds as noted in Britton [lo], p. 24, lines 13-15 - since the identity mapping verifies the isomorphism condition. Thus, since F(x, y ) is torsion free, Lemma 7 follows from Lemma 6. Having thus verified the hypothesis of Theorem IV” for 17, as specified just a moment ago, we now show the conclusion of this theorem. The argument has much the same flavor as that just completed: for we must check that certain groups of Rabin [40] are torsion free except for certain integers. The main tool is the following lemma. LEMMA 8. Let P be a group which is a free product with amalgamation. Then (8.1) P has an element ofJinite order N fi and only if some one of the factors of P has an element of order N . Thus (8.2) if each factor of P is either torsion free or torsion free except f o r N , and if some factor of P is torsion free except for N , then P is torsion free except for N . For any element of finite order in a free product with amalgamation the transform (conjugate) of an element is belonging to one of the factors. See e.g. Neumann 1341, Theorem 5.1, p. 514 for this result. LEMMA 9.

If the group

H , of Rabin [40], page 178, in his Lemma 2, is

ALGEBRAIC SYSTEMS AS A WHOLE

29

torsion free except for N , then so also is the group H4 of Rabin [40], page 182, in his Lemrna 7. We assume the reader has 9 1.4 of Rabin [40] before him. The Tietze transformation argument of Rabin [40], p. 179, lines 12-20, regarding the suppression of a dependent generator we shall call the trivial modijication. Note that (901) the group F of (1.3) on page 178 of Rabin [40] is torsion free. This follows by our Lemma 5. For let ( t ) be the free group on t . Then F has the stable letter u and corresponding basis ( t ) . Moreover, the mapping t2-t generates an isomorphism between the subgroup of ( t ) generated by t 2 and ( t ) itself. Hence Cond,,,(F; ( t ) ; u). Now since H , is torsion free except for N by assumption, Rabin’s group Hi of line 4 on p. 179 is torsion free except for N by our (901) and our Lemma 8.2 since Hi =(H,*F)u(x)=u. Rabin’s group H I defined in his Lemma 2 on p. 178 is isomorphic to H i by the trivial modification, and hence is torsion free except for N . Rabin’s group F’ of (1.4) on p. 179 differs only in notation from F. Thus Rabin’s group H i of (1.5) on p. 180 is torsion free except for N by our (901) and our Lemma 8.2 since H i =(H,*F‘),,,. Rabin’s group H2 defined in his Lemma 3 on p. 179 is isomorphic to H i by the trivial modification, and hence is torsion free except for N . At this point we must make an interpolation into Rabin’s construction to establish the torsion freeness except for N of his group H,, defined in his Lemma 4 on p. 180. Recall that his H2 is, in his Lemma 3 on p. 179, defined by ( x , t , a : r ( x ) , u ( x ) t = t 2 u ( x ) , ta = a 2 t ) . Let the group F, differing only in notation from F, be defined by ( u , s: us =

A).

By our (901), F is torsion free. What with u having infinite order in F and u ( x ) having infinite order in H,, which is torsion free except for N , we may form the free product with amalgamation

By our Lemma 8.2, K ; is torsion free except for N . Referring to the presentation of H , displayed above, let K , be defined by (x, t , a , s: r ( x ) , u ( x ) t Clearly K ,

= t2U(X),

ta = a t 2 , u ( x ) s = s’u(x)).

K ; by the trivial modification so that K , is torsion free except

30

for N . Let

W. W. BOONE

p, differing only in notation from F, be defined by (p, b: p b = b2p).

By our (9a), fi is torsion free. From considerations exactly like those about forming K ; , we see we may form the free product with amalgamation

By our Lemma 8.2, K, is torsion free except for N . Clearly H3 zK , by the trivial modification so that H3 is torsion free except for N as required. Finally, Rabin’s group F“ is defined on p. 182, line 9, as the free group on c,d - and thus is torsion free. Since H4 = (H3 * F”),=, - as noted by Rabin on p. 182, line 19 - by our Lemma 8.2 we have the torsion freeness except for N of H4, i.e., the desired Lemma 9. By Lemma 8.2, if w # 1 in 17, the group presented by IIW - described just prior to stating Theorem 111 - is torsion free except for oD(w): for 17, is torsion free (Lemma 7), I7p is torsion free except for o,(w), and the group presented by 17” is the free product of those presented by 17, and 17;. Let fiw be the presentation obtained from Ilwby adding the new generator x , , + ~ Again . by Lemma 8.2, since the infinite cyclic group is torsion free, fiwis torsion free except for oD(w). As argued by Rabin near the bottom of p. 183 of Rabin [40], X , + ~ W X ~ , ! ~has W infinite order in the group presented by f i w . Finally, the presentation we have called I7, - described just prior to stating Theorem I11 - is obtained from fib‘ by following directions (a) and (b), top of p. 183 of Rabin [40]. Thus by Lemma 9 we have shown that 17, is torsion free except for o D ( w ) :for fiwand 17, differ only as to notation from Rabin’s H,, and H4 respectively. This shows Theorem IV”. Now Theorem IV’ follows easily. Where w and w‘ are two distinct words of 17,, suppose w # 1 and w’# 1 in 17,. By Theorem IV”, 17, is torsion free except for o,(w) and II,, is torsion free except for o,(w’). But w and w’ are distinct, so o D ( w ) # D D ( w ’ ) . But from the very definition of torsion free except for N , it is not possible that a group be torsion free except for N and except for N‘, with N # N‘. Hence not h’,rn,, as claimed. As explained earlier this gives Result 4. Proofs of Results 6 and 7. As remarked in section 1, for these proofs, we d o require the (constructive) existence of a finitely presented group with word problem of arbitrarily preassigned recursively enumerable degree. We show only the case where degree D (called Do below)

31

ALGEBRAIC SYSTEMS AS A WHOLE

Let 17, be a n arbitrary finite presentation of a group, P an arbitrary Markov property of groups. By direct application of the main technical result of Rabin [40], we have that, where w is any word of no,there exists a finite presentation U,, of a group such that

w = 1 in 1 7 0 e 1 7 w ~ P .

(1)

Moreover, II, is recursively computable from w alone for fixed noand P. While in Rabin [40], IT, has an unsolvable word problem this in no way affects the correctness of (1). As we already noted in section 1, the property of groups to have a word problem of a particular recursively enumerable degree of unsolvability, say Do ZO’, is a Markov property. Thus as a special case of (1) we have w = 1 in 17, o {(?uW)[u, = 1 in

(2)

TI,]} ED,

where u, is any word of n,. But we can now iterate Rabin’s construction, i.e., from (1) we have that for each n, and any arbitrary Markov property P,one can recursively comof a group such that pute a certain finite presentation IIw,uw u, = 1 in 1 7 , 0 1 7 w . u , ~P .

(3)

By (2) and (3) we have

w

(4)

= 1 in 1 7 , 0 { ( ? u w )

[IT,,uw~P]}~DO.

Hence, by (4) where D, is any recursively enumerable degree of unsolvability,

(5)

{(?w)[w

=

1 in

no]} €D1 o [{[(?.W>

Cflw,uwEP3)

ED011ED1 *

By (5) and Fridman [19], Clapham [15], or Result G, p. 50 of Boone [71, we have the present Result 6 . To show Result 7, take the 17, in (2) torsion free so that the groups 17, l 7 The case D = 0’ is similar. But a construction uniform in D requires essentially new ideas furnished in Jockusch’s supplement to this paper. l8 By result G, p. 50 of [7], and our Lemma 7 , (t) there is a uniform construction which for any recursively enunwable degree D produces a torsion free finitely presented group with wordproblem of degree D . In the notation of p. 21, last paragraph, let 170,171and 172 all present torsion free groups; and let 171 and 172 have word problems of degrees DO# 0’, and 0 , respectively. By Lemma 9 and a simplified version of the remarks on p. 30 showing Theorem IV” from Lemma 9, the groups n ( w ) described at top p. 22 are torsion free. Take these n ( w ) as the Z7, of (2).

32

W. W. BOONE

of (2) are also.18 Then by Theorem IV of the proof of Result 4 we have for each XI, of ( 2 ) and arbitrary words u, and u; of ll,, that we can recursively compute Il,,uw and nw,u,w, finite presentations of groups such that Condl (w,u,,

(6)

UL)-~,,~,

E nw,u,w.

Here Cond, (w,u,, us) means that [u, = 1 in Ilw and u; = 1 in

n,] or [ u ,

is u;]

.

But, since for each w there is an obvious recursive procedure to determine if

u, is u; or not,

(7)

(?uW)[uw = 1 in Il,] =:(?uw, uk) Cond, (w,u,, u;)

for each w,where E indicates Turing equivalence. Directly from (6) we have for each w that

(8)

(?uw,u;) Condl (w,uw, u;)

ST

Thus by (7), (8), and the transitivity of (9)

(?uW)[u, = 1 in

n,]

(?u,, uk) =T,

[n,,,_

nw,firw]

for each w,

E ~ ( ? u , ,u;) [Il,,uw 2 llw,uew].

By ( 2 ) and (9) we have that for each w and any recursively enumerable degree of unsolvability Do, (10)

w = 1 in

no-= {(?uW, u;) [n,,,_z n,,ufw]> €DO.

Hence, for any recursively enumerable degree of unsolvability D,, (11)

{(?w)[w = 1 in no]} ED, c>

{(?4 [{(?uw, 4) [nw,uw = n w , u , w l l E Doll E Dl

*

By (f) of footnote 18, we now have Result 7 from (11). Proof of ResuEt 8. In effect, this result was shown by Rabin in Rabin [40] and it is only a matter of looking at his account in a certain way to see that this is so. Near the end of our proof for Result 1 (“For our purposes we must ...”) we discuss our point of view toward Rabin’s no,Ill,IT2, n, n,, and I l ( w ) , which we here again take. Now Rabin’s n, is obtained from his Il by adjoining seven additional generators x , + ~ t, , a, s, b, c and d independently of w, as well as certain defining relations - which depend on w - which we here call S(w).Let Il’ be the presentation obtained from Il by adjoining the generators x,+ t , a, s,

,,

ALGEBRAIC SYSTEMS AS A WHOLE

33

b, c and d, but no new defining relations; nothe presentation whose generators and whose defining relations are those of Illand Il'. Of course n, and D2 depend on the chosen Markov property P. Since IT, for each word w of IZ, can be obtained from the fixed noby adjoining the relations S(w),nocan be taken as the desired ZIP of Result 8.

Result 9. As noted in the statement of Results, this Result requires no proof. Actually, many of the earlier arguments of the paper are, in effect, applications of Result 9. Familiar arguments for unsolvability in symbolic logic can be modified so as to furnish us with the hypothesis of Result 9 in much the same way as we reinterpreted unsolvability results in algebra to obtain Results 1, 2 and 3. For example, consider the arguments for the fact that there exist partial propositional calculi with unsolvable decision problem as to theoremhood (Post and Linial [39], Davis [17], Yntema [54]). These arguments, in effect, stipulate a recursive class of well-formed formulas whose decision problem as to theoremhood is of preassigned recursively enumerable degree. For in such argument, call a well-formed formula, of the propositional calculus S being constructed, a code formula if the well-formed formula represents a word of the underlying Thue system or Post normal system T. By the results of this paper or Yasuhara [53] we may take it that T has a recursive class R of words, with a word problem of a special kind, of given recursively enumerable degree D . Then for the desired class C to satisfy the hypothesis of Result 9 we simply take those code formulas of S which stand for words of R. The authors mentioned go on to show that, since the theoremhood problem for S is unsolvable, the completeness problem for partial propositional calculi is also. This latter argument in effect furnishes us with the remainder of the hypothesis of Lemma 9, where P is completeness, etc. Thus there exists a recursive class of partial propositional calculi with completeness problem of preassigned recursively enumerable degree. Certain familiar proofs of Church's Theorem - such as given in Davis [17] and Hermes 1201- furnish us, in a similar way, with the C of Result 9 where S is the first order functional calculus. But no particularly interesting applications of Result 9 relative to this situation are known to the present author.

SUPPLEMENT TO BOONE’S “ALGEBRAIC SYSTEMS” C . G. JOCKUSCH Jr. University of Illinois

We assume complete familiarity with the arguments on pp. 31, 32 for Results 6 and 7 of Boone’s paper. These arguments apply only if the given degree D is not 0 , because one needs to know that it is a Markov property of groups to have word problem of degree D . On the other hand, it is very easy t o modify the argument for the case D=O‘ because the property of having a word problem of degree # 0’ is also Markov. (Roughly, substitute # for E.) However, as C .F. Miller I11 has pointed out, the existence of these two separate constructions does not guarantee that there is a construction uniform in D.In this note we specify a uniform construction for Result 7. The uniform construction for Result 6 is entirely analogous. Our construction is essentially the union of the two constructions - the one for D # 0’, the other for D = 0’ - previously mentioned. The proof that our construction has the desired properties hinges on determining the output of each of the two separate constructions when applied to a degree D which it was not intended to cover. We consider first Boone’s argument for Result 7 for the case D#O’. However, for the moment we make no assumptions on D , D‘ other than that they are arbitrary recursively enumerable degrees. Henceforth, we write D, D‘ as Do, D , , respectively, as in Boone’s proof of Result 7. If Z I is any finitely presented group, we write deg n for the degree of the word problem of n. We use 0 for the degree of all recursive sets and d u d * for the least upper bound of the degrees d, d*. Any unexplained notation will be found in Boone’s proof. Referring to the right-hand side of Boone’s (11) we let d(D,, 0,) be the degree of

(9

{(?.I

C{(?.W?

ul)

En,,._ = ~

w , u ~ , 1 ~’ ~ ~ 0 1 ~

Boone’s proof showed that d(D,, D,)=D,, provided D,#O’. We must now 34

SUPPLEMENT TO BOONE’S “ALGEBRAIC SYSTEMS”

35

compute d(O‘,Dl) by examining the details of Boone’s construction. However, we may not use Boone’s ( 2 ) because it holds only when the property of having word problem of degree Do is a Markov property, i.e. when Do#O. From our (i) and Boone’s (9) (whose validity depends only on each n, being torsion free, - and not on (2)) we see that d(Do, Dl) is the degree of (ii)

W W )

[deg n, = Doll.

Thus we desire to find deg Il, for each word w of Il,. Recall that Il, is obtained by Rabin’s construction from IT,, Il,, 112, where deg n , = D l , deg I l l = D o , deg I l , = O ’ . Furthermore, if w = l , I ~ , E I ~ and , , if w # 1, n, can be embedded in IT,. It follows that (iii) (iv)

w = 1 a d e g n , = d e g n , = Do ; w # 1 a d e g n , = 0’.

If we now assume that Do = 0‘ we have deg Il, = Do for all w in Il, so it follows immediately from (ii) that d(O’, Dl) = 0. We now consider the natural construction intended for the case Do = 0‘. This construction is identical to the previous one except that it starts with groups n,*, IlT, Il: rather than no,ZI,, Il,. To exploit the fact that having word problem of degree # 0‘ is a Markov property, we choose deg Ilg = D,, deg I l F = O , and deg n;=O’.Now let Il; and Ilz,,, be constructed from n,*, IlT, Ilg as before, and let d*(Do,Dl) be the starred analogue of d(Do,Dl). We now have that d*(Do,0,) is the degree of (ii)* We also have (iii)* (iv)*

w = 1 + deg Il; = deg IZT = 0 ; w # 1 =>degIlz = 0’.

From (iii)* and (iv)* it follows that w = 1 in Il,* iff deg I l z = O . Hence we see from (ii)* that d* (0, 0,) = deg Il,*=D,.Similarly, d* (0‘,Dl) = D , . On the other hand, if O have r(p)=r(p’) andp(p)=p(p‘). LEMMA4. For each ordered pair (D, 0‘)of recursively enumerable degrees of unsolvability, there exists a recursive family F(D, 0’)of fully recursive classes of finite presentations of groups, such that: The decision problem to determine, for an arbitrary member Q of F(D, 0’) whether or not the isomorphism problem f o r Q be of degree D, is itself of degree D’. 2.2. On certain special Tietze transformations We introduce now the following “elementary operations” on finite presentations of groups (compare Markov [26]): The presentation p = ( { x 1,..., x r } , {a,, ..., K ~ - ~ai-a‘a”, , m i + , , ..., a,}) (where ul,..., a,, a’,a’’ are words in the x;l’s) is replaced by the presentation pl=({xl, ..., x r } , {K,, ..., ai-l, ci‘~4x,:~a‘‘,a i + 1 7..., a,}) nherejE(1, ..., r } , E = I. The inverse of Op, (deleting a syllable xsx,YE in one relator). The presentation p=({xl ,..., x r } , {a1, ..., ai-,,ai, ..., a,}) is cil, cii+l, ..., ci,}) where the replaced by pl= ( { x l ,..., xr>,{a1,..., word a: is a cyclic permutation of the word oli, The presentation p = ( { x l ,..., x r } , {a,, ..., mi, ai+,, ..., a,}) is re-1 placed bypt=({xl,..., xr}7 {a,,..., cii-1, xi , ~ i + 1 , - - . ,a,}). The presentation p = ( { x , ,..., x r } , {a1,..., ai-,, ai, a i + l ,..., a,}) is replaced by pl=({xl ,..., xr}, {a1,..., a i - l , u p j , u i + ,,..., a,}) where j ~ { l..., , p),j#i. The presentation p=({xl, ..., xr}, {K,,..., a,}) is replaced by p l = ({xl, ..., xr, x r + , } , {al,..., a,, x,+,a}) where x , + ~ is a letter, different from xl, . . ., x,, and ci is a word in the letters x : .., xr’

’,.

Lemma 3 with (3iii) deleted is Result 4 of Boone [8].But certainly, adjoining p,, to Boone’s class C ( D ) does not change the degree of either the isomorphism problem or the triviality problem. In his Result 7, all presentations concerned may be taken to have the same set of generators. so that recursive implies fully recursive.

l1

w. w. BOONE, w. HAKEN and v. P O ~ ~ N A R U

48

Op;

: The inverse of Op, (deleting in p a generator x j and a relator a i that reads x j a where a,, ..., a c - l ,a,a i + l..., , a, do not contain letters x j or x , ~’).

Remarks. Clearly, these operations preserve the isomorphism class of p ; 8 ( p ) z 6 ( p l ) . Moreover, Op:’, Op,, Op,, Op, preserve also the group presented 8(p)=Q(pl), i.e., the alphabet is not changed and hence g(p)=%(pl) and also %(p)=%(pl). The inverse operations of Op, and Op, are again operations Op,, Op,, respectively; the inverse of Op, can be composed of one application of Op, followed by one application of Op,, several applications of Op;’, and another Op,. The following lemma generalizes a lemma of Markov [26] and is quite parallel to a well-known result of Tietze: Let p=({xl ,..., x,.), {a1,..., a,)) and p’=((yl ,..., ,vrr), be two isomorphic group presentations, and let t, t’ be any integers such that p + t - r =pf + t‘ - r‘, t > p + r‘, and t’ >p’ i-r. Then the presentation p * t can be transformed by means of a Jinite sequence of operations Op: 1 , 0 p 2 , 0 p , , 0 p 4 , 0 p ~into the presentation p‘ * t’. Proof. We may assume that p 4r‘3p‘ + r. Let I : 8 0s)- 8 (p’) be an isomorphism of 0(p) onto 6 01’). Our first objective is to transform p * t and p ’ * t ’ into presentations p # and p;, respectively, both in the same generators, say zl, ..., zr# with r # = r + r ’ , where z , ,..., z, “correspond” to xl ,..., xr and zr+l,..., z,# correspond to y l , ..., y r r . Let tl,.. ., t,.,be words in the xi’s such that the corresponding elements C j ‘ 3 ( p ) ~ Q ( p(where ) Yal,-l, ~llY:~:~l-tl,Yul,+l,.'., Y P # ' ~llY:::~;ll,*t-l});

and then by d p - ' into ( { ~ 1 , . . * z, r g ) , { ~ 1 , . . . , ~ u l l - 1 9 ~ u I 1 )~ u i l + t , . . . , ~ p r ' , ~ 1 ~ : ~ ~ ' ~ ~ ~ ~ * ~ - ~ } ) .

Then by a similar triplet of operations, we obtain ( { z 1 , . . . >Z r 5 ) ,

{~1,..-r 'ipt'

1

T,,Y",::T;~T I

FIZT-l ~ Y U I 1 ~2

*t-I 3

}).

Continuing in this way we obtain (by a, -2 further triplets - see (#)) ({zlr

..., z r # j , ( ~ 1 ,

n T,,Y",~TG',*'-'I). a1

. . I >

~ p r r

k= 1

From this we have (by (#)) by a sequence of Op:' ({Zl,

...3

Z'J>

{Yl?

...9

Yps7

81,

*'-'I).

UNSOLVABLE PROBLEMS IN TOPOLOGY

51

In a similar way we obtain further

This completes the proof of Lemma 5.

3.

Topology

3.1. Preliminaries on manifolds and handles

In this section we recall the basic topological concepts which we use in this paper. However, we expect the reader to be familiar with the elementary concepts of general topology and of algebraic topology, especially the notions of fundamental group and homology groups, as explained in the usual text books, e.g. Seifert and Threlfall [45], Cairns [14], Spanier [48]. Topological manifolds. A (topological) n-manifold M (with or without boundary) is a connected, separable, metric space each point p of which possesses a closed neighborhood N ( p ) that is homeomorphic to the compact unit n-ball x: + ... + x’,< 1 in euclidean n-space En.A point P E M is called a boundary point of M if it lies in the boundary dN(p), of its closed neighborhood N ( p ) ; otherwise p is called an interior point of M . The set aM of all boundary points of M is called the boundary of M ; Int M = M - d M is called the interior of M . If M is coinpact and d M = @ then M is called a closed n-manifold. An n-manifold D,which is homeomorphic to the compact unit n-ball is called a (compact, topological) n-ball; dD, is called a (topological) (n - 1)-sphere. Combinatorial manifolds. A topological m-simplex om is an equivalence class of homeomorphisms cp: Io,(+6, of a topological m-ball Ioml, the point set of om”,onto a rectilinear simplex 6, in En,where two homeomorphisms cp’: Iom1+66 and 9”: Iom/-f6aare equivalent if there is a linear map x:6;+6:, i.e., x is a homeomorphism given by linear equations, such that (p“=x.cp’. If 6, is a face of 6, (we permit r=m) then cp1cp-’(br) represents a face or of urn.A simplicial complex A is a set of topological simplices such that (i) if O E A then all faces of o also belong to A , (ii) if o , o ’ ~ Ao#o‘, , then lul n lo’(is either empty or the point set of a face both of o and of 0’. The point set union ] A [ of A , with the topology induced by the c’s of A , is called the polyhedron of A . A simplicia1 complex A* is called a semilinear subdivision of A if (a) ] A * /= lAl, (b) for each Q * E A * there is a ~ E A represented , by a homeomorphism cp: 1o1+6 such that 1u*I c 101 and cp1 lc*l represents o*. 6‘

52

w. w.

BOONE,

w. HAKEN and v.

POENARU

It is often convenient to consider simplicial complexes A such that the point sets 101 of the simplicies CT are (rectilinear) simplicies in a euclidean space En and such that the identity map on I C T ~ represents 0.In this situation one may identify 0 and 101 and call A a (rectilinear) simplicial complex in En. By a triangulation of the topological manifold M we mean a simplicial complex A with lAl = M . It is a famous problem in topology whether every n-manifold admits a triangulation; for n > 4 this is still an open question. Two simplicial complexes A , , A , are combinatorially equivalent if they possess semilinear subdivisions A ; , AT respectively, that are isomorphic (i.e., such that there is a 1-1 correspondence between the simplices of A : and AT that preserves the dimensions and the incidence relations). It is clear that in this situation there exists a so-called semilinear homeomorphism ;1:I A , I +Id, 1 that maps the simplices of AT linearly onto simplices of AT. A simplicial complex r is called a combinatorial n-ball if r is combinatorially equivalent to the triangulation of an n-simplex into all its faces. A combinatorial n-manifold is an n-manifold M together with a triangulation A such that for each vertex (= O-simplex) P E A the simplex star St (plA ) (=the set of all simplices of A that are incident with p , and their faces) is a combinatorial n-ball. Such a triangulation is called a combinatorial structure on M . A famous conjecture in topology is the so called Hauptvermutung; it states that two complexes A , , A , are combinatorially equivalent if lAll and ld21 are homeomorphic. The converse is trivial. This has been disproved for complexes in general (Milnor [29]), but the restriction to combinatorial n-manifolds is still an open question for n 2 4. It has been proved for n = 3 - Moise [30], Bing [4]; recently a proof was obtained for all simply connected (i.e., with trivial fundamental group) combinatorial n-manifolds with n 3 5 (Sullivan [49]). By aJLinite (non-oriented) abstract complex 8 we shall understand a finite collection of finite sets; each set consists of letters, called vertices, taken from some suitable alphabet, say pl,p,, ...,pu; and further SEO and S'c S imply that S ' E ~If. SEOcontains n+ 1 vertices, then we call S an (abstract) n-simplex of 8. And a subset of S is called a face of S. The relation of isomorphy can be defined for abstract complexes in the same way as for simplicial complexes; it is also meaningful to say that an abstract complex is isomorphic to a simplicial complex. In fact, a finite abstract complex can be regarded as a finite presentation of an isomorphism class of simplicial complexes. We call an abstract complex 8* a subdivision of the abstract complex 0 if every vertex of 0 is also a vertex of O* and if there are simplicial complexes A * , isomorphic to 8*, and A , isomorphic to 8, such

UNSOLVABLE PROBLEMS IN TOPOLOGY

53

that A * is a semilinear subdivision of A . Now we may define the combinatorial equivalence for abstract complexes in the same way as for simplicia1 complexes. Two n-manifolds M , M’ are called homotopy equivalent if there exist continuous maps f : M-, M ’ and g :M’-, M such that g of : M-+M is homotopic to the identity map on M , and f o g : M‘-,M‘ is homotopic to the identity map on M‘. Diflerentiable manifolds. A coordinate system on a (topological) n-manifold M , (with or without boundary) is a homeomorphism h: W-tH,, where W and h (W) are open sets in Mn and in H,, respectively, and where H,, is an n-dimensional euclidean half-space (i.e., the subset of euclidean n-space E, for which x,,>0). The mapping h associates with each point P E W the coordinates of h(p) in H,,. Two coordinate systems h1:Wl+H, and h,: W2-fH,, are called C-related, r e { l , 2, ..., co,w > if the corresponding “coordinate transformation” h l o h;’Ih,(W, n W,): h2(W1n W2)+ 4 h l (W, n W,) is of differentiability class C‘ (here C” is the class of analytic functions), and has non-zero Jacobian determinant in all of h, (W, n W,). A C-atlas of M,, is a system h,: W,+H,, of pairwise C‘-related coordinate systems on M,, (6 ranging over an arbitrary index set) such that the W,’s cover M,,. A C-structure on M, is a maximal C-atlas, i.e., not a proper subsystem of another C-atlas. A C‘-n-manifold is an n-manifold M,, together with a C‘-structure, and this is called a diferentiable manifold if r = co. Let M,,, M,‘ be topological manifolds; if h : W-+H, is a coordinate system on M,, and if q : M , Z M ; is a homeomorphism, then we call hocp-’:q(W)-tH,, “the coordinate system on M,‘ carried over by cp from h”. Furthermore, if S is a C-structure on M,, then the collection of all coordinate systems on Mi carried over by cp from members of S is a C‘-structure on M i ; we shall call this the C-structure carried over by q (from S ) . Two differentiable manifolds M , M‘ are called difeomorphic if there is a “difeomorphism” M-+M‘, i.e., a homeomorphism of M onto M‘ that carries the differentiable structure of A4 into that of M‘. We remark that, if aM,#8, each C‘-structure on M , “induces” a C-structure on the (n - 1)-manifold aM,,, since for each coordinate system h: W-tH,, we have h(WndM,)=h(W)nE,,-, where E,-l means the (n - 1)-space x, = 0 bounding H,, in En. A C’-imbedding of a C”-manifold M,, with r‘>r in euclidean m-space Em means a homeomorphism 9 : M,-E, that is of differentiability class C‘ with respect to the C”-structure of M,,, i.e., g0h-’:h(W)+E,,, is of class C‘ and the Jacobian matrix of g o h - 1 if of rank n, whenever h: W-,H,, is a coordinate system of the C”-structure of M,,.

54

w. w. BOONE, w. HAKEN and v. P O ~ N A R U

If M, is a C‘-manifold then a combinatorial structure A on M, is called compatible with the C‘-structure of M , if each m-simplex am€A is a so-called C-simplex, i.e., there is a coordinate system h : W-tH, in the C‘-structure of M , such that JumI c W and h(Ja,J) is a rectilinear m-simplex in H,. For more details on differential topology see for instance Munkres [32]. Concerning differentiable manifolds we quote the following famous results: (i) Each C’-structure contains a C“-structure and moreover an analytic, C”, structure (Whitney [52]). (ii) Each differentiable manifold admits a (compatible) combinatorial structure (Cairns [I I, 131; Whitehead [51]). (iii) There exist combinatorial manifolds that do not admit a (compatible) differentiable structure (Kervaire [22]). (iv) For n B 7 the n-sphere admits several different differentiable structures (Milnor [27]). Handles. For the construction of differentiable and combinatorial manifolds with prescribed fundamental groups, we shall use the operation of “handle-adding” to a manifold M with boundary which induces the operation of “Morse-surgery” on the boundary-manifold aM. For the remainder of this section, all manifolds considered are to be differentiable or combinatorial or both. Let M, be a compact manifold with aM# @.We consider an (n- 1)-dimensional submanifold Mi - c JM, and p copies of the n-ball D,: D’,,D” ,..., 0,”. Each of them is regarded as a Cartesian product of the A-dimensional ball with the (n - @dimensional ball. Passing to the boundaries, we have:

ao: = (ao;x D:-~)U(D: x =(sf-,X D ~ - ~ ) U ( DX S: ; - ~ - ~ ) Here S:-

( i = I ,..., p ) .

means the (A- 1)-sphere aD:; actually Int(S:-, x D:-,)nInt(D: x = 0; further and a(S:-, x D i - J = S:-l x a&, = S f u lx

a(D; x S ~ - , - , )

= aD: x

S,,-A-1 i

=$-I

X $ - ~ - 1

are identical. Let us consider p differentiable and/or semilinear l4 homeomorphisms q i : ~ i - x. l~ i - ~ - + I n t ~ ; - (~i = 1 ,..., p )

UNSOLVABLE PROBLEMS IN TOPOLOGY

55

such that Image cp‘nImage cpk=O for I # k . Let us consider the quotient space obtained from M,, uDj u-.-Y 0,” if every x ~ S j - x, D l - , ( i = l , . . . , p ) is identified with c p i ( x ) ~ M : - ,c d M , . The topological space we obtain in this way is an n-manifold and has a “natural” differentiable and/or combinatorial structure. (See Smale [47] ; in this paper we shall need only the cases A = 1 or 2 where n 3 5, and in these cases we shall directly present these structures.) And we denote this differential and/or combinatorial n-manifold, as Smale does, by

X(M“,M;-l; c P 1 , . . . , ’ p P , A ) . It is called the result of “adding p handles o j index I. to Mn on MA-,”. We recall the definition of “Morse surgery” (see e.g. Milnor [28]). Let M i - be a compact manifold. We consider p copies of the (n - 1)-sphere Si- .. ., S,P- 1. Each of them is regarded as a union of two Cartesian products

,

,,

i

where

=

( s ; - ~x D;-~)u(D:

d(Si-, x Df-J

x

s;,-,-,)

= d(Di x

Let us also consider p differentiable and/or semilinearl* homeomorphisms .

.

cp’:S;-, x Dk-I+Int M,,’-,

(i

=

1,..., p )

such that Image ‘p’nImage qk#O for I Z k . Let C be the closure of M ~ - , - - U f = ,cpi(Si-,xDL-,). We have

ac = a ~ ” ‘ - , u where

a($-,

u cpi(sj-,x s;-,-,> P

i= I

xD;-J=s;-]

xs:-A-l.

Let us consider the quotient space obtained from Cu

u Di x P

i= 1

,

if every x ~ S j - x, Si-,-l is identified with cp’(x)~dC.This space is in fact a Here “semilinear” means compatible with the combinatorial structures A n i of Dlli and A n of M n , i.e., pi maps simplices of a certain semilinear subdivision of A n i linearly onto simplices of a semilinear subdivision of An. l4

56

w. w.

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w.

HAKEN

and v. POENARU

manifold which we denote by

‘>

This is called the result of Morse surgeries of index A applied to M: - I ” . The definitions of handle adding and Morse surgery have been given independently of each other; however, note that the operation of handle adding t o M , on M i - induces Morse surgery to M i - Let us identify the spheres Si- of the definition of Morse surgery. with . dD;of the definition of handle adding. Further, let us regard M i - D;, Sf- SL-A- and cpi as identical in both definitions. Then we have

and in the special case that MA- I = d M , we have v = d X , which we shall speak of as “the (n - 1)-manifold obtained from dM, by the Morse surgery induced by the handle adding t o M,”. We remark that every differentiable and/or combinatorial n-manifold M , without boundary can be obtained from the n-ball D, by successive handle-addings of index A= 1, 2, ..., n (see Smale [47]), - a list of the corresponding homeomorphisms cp (for each A = 1, ..., n) being then called a handle-presentation of M,. 3.2. Finite presentation of differentiable and combinatorial manifolds. Proof of Theorem 4 Mathematicians have firmly fixed as a working concept the notion of “finite presentation of a group”. On the other hand the notion of “finite presentation of a manifold” requires here a considerable discussion. The logician will recognize it as much the same sort of analysis by which one passes from the intuitive notion of effective process to the precise technical notion of recursive process (Church’s Thesis). We contend that any definition of a “finite presentation lrJz of a differentiable and/or combinatorial n-manifold M” should satisfy the following conditions : (a) Zm is a finite notation, i.e., a finite sequence of symbols in some language; (b) there is an algorithm to determine whether or not any given finite notation in this language be a finite presentation; (c) to each finite presentation Zm there is precisely one n-manifold M(!JJl), presented by lrJz. However, a concept of finite presentation that fulfills these three necessary

UNSOLVABLE PROBLEMS IN TOPOLOGY

57

conditions may still be unsatisfactory. We are thus led to the further demand that (d) 9X describe M(9Jl) in a “natural” way. As to the interpretation of (d) we take the point of view that a finite presentation 9X of a differentiable and combinatorial manifold should have the property that (compare section 1.2) a triangulation A and a Cm-atlas 2 of M(9X) are described by 9X. Let us remark that the “handle-presentation’’ (see last paragraph of section 3.1) of a differentiable manifold, which is a very useful tool for many investigations, is not a finite presentation unless the C“-homeomorphisms cp are described by a finite notation so as t o fulfill Condition (a). We remark further that an abstract complex is a very satisfactory finite presentation of an isomorphism class of simplical complexes. However, for presenting combinatorial manifolds of dimension > 3 we need a more elaborate concept because of condition (b). There is no algorithm known that allows us to decide whether or not a given complex represent a combinatorial n-manifold (whenever n > 3 ) ; for, for such an algorithm, we should need a solution of the combinatorial equivalence problem with the (n - 1)sphere. Note that for dimension > 3 a presentation in terms of “incidence matrices” is to be rejected on the same grounds. Now we shall define the finite presentation YJl in such a way that it describes (I) a euclidean q-space E, (9X); (11) an n-dimensional, rectilinear, simplicia1 complex A ( 9 X ) in E,(9X) with rational vertices; (111) for each simplex star of A (9X) a semilinear homeomorphism into an n-dimensional subspace of E,(9X) (which makes it evident that A is a combinatorial n-manifold) ;and (IV) for each open simplex star of A (YJl) a homeomorphism into an n-subspace of E,(YJl) such that these homeomorphisms form a C“atlas q(9X) on IA(9X)l and are described by a set of algebraic equations. The algebraic equations will be derived from a q x q-matrix L and a 1 x q-matrix u whose components are polynomials in the coordinates of E,(%Jl). Here we use techniques developed by Nash [ 3 3 ] . The homeomorphisms can be interpreted as mapping each point of lA I into the nearest point of an approximating sheet g of an algebraic variety and then projecting this point into an n-subspace of E,(YJl). The matrix u approximates the component matrix of the distance vector from any point in a neighborhood of g to the nearest point on g ;the matrix L approximates the component matrix of the tensor that projects each vector into the (q-n)-dimensional normal plane to .!%. An algebraic atlas presentation YJl of a closed n-manifold with a combi-

58

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w. HAKEN and v.

POENARU

natorial anda compatible diflerentiable structure means an (ordered) collection

m = ( X I ,..., S q ; p l , ..., p s ; e; i,, ..., is; L; U; is, E , D) with the following properties: (I) x,,. .., xq are letters, called coordinate variables or simply coordinates. We denote by E,(%R), “the euclidean q-space presented by %R”, the euclidean q-space with coordinates x l , ..., xq. (11) p , , ..., p s are pairwise different 1 x q-matrices whose components are rational numbers; 8 is a finite abstract n-dimensional complex with vertices p1,..., p s . We denote by p, ,..., p, the points in Eq(%R)with coordinate matrices pl, .. ., p s , respectively. We require the following further properties of the p’s and 8: (IIa) If (p,,, ..., pjvm)~O, then p,,, .. ., pi, are in genera1 position, i.e., in E,(%R) there is a (rectilinear) m-simplex om with vertices pi,, ...,pi,,. (IIb) The set of all those simplices which correspond to the members of 8 in the sense of (IIa) is a rectilinear simplicia1 complex in E,(%R), ‘‘the simplicial complex presented by %R”, denoted by A (%R) ;moreover, the boundary complex of A(%R)is empty. (III) i, (ke{l, ..., s>) is an n-tuple of positive integers i l L < i 2 , < . . . < i f l k < q such that the (compact) simplex star St(p,lA(%R)) of pk in A(%R) projects 1-1 into the coordinate space E n ( i k )with coordinates xitr, ..., x,,,. I.e., the map z,:Eq(%R)-+Efl(i,)that maps a point with coordinates x;, ..., x,* into the point with coordinates ..., x:~ induces a semilinear homeomorphism of St(p,lA(%R)) into En&). (IV) L is a symmetric q x q-matrix, and u is a 1 x q-matrix where the components of these matrices are polynomials in the variables x,,..., xq with rational coefficients. 6, E and D are positive rational numbers, ~ < 1 / 2 n ; D < l . Let B,, k = 1,2, ..., s, denote the q-ball in E,(m) with radius 6 and with center p k . Then we require the following properties: (IVa) For each point in Ui= B,, L possesses n “small” eigenvalues whose absolute values are smaller then E/n and q - n “large” eigenvalues whose absolute values differ from 1 by less than ~ / n This . is equivalent to the condition that the coefficients of the characteristic polynomial a(],) of L (in the variable 1,with highest coefficient normed to 1) are sufficiently close to the coefficients of An(l.- l ) q - ” . (IVb) No 1-simplex of A(%R)is larger than 46. Let p(A) be that factor of ~ ( 1 that ~ ) embraces the n small eigenvalues with highest coefficient 1. By Nash [33] the coefficients of P ( A ) are real, analytic functions of the x,’s, and, by (IVa), all but one in absolute value < E .

,

UNSOLVABLE PROBLEMS IN TOPOLOGY

59

Further, let P be the q x q-matrix ,8(L), let Qi be the 1 x q-matrix Pu, and let Qi, ..., Qiq be the components of Qi. Finally, let i;,, ..., i6-n be the integers in { 1, ..., q } - i,. Then, for each k = 1, ..., s, we require the further properties: (IVc) The absolute value of the Jacobian determinant

(IVd) The system of 2q-n equations15

1

Qii,,k =...=

@i,q-nL

=0

x" - x = y1 grad @ i , , k + .--+ q4-, grad C P ~ , ~ - , , ~ , together with the inequality Ix* - XI p @ ’ ) + r ( i i ) . By Lemma 5, p = f i * t can be transformed by a finite sequence of operations Opf ..., Op:’ into the presentation f i ’ * t n . By 4” (*>

F, (p) M iFn(p’ * t”).

Hence it follows from 3” that and hence,

(t) But

and hence

t - t“ = p ( p ’ ) - r(p‘) - p ( f i ) = P(P’) - t ’ -

+ r(fi)

W )- P ( P ) + t +

t’ - t” = p ( p ’ ) - r(p’) - p ( p )

This completes the proof of (3.C). If /?2(F,,(p’))=P,(Fn&)), then t “ = t ’ b y Fn ( P )

+ r(p).

(t),hence f i ’ * f ‘ ’ = p ’ so that

iFn ( P ’ )

by (*). This completes the proof of (3.D). Proof of Lemma 6. First we shall define a special class of algebraic atlas presentations which correspond in a certain way to group presentations; moreover, there is an algorithm to decide for given algebraic atlas presentation %Jl, (i) whether %Jl belong to the special class, and (ii) if it does, to find corresponds. These all group presentations (up to congruence) to which special algebraic atlas presentations will be designed to meet the conditions 2”, 3”, 4”of Lemma 6. Then we shall define the function F,&) having special presentations as values. Our program is to define an algebraic atlas presentation %Jl as “special” if the corresponding n-dimensional complex A (fm) is “obviously” (i.e., in a certain algorithmically recognizable way) the boundary of a star-neighborhood of a 2-dimensional complex A , in E,,, = E,(fm), where A , “corresponds” to

UNSOLVABLE PROBLEMS IN TOPOLOGY

65

a group presentation p (i.e., O(p) is isomorphic to the fundamental group zl(A,) of A,, and JA,J can be decomposed into one point p,, r ( p ) open arcs Y’with all boundary points identical to p o , in 1-1 correspondence with the generators of p , a ndp(p) open disks A j with boundaries in UiLpi Y ‘ u p , in accordance with the relators of p ) . 6.1. DEFINITION. An at most 2-dimensional rectilinear simplicia1 complex A , in E n + l , n>4, with rational vertices is said to correspond to a group ..., M,}) if there exist semi-linear maps presentation p = ( { y l , ..., y,}, ‘ p i : Y’i+]A21 (i= 1, ..., r ) and $ j : A‘j-+lA21 ( j = 1, ..., p) such that: (a) The Y’l’s are oriented arcs (1-balls), and the A‘j’s are disks (2-balls) with oriented boundaries. (b) The restrictions q’lInt Y” and $jlInt A’j ( i = l , . . . , r ; j = l , ...,p) are homeomorphisms with pairwise disjoint images, say Y‘ and Aj. (c) There is a vertex p O e A , such that p,=q~’(dY’~)for all i = l , ..., r, Y’u A j = Id,/, and the closures of Y’ and A j are polyhedra p, u of subcomplexes of A,. (d) In each dA” there is a “base point” p ’ j such that: if eJ is the word y::yji;. . .yz;(gl = l), and if a point p’ runs through dA‘j, starting and finishing at p‘j, in the direction of the given orientation of aA”, then the image point $ j ( $ ) runs through the closed path p o Yk’po YkZp,...po Ykmp,so that it runs through Ykr in the sense of, or in the opposite sense of the orientation of Ykl(as carried over by qk’from Y’k’)according as g,= + 1 or - 1 .

u;= us=

6.2. LEMMA. There is an algorithm to decide f o r an arbitrary givenl6 complex A , (i) whether or not it correspond to any group presentation in the sense of 6.1, and (ii) i f A does correspond to some group presentation, to determine all congruence cIasses of group presentations to which it corresponds. This is nearly trivial since one can determine whether A is 2-dimensional, and then examine all sets of subcomplexes of A so as to determine whether they can be regarded as sets { p o , closure of Y’, closure of Ajli= 1, ..., r ; j = 1, ..., p > with the demanded properties. Here we need the fact that the semilinear homeomorphism problem with the arc or the disk has a simple recursive solution. Whenever such a set of subcomplexes is found, the corresponding group presentations p can be read from it.

Remarks. (1) If A corresponds to p then A also corresponds to all group presentations p‘ which can be derived from p by operations Op,, Op,, and Here “given” means that the rational coordinates of the vertices are explicitely given, together with a corresponding abstract complex.

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w. HAKEN and v. POBNARU

by replacing a particular generator y i everywhere in the relators by y [ I . This holds since condition (d) of 6.1 can be fulfilled with respect to p' by changing base points p'j and/or orientations of dA'j's, and/or Y'"s. ( 2 ) If A corresponds to ci then the fundamental group n,(A) of A is isomorphic to 00).This fo0;:ows immediately from the standard procedure for determining presentations of the fundamental group of a given complex; see for instance Seifert and Threlfall [45], 5 46. If p is a group presentation and E n + 1is a euclidean (n+ 1)6.3. LEMMA. space with n 3 4 then there is a rectilinear simplicia1 complex A , in E n t lthat corresponds to p in the sense of 6.1. This is a special case of a general imbedding theorem for simplicial complexes, see for instance Seifert and Threlfall [45], 8 11. We remark that A 2 could even be constructed in E4 (but in general, not in E 3 ) although we shall not use this fact.

6.4. DEFINITIONS. Let A , be an at most 2-dimensional, rectilinear complex in En+,, 7234, with rational vertices; let Pi,Kj,Th (i= 1, ..., u o ; j = 1, ..., u,; k = 1, . .., u2) be the vertices, edges, triangles, respectively, of A , . Then, by the spherical handle neighborhood of A , in with radii po, p l , p, ( p 2 < p 1 < p o ) we mean the union N of (n+ 1)-balls Ph+,, K i , , , T,k+ with the following properties (for all i,.j, k ) : (P) Pb+, is the (n+ ])-ball in En+, with radius y o and center Pi. (K) Ki+ is the (n + 1)-cylinder in En+ Tnt PL+ with radius p1 and axis K j (i.e., the set of all points in E n + ,Int Pk+, whose distance:to Kj is < p l ) ; p1 is to be so small that the Ki+,'s are pairwise disjoint. (T) Ti+ is the (n+ 1)-cylinder in E n + -1I n t ( U z , P:+,u K i + , ) with radius p, and axis T k ;p, is to be so small that the T,k+,'s are pairwise disjoint. Further, by a normed, rectilinear handle neighborhood of A , in En+,we mean a rectilinear (n + 1)-dimensional complex N* with rational vertices that contains sub-complexes

,

,

uz

Uz ,

,

urLl

and with the following properties (for all i, j , k ) : (6*) There are positive rational numbers po, pl, p,, and 6*, 26* O and for each point p of %? the 6#-neighborhood (2 of p in the normal line (to through p ) intersects dN in precisely one point, say q(p), so that the map p - + q ( p ) is a homeomorphism of %? onto dN. This manifold V can be derived from aN by “smoothing out the corners” at which the boundaries of the cylindrical (or spherical) regions dPL+ ndN, aKi+ n dN, dT;+ ndN fit together; see for instance Cairns [12]. By Whitney [52] there exists an analytic n-manifold B in En+ that approximates %?. Now the construction of the algebraic atlas presentation 1)32 follows the proof of the first sentence of Theorem 4 with q = n + 1 where we take the complex dN* to be the complex A (2R). This is permissible, provided that N* has been chosen fine enough and that all approximations have been chosen close enough. This insures that the @neighborhoods of the points of ii? in the normal lines Q to will be close to the normal-intervals Q to %?, and consequently will provide 1-1 correspondences between the points of L% and the points of a N and also the points of aN*. Thus aN* will lie in the neighborhood Jlr* of %? as considered in the proof of Theorem 4. This completes the proof.

,

6.7. Constructing F n ( p ) . Using 6.2, 6.3, 6.5, 6.6, we shall show that there exists a recursive function that associates with each group presentation p an algebraic atlas presentation llJl of a closed differentiable n-manifold and such that: (i) Eq(1)32)=En+,,(ii) A(m)is the boundary complex of a normed, rectilinear handle neighborhood, say N * , of a rectilinear complex, say A , , in and (iii) A , corresponds to p (as defined in 6.1). It is possible, by reworking the arguments of these earlier sections, to construct such a function which is “natural”, i.e., such that it is relatively easy to compute the value of the function for given p. However, the description of such a function would be rather long and tedious. So, as is sufficient for our purposes, we construct an Fn which is less convenient to compute, but very easy to define, as follows: For given n, one can recursively enumerate those algebraic atlas presenta-

UNSOLVABLE PROBLEMS IN TOPOLOGY

69

tions of closed, differentiable n-manifolds having xl, x,, ..., x,+ as coordinate variables (q=n+ 1). Let Q be such a recursive enumeration. We define F,@) to be the atlas presentation Q ( m ) with smallest m such that (i), (ii) and (iii) as stated above are fulfilled. Clearly, for given p , F,@) exists in view of 6.3 and 6.6. And F, is recursive since, by 6.2 and 6.5, we can, step by step, check Q(1), Q(2), ... until the desired Q ( m ) is found7. Of course it still remains to show that this F,, satisfies the conditions 1" through 4" of Lemma 6. 6.8. Proof of 1". The algorithm demanded follows immediately from 6.5 and 6.2. 6.9. Proof of 2". By 6.7, condition (ii), d (F,,(p)) is the boundary complex of a rectilinear handle neighborhood N* of a 2-complex d, corresponding Since . n 3 4 , we have z l ( A ( F n @ ) ) ) ~ (A,), n l as may t o p ; hence n , (A,)? 00-1) be seen from the following argument l8: Every closed curve C on A , can be homotopically deformed within N * into a curve in dN*=A(F,(p)) (a small deformation takes C into a curve in N*, disjoint from A,; then this curve may be moved into dN*), and vice versa. Similarly, if C bounds a singular disk (i.e., a continuous image of a disk), say D, in A , then the deformed curve in dN* bounds a singular disk (obtained from D by deformation) in dN*, and vice versa. From this we may conclude 2". 6.10. LEMMA.The normed, rectilinear handle neighborhood N* of A,, where A , corresponds to p and A(F,,@))=dN* may be obtained from an (n+ 1)-ball by (combinatorially) adding r ( p ) handles18 of degree 1 and then adding p @ ) handles of degree 2. Moreover, the differentiable structure of M(F,(p)) can be extended to a differentiable structure of N*. Proof In detail (now we use the notation of 6.1 and 6.4), N * is obtained as follows: We begin with the (n+ 1)-ball P,*,", containing the vertex p o of A , ; then we add handles of degree 1, say Yi+ (i= 1, ..., r ) where Yi+ is the union of all those P,*+l's,and K;+,'S, different from P,*,",, and containing points of Y'. This handle-adding yields a "handlebody" V,+ =P,*," u U;= Y:+1 . Next, we add to V,+, handles of degree 2, say A:+ where A:+ is the union of all those P,*+l's, K;+ l's, and T,*+,'s that are not contained in V,+, but contain points of A j . This shows the first sentence of the lemma. The differentiable structure of the manifold B, as considered in the proof

,

,

18 For simplicity of notation, from now on we do not always distinguish among a complex A , its point set ] A \ , and Id1 with combinatorial structure A . Similarly for N , aN, etc. What is meant will always be clear from the context.

70

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w. HAKEN and v. P O ~ N A R U

of 6.6, which is diffeomorphic to M(F,(p)) and differentiably imbedded in E n + , ,is represented by an atlas obtained by projecting certain simplex stars on .G? into n-dimensional coordinate planes of Let fl be the (n+ 1)manifold bounded by 35' in E n i l and which consequently approximates N*. The projections of the simplex stars on 35' may be extended to differentiable homeomorphisms of (n +- 1)-balls on &' into coordinate-halfspaces of En+,. This yields an atlas of a neighborhood of 93 in fl.This atlas can be completed to an atlas of flby using the identity maps on (n + 1)-balls in the interior of fl. There is a homeomorphism cp of # onto N* which is the identity outside of a neighborhood of B, such that ql35'carries the differentiable structure of 35' into that of M(F,@)). Consequently, cp carries the differentiable structure of fl into a differentiable structure of N* which extends that of M(F,(p)). This proves 6.10. 6.11. LEMMA.If A(F,(p))=aN* and A(F,(p'))=dN*' where N* and N*' are normed, rectilinear handle neighborhoods of 2-complexes A; and A,, respectively, and if N* and N*' are combinatorially equivalent then M(Fn(P))XiM(Fn(P'))

( i = 1,23334)-

Proof. This is trivial for i = 2 , 3 and 4. For i= 1 (diffeomorphism) it follows immediately from Munkres [31], his 6.5 Theorem, which asserts in particular that combinatorial equivalence implies diffeomorphy if the homology groups H,,, are trivial for m>3. But N* and N*' have trivial H, for r n > 3 since they are neighborhoods of 2-dimensional complexes (and thus homotopy equivalent to 2-dimensional complexes). 6.12. Proof of 3". Let A(F,(p))=aN* as above, then A(F,(,u*l)) is the boundary of a neighborhood, say N*', of a 2-complex, say A;, that is homeomorphic to the union of A, and an open disk Api' with boundaryp, and AP+' n 1A21 =8 (see 6.1). We choose this open disk Ap+' in En+' in such a way that it intersects En+,-IntN* in a disk, say A . Let A , + 1 be a (polyhedral) neighborhood of A in -IntN*. Then N* u A , + , is combinatorially equivalent to N*'. In fact, N* u A , + is obtained from N * by adding the handle A n f l of degree 2 to N*. Thus d ( N * u A , + , ) = ( d N * - I n t ( ~ N * n ~ A , + , ) ) u a ( A , + ,- a N * n & 4 , + , ) . (The complex d ( N * u A , + , ) is obtained by Morse surgery from aN*.) In the above equation aN*ndA,+, is homeomorphic to S 1 x D n - , and Provided that isomorphic semilinear subdivisions of the combinatorial structures are compatible with the differentiable structures.

l9

71

UNSOLVABLE PROBLEMS IN TOPOLOGY

aA,+,-Int(dN*nc?A,,+,) to D , x Sn-,; moreover, aN*naA,+, lies in an n-ball in aN*. Now p2 (alv* - Int (alv* n aA,+ 1)) = (&(aN*) 1 if n = 4 (&(aN*) if n > 4 .

+

Further, adding aA,+,-aN*naA,+, increases j?, by 1 for both n = 4 and n > 4. With some effort, all this can be verified by the methods for computing homology groups as described in Seifert and Threlfall [45], Q 22. This yields 3”. 6.13. Proof of 4“. Let A(F,(p))=aN* and A(F,@’))=aN*’ where N* and N*’ are handle neighborhoods of A , , corresponding to p, and A ; , corresponding to p’, respectively. In view of 6.1 1 it is sufficient to show that N * and N*’ are combinatorially equivalent. Case 1) p’ is obtained from p by Op, or Op,. Then A , and A ; are combinatorially equivalent (see 6.2, remark (l)), and thus so are their neighborhoods N* and N*’. Case 2) p 1 is obtained from ,u by Op:‘ or Op,, say by replacing a i by a:. Let p” be the group presentation obtained from p by deleting ai.Let A‘; correspond to p” and let N*” be a handle neighborhood of A ; in Then a complex, homeomorphic to A , (to A ; ) is obtained by adding to A’; an open disk A’ (an open disk Ali) that corresponds to ai (to a:). We choose A’ and A” in En+, so that A’n(E,+,-IntN*”) and A’in(E,+l-IntN*”) are disks, say A and A’ with aAnaA’=0. Let A,+1 and A;+, be (polyhedral) neighborhoods of A and A‘, respectively, in -IntN*“. Then N * ” u A,+1 and N*” u A;+ are combinatorially equivalent to N* and N*’, respectively. We remark that adding A,+, or to N*” means adding a handle of degree 2 to N*“ corresponding to aior a:, respectively. Now we prove that N * ” u A,,, and N * ” u A A + ~are combinatorially equivalent to each other. For this we use the method of “sliding handles” (see Smale [47], Potnaru [36]) which can be described in our case as follows: The curves aAi and aA“ are homotopic in A ; ; hence, dA and aA’ are homotopic in N*“, i.e., there exists a singular 2-dimensional annulus (a continuous image of an annulus) in N*” with boundary curves dA and dA’. Since n 2 4 , this singular annulus can be deformed into a non-singular annulus, say J c dN*“ with aJ= aA u aA’. Consequently, dA can be “moved over J” into aA’. Hence, there exists a semilinear homeomorphism of N*“ onto itself that is the identity outside of some neighborhood of J , and that maps the neighborhood aA,+l naN*“ of the curve aA (in dN*”) onto the neighborhood aAA+ naN*” of aA’ (in dN*”). This homeomorphism can be extended to a semilinear homeomorphism of N*” u A , + onto N*” u A: +

,

72

w. w. BOONE, w. HAKEN and v. P O ~ N A R U

are combinatorially equivalent. This Hence N*" u A,+1 and N*" u completes Case 2. Case 3) p' is obtained from p by Op,. Again let A(F,&))=aN*, A(F,(p')) =dN*', etc. Then a complex combinatorially equivalent to Ah is obtained from A, by adding an open arc Y"' (corresponding to the new generator) and an open disk A p f l (corresponding to the new relator), where d A p f ' contains Y r + l precisely once. We choose Y'+l and AP+' in so that Y'+'n(E,,+,-IntN*)isanarc,say Y, and ( A P + l u Y r + l ) n ( E , + , - I n t N * ) i s a disk, say A . Let A,+1 be a (polyhedral) neighborhood of A in - Int N*. Then N * U A , + ~is combinatorially equivalent to N*'. But, on the other hand, A,+1 is an (n+ 1)-ball such that dA,+, naN* is an n-ball (viz., a neighborhood of the arc i3A naN* = aA-Int Y ) . Hence N" u A , + 1 is combinatorially equivalent to N * . This settles Case 3. Case 4) p' is obtained from p by Op; I . Interchange p and p' in Case 3 for this case. This completes the proof of 4". This finishes the proof of Lemma 6 - and hence of all previously stated results. 3.4. An open question: A topological analogue of the Markov-AddisonFeeney-Adjan-Rabin Theorem In Boone [8] in this volume, the notion of a Markov property of semigroups or groups is explained. We should like to raise here the question as to whether the work of Markov [24,25], Addison [l], Feeney [18], Adjan [2] and Rabin [40] can be paralleled in topology. What one would have to do is frame a definition of "Markov property of manifolds" in such a way that "most of" the properties of manifolds which are of actual interest to topologists would be Markov under the definition. Then one would have to show that for a given Markov property, one cannot recursively recognize whether or not a given presentation presents a manifold enjoying the given property. We do not here propose a definition of "Markov property of manifolds". Indeed, finding a useful definition - a definition which does not, in a trivial way, refer matters back to group theory - seems difficult. References 1 . ADDISON,J., On some points of the theory of recursive functions, Dissertation, University of Wisconsin, 1954. 2. ADIAN, S. I., The algortihmic unsolvability of checking certain properties of groups, Dokl. Akad. Nauk SSSR 103 (1955) 533-535 (in Russian). 3. BAUMSLAG, G., W. W. BOONEand B. H. NEUMANN, Some unsolvable problems about elements and subgroups of groups, Math. Scand. 7 (1959) 191-201.

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4. BING,R. H., An alternative proof that 3-manifolds can be triangulated, Ann. Math. 69 (1959) 37-65. 5. BOONE, W. W., The word problem, Ann. Math. 70 (1959) 207-265. 6. BOONE,W. W., Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems, Ann. Math. 83 (1966) 520-571. 7. BOONE,W. W., Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups, Ann. Math. 84 (1966) 49-84. 8. BOONE, W. W., Decision problems about algebraic and logical systems as a whole and recursively enumerable degrees of unsolvability, this volume. 9. BOONE, W. W. and H. ROGERS JR., On a problem of J. H. C. Whitehead and a problem of Alonzo Church, Math. Scand. 19 (1966) 185-192. 10. BRITTON,J. L., The word problem, Ann. Math. 77 (1963) 16-32. 11. CAIRNS,S. S., Triangulation of the manifold of class one, Bull. Am. Math. SOC.41 (1935) 549-552. 12. CAIRNS,S. S., The manifold smoothing problem, Bull. Am. Math. SOC.67 (1961) 237-238. 13. CAIRNS, S. S., A simple triangulation method for smooth manifolds, Bull. Am. Math. SOC.67 (1961) 389-390. S. S., Introductory topology (New York, Ronald, 1962). 14. CAIRNS, 15. CLAPHAM, C. R. J., Finitely presented groups with word problem of arbitrary degrees of insolvability, Proc. London Math. SOC.(3) 14 (1964) 633-676. 16. COHEN,P. J., Decision procedures for real and p-adic fields (Mimeographed. Stanford University, Stanford, California, 1967). 17. DAVIS,M., Computability and unsolvability (New York, McGraw-Hill, 1958). W. J., Certain unsolvable problems in the theory of cancellation semi-groups 18. FEENEY, (Catholic University of America Press, 1954). 19. FRIDMAN, A. A., Degrees of unsolvability of the problem of identity in finitely presented groups (in Russian) (Moscow, USSR Academy of Sciences; Central Economics-Mathematics Institute; “Science” Publishing House, 1967). 20. HERMFS,H., Aufzahlbarkeit, Entscheidbarkeit, Berechenbarkeit. Einfiihrung in die Theorie der rekursiven Funktionen (Berlin, Heidelberg, New York, Springer-Verlag, 1961; English translation: Springer-Verlag, 1965). 21. IHRIG,A. H., The Post-Linial theorems for arbitrary recursively enumerable degrees of unsolvability, Notre Dame Journal of Formal Logic 6 (1965) 54-72. 22. KERVAIRE, M. A,, A manifold which does not admit any differentiable structure, Commentarii Mathematici Helvetici 34 (1960) 257-270. S. C., Introduction to metamathematics (Amsterdam, North-Holland Publ. 23. KLEENE, Co., 1952; fourth reprint 1964). 24. MARKOV, A. A., Impossibility of algorithms for recognizing some properties of associative systems (in Russian), Dokl. Akad. Nauk SSSR 77 (1951) 953-956. (This paper can be understood completely from a review in J. Symb. Logic 17 (1952) P. 151 by A. Mostowski.) 25. MARKOV, A. A., Theory of algorithms; 444 pages, published for the U. S. National Science Foundation by the Israel Program for Scientific Translation, 1961. Available from the Office of Technical Services, U. S. Department of Commerce. A. A., Insolubility of the problem of homeomorphy, Proc. Intern. Congress 26. MARKOV, of Mathematicians, 1958 (Cambridge University Press) 300-306.

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w. w. BOONE, w. HAKEN and v.

PO~NARU

27. MILNOR,J., On manifolds homeomorphic to the 7-sphere, Ann. Math. 64 (1956) 399-405. 28. MILNOR, J., A procedure for killing the homotopy groups of differentiable manifolds, Symposia in Pure Mathematics, Am. Math. SOC.,Vol. I11 (1961) 39-55. 29. MILNOR,J., Two complexes which are homeomorphic but combinatorially distinct, Ann. Math. 74 (1961) 575-590. 30. MOISE,E. E., Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. Math. 56 (1952) 96-114. 31. MUNKRES, J., Obstructions to the smoothing of piecewise-differentiable homeomorphisms, Ann. Math. 72 (1960) 521-554. J., Elementary differential topology, Ann. Math. Studies No. 54 (Prince32. MUNKRES, ton University Press, 1966). 33. NASH,J., Real algebraic manifolds, Ann. Math. 56 (1952) 405421. 34. NEUMANN, B. H., An essay on free products of groups with amalgamations, Phil. Trans. Roy. SOC.London, Ser. A 246, No. 919 (1954) 503-554. C. D., Some problems on 3-dimensional manifolds, Bull. Am. 35. PAPAKYRIAKOPOULOS, Mat?. SOC.64 (1958) 317-335. 36. POENARU, V., Sur la theorie des immersions, Topology 1 (1966) 81-100. 37. POST,E. L., Recursively enumerable sets of positive integers and their decision problems, Bull. Am. Math. SOC.50 (1944) 284-316. ofaproblemofThue, J. Symb. Logic 11 (1947)l-11. 38. P~~~,E.L.,Recursiveunsolvability 39. POST,E. L. and S. LINIAL,Abstract, Bull. Am. Math. SOC.55 (1949) p. 50. 40. RABIN,M. O., Recursive unsolvability of group theoretic problems, Ann. Math. 67 (1958) 172-194. 41. REIDEMEISTER, K., Topologie der Polyeder und kombinatorische Topologie der Komplexe (Leipzig, Akademischer Verlag, 1953). 42. ROGERS, H., JR.,Theory of recursive functions and effective computability (New York, McGraw-Hill, 1967). 43. ROTMAN, J. J., The theory of groups. An introduction (Boston, Allyn and Bacon, Inc., 1965). 44. SACKS,G. E., Degrees of unsolvability, Ann. Math. Studies No. 55 (Princeton University Press, 1963). H. and W. THRELFALL, Lehrbuch der Topologie (Leipzig, Teubner, 1934). 45. SEIFERT, 46. SINGLETARY, W. E., Recursive unsolvability of a complex of problems proposed by Post, J. Faculty of Science, Univ. Tokyo, Sec. I, 14 (1967) 25-58. 47. SMALE,S., Generalized Poincark conjecture in dimensions greater than four, Ann. Math. 74 (1961) 391406. E. H., Algebraic topology (New York, McGraw-Hill, 1966). 48. SPANIER, 49. SULLIVAN, in preparation. 50. TARSKI, A., A decision method for elementary algebra and geometry (Santa Monica, Rand, 1948; Paris, Institut Blaise Pascal, 1967). 51. WHITEHEAD, J. H. C., On Ckomplexes, Ann. Math. 41 (1940) 809-824. 52. WHITNEY, H., Differentiable manifolds, Ann. Math. 37 (1936) 645-680. 53. YASUHARA, A. H., A remark on Post normal systems, J. Assoc. Computing Machinery 14 (1967) 167-171. 54. YNTEMA, M. K., A detailed argument for the Post-Linial theorems, Notre Dame Journal of Formal Logic 5 (1964) 37-50.

CONSTRUCTIVE THERMODYNAMICS W. K. BURTON Department of Natural Philosophy, The University, Glasgow 1. The purpose of this note is to discuss the feasibility of formulating a fundamental part of physics in a constructive manner. As a starting point we take the formulation of thermodynamics given by Robin Giles [2]. In this book, Giles effects a complete separation between the physical and the mathematical aspects of the theory, and presents the latter as an informal axiomatic theory measuring up fully to the standards of rigour customary in contemporary mathematics. Its reformulation as a formal theory would present no particular difficulty, but there are reasons for believing it to be worth while to attempt this in a constructive sense, making slight modifications in the original theory if necessary. These reasons stem from the physical aspects of the theory. In addition to the various mechanisms for producing theorems (derived formulae) it is necessary, in a physical theory, to lay down certain rules of interpretation which connect at least some of the formulae with practical actions. In the past this kind of problem has not received much attention, and the further great merit of Giles’s approach is that for the first time questions of this sort are submitted to a precise analysis. The axioms of the theory contain just four primitive concepts which are called ‘state’, ‘union’ of states, the relation of a state ‘going to’ a state, and the relation of a state being ‘equal’ to a state. Giles’s theory being informal, there will of course be further primitive concepts, for example logical ones, which will have to be taken into account in a complete formalisation. As a matter of fact Giles himself appears not quite to count equality between states as one of his primitive concepts, perhaps feeling that it belongs to a different level from the others. Denoting states by small Roman letters with or without subscripts, we have primitive formulae of the form a = b, a + b = c , a + b (read as ‘state a equals state by,‘state a plus (union) state b equals state c’, and ‘state a goes to state b’, respectively). Formulae then result by combining primitive formulae by means of the logical particles. Giles’s idea is now the following: if rules 15

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are laid down which attribute unique meanings to the primitive formulae, all the formulae will acquire unique meanings. These rules, which he calls primitive rules of interpretation, permit other derived concepts to be introduced by means of explicit definitions, and these derived concepts are thereby ‘explained’ in terms of the primitive ones. No other concepts besides primitive and derived ones appear. The axioms of the theory, being formulae, also acquire an interpretation, and the question arises as to whether the axioms are true under this interpretation. If they are, then the theorems will also be true, providing the rules of inference lead from true formulae to true formulae. Giles selects the aspects of experience which are linked to the primitive concepts in the mathematical theory by the primitive rules of interpretation to be as ‘direct’ as possible. An experience is direct to the extent that it can be demonstrated rather than explained in terms of other (more direct) ones. The implied ordering of experience according to directness is admittedly rather crude: it corresponds roughly to an order of concept formation in a child as it matures. On the theoretical level the direct experiences are supposed to correspond in some way with primitive concepts in a theory, and the less direct ones to derived concepts. The theory then, as it were, ‘explains’ the indirect aspects of experience in terms of the direct ones. 2. Before presenting the axioms of Giles’s theory, we wish to summarise Giles’s own discussion of his rules of interpretation. We do this not only to give the theory some intuitive content, but also because we wish to consider later on some modifications in these rules. The main purpose of a physical theory is to make predictions. The basis on which these predictions are made consists of prior knowledge about the ‘system’ which is under investigation. This knowledge, in its turn, consists of information about what has happened to the system in the past: in other words of how the system has been prepared, Thus we consider that the basis on which predictions are made is the method of preparation (of a system), and it is this which we wish to call the state (of a system). We use capital Roman letters A, B, ..., to denote systems. Then a state a of a system A may be designated by adding a subscript to A: thus A,, A,, ..., are states of the system A. In the mathematical theory, systems are not alluded to at all, the method of preparation being taken as including a specification of how the system is selected or produced. Thus ‘system’will appear only, if at all, as an arbitrary collection of states. If we have two systems A and B then we can conceive of them jointly as

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forming a compound system, denoted by ‘A + B’, consisting of the conceptual union of systems A and B. In this union A and B are both considered as isolated. In fact a system can only be prepared in isolation, for if the method of preparation produced the system together with some ‘environment’, the position of the boundary between system and environment would have to be explained, and then the state would no longer be determined by the method of preparation alone. Accordingly, the term ‘state’ can only refer to conditions in which the system concerned is isolated. It is clear that + is associative and commutative. Given any system A it is possible in principle to construct a finite number of replicas of A. Thus ‘A + A’ has a meaning: it is the union of A with a replica of A. We denote it by ‘2A’. Similarly if m is a positive integer, ‘mA’ denotes the union of m replicas of A. Just as we can add systems, SO we can form in a natural way the union A, B, of any two states A, and B, of systems A and B. We define A, B, to be the state of the system A + B in which A and B are isolated and in the states A, and B, respectively. The addition of states is also associative and commutative, and as in the case of systems we can add replicas of the same state: we denote the union of m replicas of A, by ‘mA,’. Although A, +B, is always a state of the system A+B, not every state of A + B is of this form; only those in which the parts A and B are isolated. Thus the rule of interpretation for a + b is to be: $ a and b are states, then a b is that state whose method of preparation consists in the simultaneous and independent performance of the methods of preparation corresponding to the states a and b. The operation of addition of states may be regarded as defining a relation a + b = c between three states a, b and c. We now consider another relation between states connected with the natural evolution of a state with time. If, during some time interval, the state of a system A changes, a natural process is said to have occurred. In general, A will interact with other systems during such a process. Suppose A is part of a larger system I which remains isolated throughout the process. Thus I contains, together with A, every system with which A interacts during the process. Although these systems do not remain isolated during the process, it is possible that, for some of them, the initial and final states may coincide. If so, we say that they are not involved in the process. A system is involved in a process if and only if its initial and final states differ. If there exists a natural process involving only a system A which has initial and final states A, and A, respectively, then we write ‘A,-’A2’ (read “A1 goes to Az’’).

+

+

+

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Thus the rule of interpretation for a + b is to be: a-tb ifand only ifthere is a state k and a time interval z such that a + k evolves in isolation in the time z into the state b + k. With these explanations we have arrived at rules of interpretation for ‘state’, ‘+’ and ‘-+’.When are two states to be regarded as equal? Clearly if two states are prepared in the same way they should be regarded as equal. However, even if two states are not equal in this sense, but nevertheless any two experiments applied to these two states yield the same result (or rather the same statistical distribution of results) then these states need not be distinguished. This gives rise to a wider notion of equality, which in fact is the one which Giles uses in his book.

3. It is convenient [2]to characterise thermodynamics by making use of the concept of a primitive observer (for thermodynamics). Such an observer is a being whose direct experience embraces only the physical aspects of experience associated by the primitive rules of interpretation with the primitive concepts “state”, +, -+ and = . That is, he is directly aware of states and relations among them of the forms a = b, a + b = c and a-tb, but of nothing else. Thermodynamics can now be characterised as a physical theory which is meaningful to such an observer, and which could, indeed, have been developed by him. The specification of the concept of primitive observer for a theory amounts to the specification of a range of observational powers sufficient to guarantee that the theory can actually be applied in practice. As we shall see later, meagre though the powers of a primitive observer for thermodynamics may look, they transcend in important respects the powers of human observers. 4. We now present Giles’s axioms for thermodynamics as given in Appendix A of his book [ 2 ] . Consider a non empty set 6whose elements will be called states. We postulate in G an operation and a relation +. satisfying the following axioms. AXIOM1. In 6 (i) if a, be 6then a + b e 6 , a + b = b + a, and if a, b, c e 6 then a+(b+c) = (a b) c ; (ii) a-+a (iii) a+bA b-tc=>a+c a, b, CEG. (iv) a + c + b + c o a + b AXIOM2. If a, b, CEG a+b Aa+c=>b+cv c-tb.

+

+ +

1

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79

DEFINITION 1. A process is an ordered pair of states (a, b). Denote the set of all processes by 13; denote the elements of !@ by small Greek letters, a,P,y,....

Define an operation

a relation

--*

in ’p by

-

+ in ‘p by

+ (c, d) = (a + c, b + d) (a, b) (c, d ) o a + d b + c (a, b)

-+

+

in ‘p by setting (a, b)-(c, d) whenever there is a state and a relation x such that a + d + x = b + c + x . is an equivalence relation with respect to which It is easily shown that + and + are compatible. Henceforth equivalent elements in !@ are identified. In particular all processes of the form (a, a) are equal: denote any such process by 0. If CI is the process (a, b), denote the process (b, a) by - CI.Then 0 + a= CI and a+ (- a) = 0, and ‘p turns out to be an abelian group under + with zero element 0.

-

DEFINITION 2. CI isnaturalif a+O, antinaturalifO-+a,possibleifa-+O v O+a, reversible if a-0 A 0-m. It is irreversible if it is possible but not reversible, and impossible if it is not possible. The set of all natural (antinatural, possible, reversible) processes is denoted by ‘pN(’pA, pp,‘p,). It is easily shown that Ppand ‘p, are subgroups of ‘p.

DEFINITION 3. Given states a and b, if there exists a positive integer n and a state c such that (na + c, nb)E ‘ppwe write a c b (read “a is contained in b”). 4. A state e is an internal state if, given any state x, there DEFINITION exists a positive integer n such that x c n e .

AXIOM3. There exists an internal state. AXIOM4. Given a process a, if there exists a state c such that for any positive real number E there exist positive integers m, n and states x, y such that m/nb+ there is a state c such that for any positive real (D,) number E , there are positive integers m, n and states x, y such that m/nO ist, beginnt q mit einem Funktional. 2.1. Es ist k=O. Dann ist x[q] mit q identisch, also regular. 2.2. Es ist k>O, und q beginnt mit einem Funktional. Dann sind nach (JV1) die Terme xk[q] und qx,[q] ... x,-,[q], also auch der hiermit subtermgleiche Term q x , [ g ] ...x k - [ q ] uk mit einer Variablen uk regular. Der Typ von uk ist aber ein direkter Subtyp von T, so da13 nach (JV2) auch q x , [q] ...xk[q],und das ist x[q], regular ist. Durch Bar-Induktion nach der Lange der Subtermketten beseitigt man zunachst (JVl), danach durch Induktion nach dem Typ (JV2). Damit ist Satz 1 bewiesen.

SATZ2. Jeder Term p ist regular. Beweis durch Induktion nach der Definition von p . 1. Fall. Fur Terme 0 und U' und, falls t bzw. t [ u , , ..., u,] regular ist, auch fur (t)' bzw. das durch Abstraktion definierte Funktional f ist die Behauptung trivial. 2. Fall. Sind die Terme p und q regular, so auch p u und q; also ist nach Satz 1 auch pq regular. 3. Fall. Sei p ein durch Rekursion definiertes Funktional g, und die definierenden Terme t [ u l , . .., un] und 5 [u, a, u l , ..., u,,] seien regular. 3.1. Dann ist auch go und nach Satz 1 s[gO, 0, u l , . .., un] regular. 3.2. 1st s [ g z , z, u l , ..., u,] regular, so sind auch gz' und nach Satz 1 s[gz', z', u,, ..., u,] regular. Mit Induktion nach z folgt aus 3.1 und 3.2, da13 jeder direkte Subterm von g, also auch g selbst regular ist. Damit ist Satz 2 vollstandig bewiesen. SATZ 3. Jeder geschlossene Term t vom Typ o ist berechenbar. Beweis. Da jeder direkte Subterm eines geschlossenen Terms vom Typ o wieder ein geschlossener Term vom Typ o ist, besteht der Subtermbaum von t nur aus geschlossenen Termen vom Typ 0.

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1 . Fall. t hat keinen direkten Subterm. Dann ist t der Term 0, also berechenbar. (Denn die Reflexivitat der Gleichheit folgt mit (A) aus der Komparativitat .) 2. Fall. t hat genau einen direkten Subterms. Dannist t entweders‘, und mit s = z ist auch t =z‘herleitbar, oder t = s ist ein Axiom, und aus s =z folgt t = z. 3. Fall. t hat unendlich viele direkte Subterme. Dann ist t ein Term gsp I .. .pn. Aus s = z folgt t =gzp, .pn, und gzp, ...pn ist nach einem Axiom (R) gleich einem direkten Subterm von t. 1st dieser gleich 2, so ist auch t = P herleitbar. Also ist mit den direkten Subtermen von t auch t berechenbar, und durch Bar-Induktion folgt mit Satz 2, daB t berechenbar ist. Der Beweis von Satz 3 zeigt, daB die Definition der direkten Subterme fur jeden geschlossenen Grundterm ein Berechnungsverfahren liefert, das von auBen nach innen fortschreitet, im Falle eines Terms gtp, ...pn jedoch erst den Rekursionsterm t ausrechnet. Nach Satz 2 bricht dieses Standardverfahren nach endlich vielen Schritten ab.

..

3. Eine Hierarchie arithmetischer Operationen Um die hier verwendete Bar-Induktion durch eine transfinite Induktion zu ersetzen, betrachten wir die folgendermaBen durch eingeschachtelte transfinite Rekursion definierte dreistellige Ordinalzahlfunktion a. (1) acrpo=p’. (2.1) aopy’ = (aopy)’. (2.2) occ’py’ = ocC(0a’py) (aa’py). (2.3) o@y’=supaa(oApy) (aiby), falls 2 eine Limeszahl ist. a% Beweis durch Induktion nach y. 1 . Nach Definition ist y ' " ' ( ~ ) O ~ q ( a ' a ~ A ' ( fur ~ ) Oalle &dim(d). Let W c X be a set such that e,(r)Eu(H(A, X ; W ) ) and card(W)=eA(v').Let M be the set of all elements u ~ u ( Phaving ) this property; since X c M , it suffices to prove that M is closed in P. If r E M P J ,then, by definition of M , a sequence x ~ u ( P ' ) ' j can be found such that e , . y = e A . X . If hr corresponds to j in H ( A , X ) , this shows eA(hj(q))= h r (e,. y) =hr(e, .x)= eA(h,(x)). Therefore, j ez *( Z ) implies h.( ) E U ( P ' ) whence , h j ( y ) € M . If not j e z * ( Z ) , define I e X X xl by I tfl*(yj)=X*p,; t p * ( p j ) and /z t X - P * ( p j ) the identity. Let g and g 1 be the extensions of I in Hom(P, P ) and Hom(P', P'); then g1= g ru(P1)and

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139

Since with ( s j i h j ( P j ) ) a's0 ( g ( s j ) , S(hj(Pj))> holds in A , one obtains eA(hi (q))= eA( h j ( ~ = ) )eA( g (hj(pj))) = eA( g (sj)). Then s j e u ( P ' ) implies g(si)=g' (sj), g ( s i ) E u ( P 1 ) ,whence h , ( q ) ~ M .

gfhj)=u(H($(B),X ) ) . Since Y , is equational, also A = Y,(B)is functionally free for A and u(H(B,X ) ) = u ( H ( A , X ) ) holds; hence u ( H ( A , X))=u(H(AlK, X)). Now Lemma 8 gives the existence of a sequence ( t i l i E Z ) in u(P') such that, for every iEZ, the equation (hi(Pi),t i ) holds in A and, therefore, in every GEA. Hence P*(ni)ESupp(eG(ti)) for every GEA and every ~ E Z ;since e G ( t , ) = e G I K ( t one i ) obtains that C is admissible for ( t J i E Z ) . Let Y , be the equational equivalence from C onto the class K constructed from C and ( t i \ i E l ) . Since the equations ((hi(Pi),t , ) l i € Z } hold in every GEA, Lemma 11 (b) gives G = Y , ( G l K ) for every GEA, i.e. A = K . Therefore 4 = Y, * Y, is a bijection and, moreover, an equational equivalence.

m.

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A rather peculiar criterion for completeness is given by

. ( s k l k e K )is complete LEMMA 13. Let B be admissible for ( s , ( k ~ K )Then with respect to B if there exists a function ( i ( k ) l k E K ) from K onto l a n d if, for every kEK, there exists an automorphism gk of P' such that g k ( f i ( k ) (Pi(k)))=Sk.

Observe first that gk YX is a bijection of X onto X,because P' is absolutely freely generated by X . Hence (gk X)-' induces automorphisms p k , p : , p t of P, P I , P 2 respectively such that p : = p k ru(Pi),p : = p k r u ( P 2 ) and p : =g;l. Since h k ( P k ) ~ u ( P 2it) , follows that the element ~ ~ ( ~ ) = p : ( h ~ ( P ~ ) ) holds. Since (sk, hk(Pk)) holds in lies in u ( P 2 ) . Further every AEA, also (fi(k)(Pi(k)), titk,) holds in every AEA for every k e K . Now let ( k ( i ) l i E l ) be a function from I i n t o K such that i(k(i))=ifor every ieZ (the axiom of choice may have to be used here). Defining t i = t i ( k ( i ) )one , obtains a sequence ( t i l i e l ) of elements of u ( P 2 ) such that ( f i ( P i ) , t i ) holds in every A E A for every i E I . Now the same reasoning as in the proof of Lemma 12 can be applied. Let ( s , l k ~ K )be a sequence of elements of u ( P ' ) ; let B be the class of all algebras of type A', which are admissible for ( s , l k e K ) ; let Y , be the equational equivalence from B onto the class A constructed from B and ( s , l k e K ) . A function g from u ( P ) into u ( P ' ) is called reductivefor ( s , l k ~ K ) if, for every r e u ( P ) , the equation ( r , g ( r ) ) belongs to Q(A). It follows from Lemma 9 that reductive functions always exist. In case B is the class of all non-empty algebras of type A', the algebra D, considered in Lemma 9, simply becomes Y , (P'), and the proof then can be simplified considerably. THEOREM 4. Let 4 be an equational equivalence from B onto C,given by equations { ( s k , hk(Pk))lkeK}for 4 and { ( h i ( P i ) , t i ) l i E I } for 4-l. Let g be a function from u ( P ) into u(P'), reductive for ( t i l i e l ) . Let B be strictly equational and defined by a set A4 of A'-X-P'-equations. Let g * ( M ) be the set of all d2-X-P2-equations (g(u), g(v)) for (u, u ) E M . Then C consists precisely of the non-empty algebras C of type A' such that (i) C is admissible for ( t i l i ~ Z ) , (ii) the equations from g * ( M ) hold in C , (iii) the equations {(g(sk), h k ( P k ) ) l k ~ Khold } in C . Since due t o Lemma 3 also property (i) can be expressed with help of equations, one obtains in this way a set of defining equations for C. For a proof, let A be the class of mixed algebras determined by 4. Since A is also the class constructed from B and (s,lkeK), A is strictly equational and defined by M u { ( s k , hk(&))lkeK}. On the other hand, A is the class

EQUATIONAL MAPS

149

constructed from C and (t,\iEZ), whence (g(sk), sk) for kEKand (g(u), u), ( g ( v ) , u ) for (u, U ) E Mbelong to Q(A). By transitivity then the equations in (ii), (iii) belong to Q(A) and, since C=AlK, to Q(C). Conversely, let C be a non-empty algebra of type A 2 with properties (i), (ii), (iii). Since (i) holds, Y , ( C ) can be defined; since C = Y,(C)IK, it will be sufficient to show that Y , ( C ) E A . As g is reductive for (tJiEZ), in Y , ( C ) the equations (g(sk),sk) for kEK and (g(u), u), (g(u), v) for (u, V ) E Mhold. Since the equations holding in C also hold in Y 2 ( C ) ,one obtains that the equations from M u { ( s k , hk(Pk))IkEM}hold in Y , ( C ) . In concluding this paragraph, a theorem will be formulated for which the type A' shall begiven, while the type A' is to be determined in a particular way: THEOREM 5. Let B be a class of non-empty algebras of type A'. Then an ordinal type A 2 and a class C of non-empty algebras of type A' can be found such that (i) B and C are equationally equivalent; (ii) for every kEK: the ordinal number mk is a cardinal number; (iii) there exists a bijection (k(i)li€Z) of I onto K such that, for every i E I , mk(j) is the smallest cardinal number w ifor which a set Y,EX exists such that card(Yi)=wiand,foreveryBEB, YiESupp(eB(fi(Pi))). Moreover, if the type A' is ordinal, then the injections jk,kEK, of the A-coordinate system may be chosen such that pi( j ) =pi 1mk(j ) for every i e I. For a proof, let K be such that ZnK=O and let there exist a bijection (k(i)liEZ) from Z onto K. Since A' is given, there exist fixed injections p i from n, into X where card(X)=rank(Al). Since p*(ni)€Supp(e,(f.(pi))) for every BEB, sets Yimay be chosen such that Yicj3*(ni) and card(Yi)= mk(,), where mk(i)is determined in (iii). In order to treat the general case, define & j ) as an arbitrary bijection from mk(i) onto Y, and define sk(i)= f i ( p i ) . Then B becomes admissible for (sk(illiEZ), and if A is the class constructed from B and ( $ k ( j ) I i E Z ) then (hi(fli),hk(i)(&(j)))holds in every A E A . Hence (s,,,)JiEZ) is complete with respect to B. Turning to the case that d l is an ordinal type, let the sets Yi again be chosen such that Y,Ej?*(ni) and card( Yi)= mk( i). Since mk( i) d IE j , one can define pk(,)= p i r Y ) ? k ( j ) . If mk(j)= O let g i be the identical automorphism of P'. Assume now that mk(j)>O. Let p i be a bijection from ( p i ' ) * ( Y i ) onto mk(,.);since both these sets are contained in ni, p i may be extended to a bijection hi of n, onto itself. Then the bijection piS,pLr of p* (n,) onto itself can be extended to a bijection y i of X onto itself which maps Yi onto P*(mk(,,). Now let g i be the automorphism of P' induced by yi, and define

150

W.FELSCHER

~ ~ ( ~ ) = g , ( f , (ItP now ~ ) ) .suffices to show that B is admissible for (&(i)IiEI), since Lemma 13 then will ensure that ( s k ( , ) 1 i ~ is I )complete with respect to €3. Let B be in B. Since P*(mk(i))ESupp(eB(sk(i,)) is clear if mk(i)= O or if B is singular, assume that rn,(,,>O and let B be not singular. Then H (B, X ) is E((B})-freely generated by (e;'(xEX}; since H(B, X ) itself belongs to E ( ( B } ) , yi determines an automorphism g f of H ( B , X ) such that g F ( e y ) = e:,& for X E X . Then e,.g,=g"e, since these homomorphisms coincide on X.Now g? ( e B ( f i ( P i ) ) ) = e B ( g i ( f i ( P i ) ) ) = e B ( s , ( i ) > ; hence yiESuPp(eB(fi(Pi))), i.e. e,(f,(P,))Eu(H(B,X ; Yi)), implies that eB(Sk(i)) belongs to the image of u(H(B, X ; Y,)) under g y . Since Y,#O, H ( B , X;Yi) is generated by (e,BXlxEY,}; hence g f maps H ( B , X;Yi) onto H(B, X ; y: ( Yi))= H ( B , P*(mk(i))).Thus e B ( s k ( i ) ) E U ( H ( B , X;P*(rnk(i)))),p*(mk(i))E Supp (eB (sk( i)).

x;

6. Definable maps and syntactical equivalences

A relational type shall be an ordinal type A = ( n, li€I ) such that 0 . Proof. Assume V,uEpart(r) and X E fr(Vwu). Since W c p a r t ( r ) implies Z n W=O, it follows from not X E W that not ~ ( x )W. E Thus r, q are compatible. Now qEe,(V,r) i f ffor every x, x 1- Y = q Y implies xEe,(r); likewisecp Ee,(Vz rep,r)iff for every$, $ 1 -Z= -2implies $eeA(rep,r), i.e. h,.qEe,(r). Then h,-q 1- Y=$t-- Y and Z C - p a r t ( r ) shows that h @ - qr f r ( r ) - Y = q r f r ( r ) - Y. Thus every $ determines a such that x 1- Y=cp t - Y and x r f r ( r ) = h , * q r f r ( r ) : define x [ Y=h,.q 1 Y . Conversely, every x determines a $ such that $ 1- Z = q 1-2 and x r f r ( r ) = h , . ~r f t ( r ) : define $ rZ=x-fj-l / Z . Let q be in U ( T )and ~ let r be a formula. Definej(r, q ) to be the smallest ordinal number a*$I>

which is a p-formula if cp and $ are p-formulas. Every formula is therefore equivalent t o some p-formula. 3. If cp is a formula, cp

xi,...xi,

o...o

is the formula obtained from cp by substituting 0 for the variables x i , , ..., xi,. If no other variables but x i l ,..., xi,,,, x j , ,..., x j noccur in cp and if i p # j q for x i , .. .xi, all p , q (1 Q p < m, 1 < q < n) then cp/xj,.. . x j nis the formula cp

1o...o

.

If x is the sequence (x,,, ..., x i , ) then cp/x is cp/xj,...x j n . Example. If cp is the formula ( x t + x 2 ) . ( x 2+ x 3 ) and x is the sequence ( x l , x 3 ) then cp/x is ( x l +O).(O+x,). The theorems of the paper are based on the following

177

LENGTHS OF FORMULAS

MAINLEMMA.For all integers m, k there exists an integer no such that for a11 n, zz>n0, the following holds: If cp is a formula in the n variables xl,. . ., x,,, none of which occurs more than k times in cp, then there exist m distinct integers k,, .. ., k , (1 >>

= R f : u{s +,ot:

s, t E R ; }

u { ~ , , ( t )t:E R i ) u (an*( t ) : t E Rh,) u {a,(r): r E R f : } , R , = R r , where rn=ph(Rf:+'=Rf:). LEMMA 10. t t M = R,. Proof by Corollary 2.1, Lemmas 7 and 9. Since all operations introduced above are effective, this completes the proof of Theorem 1. 4. Proof of Theorem 2

Essentially, the proof can be found in [ 5 ] . pp. 111, 114, 11.5. We reformulate the necessary lemmas in the present notation and point out if neces-

196

H . LAUCHLI

sary how the proofs have to be modified. The numbering of the lemmas corresponds to the numbering in [ 5 ] . LEMMA 2‘. T, isfinite for each n. Let A , E G t for Z E I Given . a linear order relation A , is defined in a natural way.

< on I, the order-sum

XI

LEMMA 3’. Zft,(A,)=t,(B,), all ~ € 1then , t,(xIA , ) = t , ( x , B,). We prove the following generalization: Let x, y be k-sequences with x i ~ I C A l yi€ICB,l. l, Then (*)

if

tnk(A,,xlA,)

then

= tnk(B,,

ylB,),

=tnk(x

tnk(x

B,?Y ) .

(The lemma is the special case k=O, x = y = A . ) Proof. For n=O, (*) follows from t o k ( C A,, x) = { t O k ( A lxlA,):z~Z}u , {“uiOui)’: there is ~ ~such € that 1 “UjOui”$tOk(A,,,, xlA,,) and “ t @ j ” E t O k x x (A,, xlA,) for all z 2 l o } , which is an easy consequence of the definitions involved and the fact that the sets x i are finite. Induction step. Let t n + l , k ~ x ( A , , x l A , ) = t , + l , k ( B , , ylB,), all ZEZ. Then for all I and for every a , ~ l A , l there is b,EIB,I such that

n

(I)

tn,k+ 1

*

=

tn,k+ 1

(Bi,(ylBi) * b i )

Using the fact that “E” is one of our primitive predicates, an induction shows that in (l), either both or none of a,, b, are empty. Therefore, if a,=alA,, all z, for some a ~ l C A , [ ,and the b,’s are chosen to satisfy (l), then there is aJinite set b such that b,=blB,, all z. Therefore, for every a ~ l x A , I there is b E l x B,I such that for all ZEZ, tn,k+l(A,,(xlA,)*(aIA,))

= tn,k+l

( B ~ 3

(YIBc)*(blB1)).

Hence, by induction hypothesis, for every a ~ l A,I x there is b ~ l B,I x such that t n , k + l ( x A c ) X * a ) = t n , k + l ( C B , , y * b ) . Therefore t n + l , k ( C A t , X ) = t n + l , k X x 1 erweisen sich fast alle solche Relationssysteme als starr, d.h. sie besitzen nur die triviale (einelementige) Automorphismengruppe. Sei Z(n, z) die Zahl der Relationssysteme uber N vom Typ z ohne Identifikation isomorpher Systeme. Z ( n , z) ist gleich der Zahl der Zustandsbeschreibungen (state descriptions) von Carnap (z.B. in [4], ff IBA). Trivialerweise gilt die Formel

n,

1 Smnm Z ( n , z) = 2O=‘ , wenn s = [pl,. . . , , u r n ] .

Fur die durchschnittliche Zahl s(n, z) zueinander isomorpher Relations-

Die Automorphismengruppe 9%von %, eine Untergruppe der symmetrischen Gruppe 6,mit der Ordnung g , zerlegt die 6,in n ! / g Nebenklassen. Alle zueinander isomorphen verschiedenen Relationssysteme entstehen aus einem durch Ausubung je einer Permutation a m genau einer Nebenklasse auf dieses eine Relationssystem. Mithin gilt fur die “mittlere Ordnung” g(n, z) von 9%die Formel g(n, z ) = n ! / s ( n , 7). Da l, 1 zeigt man, indem man nachweist, dalj hier fur n+ co die asymptotische Beziehung g(n, z)-1, also S(n, z ) - Z ( n , T ) / n ! gilt. Es sei noch erwahnt, dalj unsere Uberlegungen aufgefaljt werden konnen als eine quantitativ-finite Variante von Bestrebungen der modernen MetaMathematik, welche sich das Auffinden von Modellen mit grol3er Automorphismengruppe zum Ziel gesetzt haben *.

2. Anwendung des Polyaschen Satzes auf das Anzahlproblem Man kann ein reines Relationssystem % vom Typ z= (a, p) uber N vollstandig beschreiben durch ein Diagramtn in Form einer p-zeiligen und nu-

* Man vergleiche z.B. [ 6 ] .

201

STRUKTURZAHLEN

spaltigen Inzidenzmatrix in den Zahlen 0 und 1 . Die Zeilen entsprechen den o-stelligen Relationen R,,..., R, aus %, die Spalten den o-Tupeln xl, ..., x,, von Zahlen aus N in irgendeiner festen lexikographischen Anordnung. uik= 1 bedeute, dafi die Relation Riauf das k-te o-Tupel zutrifft, entsprechend bedeute 0 das Nicht-Zutreffen. Bei dieser Darstellung erscheint ein reines Relationssystem als eine Folge von sog. Elementarkonjigurationen, in diesem Falle von Spalten. Unter der Sturke einer Spalte sei die Komponentensumme verstanden, under der Starke eines Relationssystems die Summe der Starken aller Spalten. x , ...x,, Rl

3

RP

Das Efementarpolynom ist eine (formal bis Unendlich erstreckbare) Potenzm reihe E(z) = evzy,

1

v=o

deren Koeffizienten e , die Anzahl der Moglichkeiten angeben, eine Elementarkonfiguration (Spalte) der Starke v herzustellen. Da dies offenbar auf (t) Weisen geht, gilt

c (3 m

E(2) =

zv = (1

+ z)".

v=o

Da sich jedes reine Relationssystem aus nu solcher Spalten bestimmt, wiirde sich ohne Rucksicht auf Isomorphie-Identifikationen ein Polynom A n , , ( z )= E(z)"O -- (1

+

c m

=

Z)@

v=o

u,zv

mit

a, =

re">

ergeben, wobei a, die Zahl der verschiedenen Relationssysteme der Starke v angibt. Wir interessieren uns hier jedoch fur die Zahl der Isomorphie-Klassen der Starke v, d.h. fur das Polynom m

wobei S,(n, T) die Zahl der nichtisomorphen Relationssysteme uber N vom Typ z mit der Starke v angibt. Allerdings sol1 hier das asymptotische Verhalten dieser Zahlen selbst nicht untersucht werden *, sondern nur die Ge-

*

Vgl. hierzu irn Falle r = (2, l > in [lo], § 5.

202

W. OBERSCHELP

samtzahlen

c S,(n,

Bna

S ( n , 7 ) = &(1)

=

v=o

z)

sind Gegenstand dieser Untersuchung. Durch ubergang zu einem isomorphen Relationssystem vermoge einer Permutation ~ € 6 erfolgt , ein Austausch gewisser Spalten von '21, also eine Permutation II uber einer Menge von nu Elementen. Diese Permutationen IZ bilden die sog. a-Tupelgruppe, bezeichnet mit Gz, offenbar wie die 6, eine Permutationsgruppe der Ordnung n!. Zur Anwendung der Polyaschen Theorie hat man den sog. Zykelindex der 6:zu betrachten. Unter dem Zykelindex Z ( 9 ) einer Permutationsgruppe 9 der Ordnung g iiber n Elementen versteht man ein formales Polynom in n Variablenf,, ...,f, Z ( 9 ) :=

1

-

9

1

f P ' ...f,"".

R

€9

Dabei ist p i die Zahl der Zykeln von n mit der Lange i. Fur die der Permutation 71 zugeordnete Partition der Zahl n, geschrieben als p (n): = (pi, ..., p n ) gilt also ipi=n. Fur das gesuchte Polynom B,,,(z) liefert nun die Polyasche Theorie im Fall reiner Relationssysteme den

xr=,

SATZ1: Bn,l-----.. 0

2

1 (a - 1) (a - 2) a2

____

2-3

IC

und n unabhangige

1 (a - 1) (a - 2) (c - 3)

... >

2-3.4 1 1 1 1 1 I--+---- ...--~ > o . 2 2 ~ 4 8 20

a3

p) Sei K >n/a. Zunachst ist wegen n/lc 2 1 : K > (1 - (1 - k-/n)")/o.Nach der Formel (1 - ~ / y K > 0 :(1 - Ic/n)"<e - K ,also (1 - ~ / n ) " = ( l --~/n)"~'~n<e-uK'n. Also ist K>(1 -e-uK/n)/a. Da x/n> l/o, so ist e-uK'"<e-l, also 1 -e-uK'n> 1 -e-'. Damit konnen wir die obige Ungleichungskette fortsetzen : K > (1 - e- ')/a > 1/2a, denn es gilt e-' 1/2a allgemein bewiesen. Insgesamt gilt, daI3 der Exponent im Nenner der Abschatzung fur ,ZK,n groDer oder gleich $p"-ist fur alle IC mit 0 < IC +t-cSbQ+

P,NP 2 NQ tc,,Q. 5. Proofs

In the proofs of the theorems above, we will assume acquaintance with Prawitz [lo]. At some places, we will make essential use of the fact that the deductions in the systems involved can be written in a certain normal form as described in [lo]. For convenience, theorems about normal deductions are provided in [lo] for c’,a reduced form of c where v and 3 are omitted. In this context, we need similar results for C. However, if ’ stands for a transformation by which disjunctions and existential formulas are replaced by equivalent formulas with and & or and V respectively, it can be seen that a normal deduction in C’ (or C;,) of A’ from r‘ goes over to a normal deduction in C (or Cs4)of A from r by an obvious transformation. N

-

PROOFOF THEOREM A. As pointed out in the discussion above, the theorem can be proved by showing that each classical inference rule goes over to an intuitionistically valid inference after the -transformation of both premises and conclusion. Instead of considering all inference rules in this way, one may however obtain the theorem directly from the following two lemmata : N N

CLASSICAL, INTUITIONISTIC AND MINIMAL LOGIC

223

LEMMA1. T t , A if and only i f r - " t,A"". LEMMA2. (a) r""t,A"" if and only if r"" t,A"". (b) r-" k,A"" if and only if r"" t,A"". The only part of the lemmata that is not trivial is the implication from left to right in Lemma 2. It suffices to prove this implication in Lemma 2 (b), i.e. to show that applications of the A-rule in deductions of A"" from r"" are superfluous. To this end let 17 be a normal deduction in c of A"" from r" " with a conclusion B of an application of the A-rule. It may be assumed that no conclusion of an application of the A-rule in Il has the form of an implication ([lo] p. 39, Th. 1). We first assume that B is not minor premiss of an application of the v E- or 3E-rule. Then Il has the form shown to the left below. [- BI [- BI __

c

__

A

(A)

c

That Il has this form follows from the following three facts. (1) B must be the minor premiss of an application of the I> E-rule because (i) B cannot be a major premiss of an application of an E- or A-rule, since Il is normal, and (ii) B, which by assumption does not have the form of a negation, cannot be premiss of an application of an I-rule, since the conclusion of such an application cannot be subformula of a --transformed formula as required by the Subformula Principle ([lo] p. 42). (2) By the same argument (i.e. the Subformula Principle), the major premiss of this application of the 3 E-rule must have the form B. (3) This premiss B must be an assumption; it can be a conclusion neither of an I-rule, since Il is normal, nor of an E- or A rule, since it would then stand below a formula that could not be a subformula of the required kind. An application of the A-rule of the kind above is clearly superfluous. It can be removed simply by transforming the deduction as shown to the right above. The situation is similar when B is minor premiss of an application of the v E- or 3E-rule. By the same arguments, it is then seen that the segment to which B belongs is minor premiss of an application of the 2E-rule, whose

-

-

-

224

D. PRAWITZ

and P.-E.

MALMNAS

-

major premiss is an assumption of the form B. This case can then easily be reduced to the first case by moving the application of the 2 E-rule upwards. PROOFOF COROLLARY A I . The corollary is obtained by proving that

t A“”

Ez

A*

holds for both intuitionistic and minimal logic. This fact is proved by induction over the degree of A . The base is trivial. For the induction step, it suffices to show that for both I and M: (a) t-A= -A, (b) F - ( A ” ” & B ” ” ) s A “ “ &B““, (c) t- - ( A v B)= - ( - A & W B ) , (d) I- - ( A “ “ ~ B ” “ ) ~ A ” ” ~ B “ “ , (e) t- - V x A “ ” = V x A “ “ , (f) I- - 3 x A - - v x - A . To prove (b), it is convenient to show I- - - ( A & B ) = -A&--B and then apply (a). Similar remarks holds for (d) and (e).

-- -

N

N

--

-

--

PROOFOF COROLLARY A 2. If T is inconsistent, then r k C A and hence A (or r*I-, A). Since I-, A = A , we have that r““(or r*) is inconsistent also by intuitionistic reasoning and that hence U is inconsistent.

r““tl

PROOFOF COROLLARY A 3. The axioms for classical and intuitionistic natural number theory are the same. Let r be the set of these axioms. Now, if A is an induction axiom, then so is A*, and if A is some other axiom, then A* = A . Hence, each member of r* is valid in intuitionistic natural number theory, and Corollary 2 then applies. PROOFOF THEOREM B. Clearly, t - , A - A ’ . Hence, if r’t-,A’, then Tt-,A. The converse is easily shown, using the fact that A t,A’. PROOFOF COROLLARY B 1. One may show by induction over the degree of the formulas that I-,A’=(A v A). Now, suppose that t , A . By the theorem, t-,,,A’.Hence, t , A x v A . It follows that either I-,A” or I-,,, A ([lo], p. 55, Corollary 6), but the last alternative k,., A is false. PROOFOF THEOREM C . Given a proof in M of A , we replace each occurrence of A with P. Since no inference rule in M involves A in an essential way, we then obtain a proof of A“ in M and hence also in I. For the converse, we note that since A does not occur in A“, it follows from the separation theorem ([lo], p. 54) that F,A” implies I-,A”. Of course the substitution of

225

CLASSICAL, INTWITIONISTIC AND MINIMAL LOGIC

A for P in the proof of A" does not change the validity of the proof, and since P does not occur in A , this substitution transforms A" to A. PROOF OF THEOREM D. The theorem follows from the following two lemmata (and the converse of Lemma 2), which may be of som eindependent interest : LEMMA 1. T t , A if and only ifrNkIS4AN. LEMMA 2. If TNtCS4AN, then TNtlS4AN. Lemma 1 is easily proved by induction over the length of the deductions. In the direction from right to left, it holds also for classical logic. In the direction from left to right, it is a peculiarity for intuitionistic logic (e.g., k C A v -A but not t,,,N(NAvN-NA)). Lemma 2 is the crucial step, and is proved by essential use of the theorem on normal deductions for modal logic. Let ( n , 9 )be a normal deduction with pure parameters in C,, of A N from TN.We assume that (l7,F)contains some application a of the A-rule by which an assumption B is discharged, where B is not an implication ([lo], p. 39, Th. I). The assumption B must be major premiss of an application of the 2 E-rule. Clearly, it can not be premiss of an application of the NI-rule, since that would break the restrictions on that rule. Other possibilities are ruled out because they would involve formulas that can not be subformulas of N-transformed formulas, contradicting the Subformula Principle. Hence, 17 has the form shown to the left below

-

-

c

B

-B

c

a is to be chosen so that there is no other application of the Ac-rule above a that discharges an assumption (applications that d o not discharge assumptions, are also applications of the A,-rule). We will show that a can be re-

moved, and assume for induction that this holds true for every application of this kind having a lower number of formula occurrences above its conclusion.

226

-

D. PRAWITZ and P.-E. MALMNAS

The assumption B that we are considering is to be chosen so that there is no other assumption of this form discharged by a that (i) stands in C or (ii) stands above the major premiss of an application of the v E- or 3E-rule, the minor premiss of which stands in C, below B.

-

Main case: There is no assumption in C that is discharged in .XI at the minor premiss of an application of the v E- or 3E-rule. We will show that in this case there is no assumption in C that is discharged in C, which shows that a is superfluous; we can simplify Ll as shown to the right above (and .F accordingly). (Note that applications of the NI-rule in 17, cannot be disturbed by removing Cl,because formulas in C, that stand below C, depend on -B, and can then not satisfy clause 1 in the restrictions on the NI-rule (POI P. 791.) Indeed, assume that there were an assumption C in C discharged in C,. It would then have to be discharged by an application of the 3 I-rule having a conclusion C I D (the A ,-rule is excluded because of the way a was chosen). But this is impossible. The segment c to which C I D belongs cannot be a major premiss of an E-rule, since the deduction is normal. Nor can c be premiss of an application of the NI-rule, because according to the restrictions on this rule ([lo] p. 79), there should then be an essentially modal formula between B and C x D that depends only on assumptions on which C x D depends (note that B and C 2 D is not essentially modal). But every formula between - B and C x D depends on C, which C I D does not depend on. Finally, c can not be premiss of an application of some other I-rule or minor premiss of an application of an E-rule, because that would again involve formulas that can not be subformulas of N-transformed formulas, contradicting the Subformula Principle; note that c cannot be minor premiss of an application of the 3 E-rule, where the major premiss is an assumption discharged by an application of the Ac-rule, since we have assumed that such assumptions do not have the form of implications.

-

-

Special case: There is some assumption in C that is discharged in Z,at the minor premiss of an application of the v E- or 3E-rule. In this case, Ll has one of the forms shown below and we will choose an application p of the v E or 3E-rule in C, and move it down to 17,.

CLASSICAL, INTUITIONISTIC AND MINIMAL LOGIC

We want to transform

227

n in respective case to ,z

z

~

B

-B

A

B

-B

A

However, this transformation can cause certain disturbances. Among other things, it is necessary (1) that C , v C2 or 3xC does not depend in (n,S ) on some assumption that is discharged by an application of the vE- or 3E-rule at some place in Z5, and (2) that C, or C, or C,X (where a is the proper parameter in question) does not contain some proper parameter of an application of the 3E-rule in Z5. fl will be chosen so that it satisfies (1) and (2). Let p, be an application of the v E - or 3E-rule as provided by the special case we are considering. If p, satisfies (1) and (2), we set /3 = p,. Otherwise, we consider an application p2 of the v E- or 3E-rule that makes (1) or (2) to fail for PI. If p 2 satisfies (1) and (2) we set p = p2 ; otherwise we consider a p3 that makes (1) or (2) to fail for p 2 and so on. By this process, we obtain finally a p, which satisfies (1) and (2), and we then set p=p,. Having chosen p in this way, it can be seen that the deduction obtained

228

D. PRAWITZ

and

P.-E. MALMNAS

by the transformation described above (where F is to be modified accordingly in the obvious way) is a correct deduction (of AN from I").To realize this, one has to check the following facts : (a) C , v C , or 3xC does not depend on any assumption discharged by an application of the 21- or Ac-rule in C, ; (b) C, or C , or C,X (where a is the proper parameter in question) does not contain the proper parameter of an application of the VI-rule in C, ; (c) no application of the NI-rule in Z12 is disturbed by the transformation. The arguments involved to prove these facts are similar to those in the main case, though we have also to utilize clause 2 in the restriction on the NI-rule ([lo] p. 79) and the (full) lemma on parameters ([lo] p. 29). By the transformation, IX is replaced by some other applications of the A ,-rule, but to all those applications, the induction assumption applies, and they can thus be removed.

PROOFOF COROLLARY D 1. We observe that each part B of A" that has the form of a conjunction, disjunction, or existential formula is essentially modal as defined in [lo] (p. 77). Hence, if B is a part of Ao,kCS4B-NB([lo] p. 77, Lemma), which gives the corollary. (One may also observe that the proof of Theorem D goes through without change when the formulas are "-transformed instead of N-transformed.) PROOF OF COROLLARY D 2. One proves easily that kcs4AN=NA+ using l-

cs4N (NA

& NB)

= N ( A& B )

and

kcs4 NVxNA

= NVxA

and then applies the remark in the discussion. References 1. CHURCH,Introduction to mathematical logic (Princeton, 1956). 2. GENTZEN, Uber das Verhaltnis zwischen intuitionistischen und klassischen Arithmetik, Manuscript set in type by Mathematische Annalen but not published (eingegangen am 15.3.1933). 3. GENTZEN, Untersuchungen uber das logische Schliessen, Math. Z . 39 (1934) 176-210. 4. GODEL,Zur intuitionistischen Arithmetik und Zahlentheorie, Ergeb. math. Kolloquiurn, Heft 4 (1932-33) 34-38. 5. GODEL,Eine Interpretation des intuitionistischen Aussagenkalkuls, Ergeb. math. Kolloquium, Heft 4 (1932-33) 3940. 6. HACKING, What is strict implication? J. Symb. Logic 28 (1963) 51-71. 7. KOLMOGOROFF, 0 principk tertium non datur (Sur le principe de tertium non datur), Mat. Sb. (Recueil mathkrnatique de la SociCt6 MathCmatique de Moscou) 32 (1925) 646-667.

CLASSICAL, INTUITIONISTIC AND MINIMAL LOGIC

229

8. S. C. KLEENE, Introduction to metamathematics (Amsterdam, North-Holland Publ. Co., 1952). 9. MCKINSEY and A. TARSKI, Some theorems about the sentential calculi of Lewis and Heyting, J. Symb. Logic 13 (1948) 1-15. 10. D. PRAWITZ, Natural deduction, A proof theoretical study (Stockholm, 1965). 11. A. TARSKI, A. MOSTOWSKI and A. ROBINSON, Undecidable theories (Amsterdam, North-Holland Publ. Co., 1953).

ZUR SEMANTIK DER INTUITIONISTISCHEN AUSSAGENLOGIK K. SCHUTTE Universitat Miinchen Als Grundzeichen zur Bildung von Formeln der intuitionistischen Aussagenlogik verwenden wir Aussagenvariablen, das Symbol A (fur die falsche Aussage) und die Junktoren A , v und +. Wahrheitswerte bezeichnen wir mit w (wahr) und f (falsch). Nach S. A. Kripke" hat man folgenden Modellbegriff fur die intuitionistische Aussagenlogik. Ein Modell ( M , R, W ) ist gegeben durch eine nichtleere Menge M , eine reflexive und transitive binare Relation R auf M und eine Zuordnung W von je einem Wahrheitswert W(v, a) zu jeder Aussagenvariablen v und jedem Element a g M mit der Eigenschaft

, = w. W ( v , a) = W, aRP* W ( V p) In einem derartigen Modell ordnet man jeder Formal F fur jedes Element EM nach der folgenden induktiven Definition einen Wahrheitswert W(F,a) zu: 1. W(u,a) ist fur jede Aussagenvariable v durch das Modell gegeben; 2. W ( h ,a ) = f; 3. W ( AA B, u) = w genau dann, wenn W ( A , a ) = w und W( B , u) = w ist; 4. W ( A v B, a ) =w genau dann, wenn W ( A ,a) = w oder W ( B ,a) = w ist; 5. W(A+B, a)=w genau dann, wenn fur jedes P E M , fur das aRp gilt, W ( A ,p)=f oder W ( B ,p)=w ist. Eine Formel F heiJ3e giiltig im Modell ( M , R, W ) , wenn W(F,a)=w fur alle a E M gilt. Eine Formel heiBe intuitionistisch allgemeingiiltig, wenn sie in jedem Modell ( M , R, W ) gultig ist. Durch Herleitungsinduktion beweist man :

KONSISTENZSATZ. Jede herleitbare Formel der intuitionistischen Aussagenlogik ist intuitionistisch allgemeingultig. * S. A. Kripke: Semantical analysis of intuitionistic Iogik I, in: Formal systems and recursive functions, eds. J. N. Crossley and M. A. E. Dummett (Amsterdam, NorthHolland Publ. Co., 1965) Seite 92-129. 231

232

K. SCHUTTE

Das Ziel dieser Note ist ein einfacher Beweis fur die Umkehrung: VOLLSTANDIGKEITSSATZ. Jede intuitionistisch allgemeingultige Formel der Aussagenlogik ist intuitionistisch herleitbar. Zum Beweis dieses Satzes verwenden wir folgende Bezeichnungen : Kleine griechische Buchstaben bezeichnen endliche (eventuell leere) Mengen aussagenlogischer Formeln. a-+B bezeichne die Formel A,

A

... A A,-+B,

... v B,,,

wenn @ = ( A ,,..., A,,,) und P={B, ,..., B,) ist, El v ... v En, wenn a leer und fi= { E l ,..., B,,} ist, A,A...AA,+A, wenn a={A,, ..., A,} undpleer ist, v

A,

wenn a und

p leer sind.

Hierbei sol1 es nicht auf eine Reihenfolge der Formeln in den Mengen a und B ankommen. Im folgenden sei F eine festgehaltene aussagenlogische Formel. T ( F ) sei die endliche Menge aller Teilformeln von F. Ein geordnetes Paar (s(, p) von Teilmengen a,p der Menge T ( F ) heiBe konsistent, wenn die Formel a+B nicht intuitionistisch herleitbar ist. Offenbar ist dann a n /3 leer. Das Paar (a, p) heil3e F-vollstandig, wenn a u fi = T ( F ) ist. Ein Mengenpaar (a*, p*) heilje eine Erweiterung von (a, p), wenn a ~ a "und B ~ f l *ist. Fur jede Formel C gilt:

1. 1st (cx,p) konsistent, so ist auch ( a u { C ) , p) oder ( a , p u { C ) ) LEMMA konsistent. Beweis. Sind (a, fiu{C}) und ( a u { C } ,p) inkonsistent, so sind die Formeln a-t p u{C> und C-+(a+P) intuitionistisch herleitbar. Dann ist auch a-+p intuitionistisch herleitbar, also (a, p) inkonsistent. Aus Lemma 1 folgt: 2. Jedes konsistente Paar (a,p) von Teilmengen a,p der Menge LEMMA T ( F ) 1aBt sich zu einem F-vollstandigen konsistenten Mengenpaar erweitern. Eine Teilmenge a der Menge T ( F ) heil3e F-ausgezeichnet, wenn das F-vollstandige Mengenpaar (a,T ( F )- a) konsistent ist. U ( F ) sei die Menge aller F-ausgezeichneten Teilmengen von T ( F ) .

LEMMA 3. U ( F ) ist nicht leer. Beweis. Das Mengenpaar (0,0) ist konsistent, da die Formel A nicht intuitionistisch herleitbar ist. Mit Lemma 2 folgt, da13 es ein F-vollstandiges konsistentes Mengenpaar (a,p) gibt. Hiermit hat man ein a~ U ( F ) .

233

INTUITIONISTISCHE AUSSAGENLOGIK

Anmerkung. Es kann sein, daD U ( F ) = (8) ist. Z.B. ist T ( A )= { A } und (8, A) das einzige A-vollstandige konsistente Mengenpaar, also U ( A )= (8).

LEMMA^. Eine Formel C E T ( F ) gehort genau dann zu einer Menge die Formel a+ C intuitionistisch herleitbar ist. Beweis. Fur jede Formel C E ist ~ trivialerweise a+ C intuitionistisch herleitbar. 1st C$LY, so ist C € T ( F ) - a und, da (a,T ( F ) - a ) konsistent ist, auch ( a , ( C } )konsistent, also a+C nicht intuitionistisch herleitbar. Definition des ausgezeichneten Modells ( U ( F ) ,G , W ) :Fur jede Aussagenvariable v und jedes Element C I EU ( F ) sei CIEU ( F ) , wenn

w ( u , a) =

w, wenn v E a ist, f, wenn v $ a ist.

Hiermit ist ein Modell gegeben, da U ( F ) nicht leer, transitive Relation auf U ( F ) ist und definitionsgemafi

c eine reflexive und

W ( v ,a) = w , a G p=. W ( v , p) = w gilt. Wir werden sehen, daD die Formel F in diesem Modell ungultig ist, falls sie nicht intuitionistisch herleitbar ist. Hierzu beweisen wir, daD das ausgezeichnete Modell ( U ( F ) ,c , W ) folgende Eigenschaft hat:

LEMMA 5. Fur C E T ( F )und

C&a ist.

M E U ( F ) gilt

W ( C , a) = w genau dann, wenn

Beweis durch Induktion nach der Lange der Formel C. 1 . C sei eine Aussagenvariable. Dann gilt die Behauptung nach der Definition des ausgezeichneten Modells. 2. C sei die Formel A. Dann ist ({ C),8) inkonsistent, folglich C$a und definitionsgemafl W(C,a)= f. 3. C sei eine Formel A A B. Dann hat man W ( AA B, a ) = w-W(A, a) = w und W ( B ,LY) =w ~ A E und M B E U(nach Induktionsvoraussetzung) o a - t A und a+B intuitionistisch herleitbar (nach Lemma 4) -a+A A B intuitionistisch herleitbar o A A B E E(nach Lemma 4). 4. C sei eine Formel A v B. Dann hat man W ( Av B, a) = w e W ( A , a ) = w oder W(B, a ) = w o A E M oder B E N(nach Induktionsvoraussetzung) o a + A oder a+B intuitionistisch herleitbar (nach Lemma 4) =a+A v B intuitionistisch herleitbar e A v BE^ (nach Lemma 4).

234

K. SCHUTTE

Umgekehrt gilt: 1st a-+A v B intuitionistisch herleitbar, so ist (a, { A , B } ) inkonsistent. Da { A , B } E T ( F ) ist, folgt dann A E X oder B E E . Hiermit ergibt sich W ( Av B, a)= w-A v BE&. 5. C sei eine Formel A+B. Dann hat man

A + B $ ~ ~ D L - + ( A -nicht + B ) intuitionistisch herleitbar (nach Lemma 4) o ( a u { A } ,( B } )konsistent ~ A E und P B $ p fur ein BE U ( F )mit a EP (nach Lemma 2) e W ( A , p) = w und W ( B , p) =f fur ein PE U ( F ) mit a E p (nach Induktionsvoraussetzung) *W(A+B, a)=$ Beweis des Vollstandigkeitssatzes. Die aussagenlogische Formel F sei nicht intuitionistisch herleitbar. Dann ist (0, { F } ) ein konsistentes Mengenpaar. Mit Lemma 2 folgt, daB es ein F-vollstandiges konsistentes Mengenpaar (DL, p) mit FED gibt. Hierfiir gilt cieU(F) und F$a, also nach Lemma 5 W(F, a) = f . Somit ist F nicht intuitionistisch allgerneingultig. Anmerkung. Mit diesem Beweis ergibt sich auch ein Entsclieidungsvevfahren fur die intuitionistische Aussagenlogik. Der Vollstandigkeitssatz IaBt sich in folgender Weise auf beliebige (auch unendliche) Formelmengen verallgemeinern. Ein Paar (a, p) von Formelmengen a, p heifie konsisterzt, wenn es keine endlichen Teilmengen U,ECY und p 0 c p gibt, 5 0 da13 die Formel ao+Po intuitionistisch herleitbar ist. Das Mengenpaar (a, p) heipe interpretierbar, wenn es ein Model1 ( M , R, W ) und ein EM gibt mit

W ( A , 5 ) = w fur alle A E ~ , W ( B , 5) = f fur alie B E @ . Aus dem Konsistenzsatz folgt: Jedes interpretierbare Paar (a,p) ist konsistent. Umgekehrt gilt auch: VERALLGEMEINERTER VOLLSTANDIGKEITSSATZ. Jedes konsistente Paar (a, p) ist interpretierbar. Dieser Satz 1aDt sich folgendermafien beweisen. Entsprechend wie Lemma 1 beweist man fur beliebige Formelmengen M , fl und fur jede Formel C :

LEMMA 1". 1st (a, p) konsistent, so ist auch ( a u { C } , p) oder (u, pu{C}) konsistent. Ein Paar (a, p) heifie maximal-konsistent, wenn es konsistent ist und DL v ,l? die Menge aller Formeln ist. Aus Lemma I * folgt:

INTUITIONISTISCHE AUSSAGENLOGIK

235

LEMMA 2". Jedes konsistente Paar (a,/) 1aDt sich zu einem maximalkonsistenten Mengenpaar erweitern. (ao, Po) sei ein gegebenes konsistentes Paar. (al, PI) sei eine maximalkonsistente Erweiterung von (ao, Po). Wir konstruieren eine Menge M von Formelmengen mit folgenden Eigenschaften : 1. U I E M . 2. Fur jedes N E Mist (a,E ) konsistent. ( E sei die Komplementarmenge von a.) 3. Zu jedem EM und zu je zwei Formeln A , B, fur die das Paar ( a u { A } ,( B } )konsistent ist, gibt es P E M mit a c P , A E P und BCP. Ein Modell ( M , E,W ) wird dann so definiert, daD fur jede Aussagenvariable v und jedes a ~ A genau 4 dann W(v,a ) = w ist, wenn V E E ist. Entsprechend wie Lemma 5 beweist man: Fur jede Formel C und jedes @ E Mgilt W(C, a)= w genau dann, wenn C E ist. ~ . Es folgt W ( A ,a,)= w fur alle A € a Ound W(B,a,)=f fur alle B E P ~Hiermit ist der verallgemeinerte Vollstandigkeitssatz bewiesen. Eine Folgerung ist der KOMPAKTHEITSSATZ. 1st (ao, Po) fur alle endlichen Teilmengen a. so ist (a,P) interpretierbar.

Po cP interpretierbar,

c a und

RECURSION THEORY AND THE THEOREM OF RAMSEY IN ONE-PLACE SECOND ORDER SUCCESSOR ARITHMETIC D. SIEFKES Mathematisches Institut der Universitat Heidelberg

In his paper [l] Buchi gives a decision method for a system of arithmetic which has the successor function as only operation, but is built up in the strong logical frame of second order one-place predicate calculus. This system is a very suitable tool in examining the behaviour of finite automata (sequential machines, cf. e.g. Rabin and Scott [7]). From its decidability follows the solvability of the automata decision problem for this language (cf. Church [ 2 ] ) .On the other hand, Buchi uses a good part of the theory of automata for his decision procedure. In view of this close connection he calls the system sequential calculus (SC). Buchi can use the means of the theory of automata, since he sets up semantically both the system SC and the decision procedure. If one wants to have a formal approach, one has to analyze the decision procedure - as already Buchi suggests - to get a complete axiom system for SC. To do so, we eliminate in [6] the theory of automata from the decision procedure and show that a certain kind of formulae works as finite automata within the system. To this end we set up an axiom system for second order one-place predicate calculus, and from these axioms and the Peano axioms for the successor function we build up primitive recursion theory as far as it is expressible in the language of SC. With the help of recursion theory we further derive theorem A of Ramsey [8] which was used by Buchi as a second help from outside and which he proposed to be the most interesting candidate for an axiom schema for SC. A careful examination shows that the remaining steps of the decision procedure are derivable; thus this very simple axiom system is complete. In fact it is the idea of Church (cf. e.g. [2]) to handle automata problems by recursion ; for this purpose he uses open recursive theories (quantifiers are excluded, but the introduction of new predicates by recursion equivalences is allowed). Therefore recursion theory suggests itself as a compen237

238

D. SIEFKES

sation of automata theory; but it is surprising that the whole theory of primitive recursion does not exceed the power of SC. - In the sequel we shall speak simply of “recursion”, omitting the word “primitive”. In this paper we shall give the derivation of the theorem of Ramsey within the system SC. Since Ramsey uses metamathematical recursion which is not available in SC, the translation of the original proof into the formal language is one of the central points in establishing the completeness of SC. Thus this topic seems to be worth a separate treatment. - In Section 1 we state the language and the axioms of SC; in Section 2 we derive recursion theory and from this in Section 3 the Ramsey theorem. A presentation of our version of the decision procedure and of the connections between SC and the theory of automata will be given in full detail in [6]. - I wish to thank Prof. G. H. Miiller for his kind criticism and helpful remarks in writing down the paper. 1. The system SC

Since we want to derive the theorem of Ramsey within the system SC, we describe the language and the logical frame before giving the non-logical axioms. Object language: As individual variables we use small Latin letters, a, b, c,... as free, t , x,y , z as bound variables. Analogous A , B, C,... resp. P,Q, R,S as free resp. bound one-place predicate variables. The quantifiers V, 3 serve for both types of variables. Further we use sentential connectives, T and F for the truth values “true” and “false”, brackets and dots for bracketing of formulae (dots extend over brackets). As only non-logical signs we have the individual constant 0 to denote the zero-element and the one-place function symbol ‘ for the successor function. Metalanguage: Formulae we denote by German capitals, natural numbers (for indices etc.) by small German letters. As for the rest we use the signs of the object language in the metalanguage. Logical axioms: We use freely rules of propositional calculus without mentioning. The other axioms and rules we state by pairs (respectively for individual and predicate variables). It is to be understood that one has to avoid collision of variables. 1) Substitution rule:

(a term).

(8 formula with one marked free individual variable).

239

ONE-PLACE SECOND ORDER SUCCESSOR ARITHMETIC

2) Changing of bound variables:

( Q quantifier). 3) Axioms for quantifiers:

(AQI1) (Vx) % (x) + % ( a ) (AQ12) %((a) + (3x1 %(x)

(AQPl) (VP) % ( P ) + % ( A ) (AQP2) % ( A ) + ( 3 P ) % ( P )

4) Rules for quantifiers: (RQIl)

B +%(a) 23 4 (VX) % (x) ~~~~

2l(a) + B (RQ12) (3x) %(x) + B

(RQPl)

(RQP2)

B +%(A) B + (VP) % ( P ) % ( A )+ B ( 3 P )% ( P )+ B

( A not in 23). (a not in 23). We call this logical frame P'K(2): second order predicate calculus with one-place predicate variables. Evidently the axiom system is not independent. It is wellknown that within this frame equality is definable by a = b H~~(VP) T P ( a ) -+ P ( b ) l

.

Further a special form of the replacement theorem is derivable which we call principle of extensionality: (EXTI

(vX) r A (x)

B (x)i.+. 3 ( A )+a ( B ).

The derivation is by induction over the length of the formula % and does not use (SP). By the same method we get the generalization

(vX) r A (x) -8 (x)i.+. % ( A )+ q ~ ) . With the help of this formula we show that the substitution rule (SP) is equivalent to the principle of comprehension : (COMP)

( 3 ~(VX) ) rP(x) -%(x)l

( P not in

a).

This shows that P'K(2) may be considered as a fragment of set theory (cf. Robinson [9], Hasenjaeger [3], McNaughton [ 5 ] ) . In most derivations we will use (COMP) instead of (SP) and it is in fact this highly impredicative comprehension rule which gives together with the induction axiom the strongness of SC.

240

D. SlEFKES

Non-logical axioms: The three Peano axioms for successor are sufficient. We need no schema in view of (SP):

a' = b' -+ a = b , a' # 0 , A (0) A (vt) rA ( t ) + A ( t ' ) ~-,(vt) A ( t ) .

(All ('42) (1)

The system built up in P'K(2) by these non-logical axioms is called sequential calculus SC. First of all we get from (I) by (SP) the induction schema

'u(o)

(1s)

A

(vt)

r'u ( t i -,'u( t ' ) ~

-+

(vt) 'u ( t )

.

Further it is known (cf. Hilbert and Bernays [4], p. 490-491) that order is definable by U

< b ++df(3P) [ P ( a ) A

(vt)

rP(t')

-+

p(t)l

A 1P ( b ) ] .

We use the following abbreviations (cf. Buchi [l]): 1)

2) 3) 4) 5)

( I t ) : % ( t ) ++df(3t)r U < t < b A %(t)l , (vt): % ( t ) ++df(v't)ra < t < b 'u(t)l , (3"t) Hdr(vx)( 3 t ) < t A 301, (V'"t> % ( t ) Hdf(3X) ( V t ) Tx < t + %(t>l , (3P)"'U(P)++df(3P) r ( 3 v ~ ( t A) ~ P ) . I

rx

-+

1) and 2 ) are the familiar restrictions of quantifiers; sometimes we use also (3t), % ( t ) and (Vt), %(I), if there is only a lower bound. 3) is to be read as "there are infinitely many t", 4) as "for ultimately all t", 5 ) as "there is an infinite P". Remark that l ( 3 " t ) 'u(t)++(V"t) 1% ( t ) is derivable. For recursion theory we need still another abbreviation: Let %(a, E ( t ) ; t < a ) mean that in the formula 'u the predicate variable E is contained only with bound arguments restricted by a. For such a formula '%(a, E ) we have a restricted form of (EXT):

(REXT)

(vX);;rA(x)oB(x)i.-+.%(a,

B).

2. Recursion theory

In his paper [2] Church considers "wider restricted recursive arithmetic", a numbertheoretic system similar to SC, which has no quantifiers, but allows

241

ONE-PLACE SECOND ORDER SUCCESSOR ARITHMETIC

the introduction of predicates by a certain recursion rule. A simple instance of this recursion schema would be

--

A (0) A (a’)

3 [ B ,(O), .. ., B,, (011

9

23 [ B ,(a), ..., B” (a), A (a)] .

In fact Church considers multiple recursion with any (fixed) number (not only 1) as step distance, but we want to generalize it slightly in another direction : For the whole section let B(a7E ( t ) ;?; rD(z)ttqz, D)I

+piz):

A

bi

rQ (XI

A

Q (v)

+

-I

@ (x, y)l .

The first step in formalizing the proof given above consists in replacing sets by predicates: We want t o define recursively a predicate E determined just by the sequence a,, a,, .. . . Thus E(ai) holds if and only if there exists an infinite predicate Ci with

ci-

(ai>

A

r(vx) ci(x>-,ai < x A @(ai,

x)

A

ci- (X)I .

We have seen in Section 2 that we are able to introduce in SC a predicate by recursion over the arguments, but is impossible to introduce a sequence of predicates by recursion over the indices. Thus we avoid the explicit construction of the sequence C, C, ... ; we use only the existence of such a predicate Cifor every i, O < i < i . Let us abbreviate the condition A ( b )+ a < b

A

@ ( a , b)

by 8 ' ( A , a, bj and the formula ( g u t ) ~ ( t A) ~ ( a A) (vx) [rA(x)

+

B ( ~ ) IA @ + ( B , b, x)]

246

D. SIEFKES

by D+( A , B, b, a). If we replace in these formulae A and B by Ci and Ci respectively, then we have the conditions upon Ci and Ci of the last paragraph. Thus we may express the above considerations by the following formula : (1)

~ ( a-VQY ) [(vx) Q + (Q, a ,

A

(w: my) -+

+

For the following proofs we abbreviate this by

(W D+(Q, P , Y , all 1.

E ( a ) -FQ) 23 (E,Q, 0). As for the second case we introduce analogous a predicate H determined by the sequence b,, b,, ... . Again we avoid the predicates Di which are determined by the sets { d i , d i , ...}. Let 6- and 9- be the same formulae as 8 ' and 3 ' with 7 s ) instead of s). Then we formalize the second case by

(2) H ( a ) -(vt),

1E ( t ) A A

A

PQ)"[@XI

8-(Q, a , x)

A

(w'or - w ( ] P I a+(Q, P , y, a ) i (wy,rH( Y ) .+ ( 3 ~ID) - (Q, P , Y , a ) i I . --f

A

As abbreviations for the right side of the equivalence we use (Vt), 1E ( t ) A

(3Q) @ ( E ,ff, Q, a )

or even shorter C(E,H , a). Since the recursions (1) and (2) are of type (R) in Section 2, we have at once the existence of E and H : LEMMA 1.

Further it is very easy to see that the recursions (1) and (2) are good for our purpose: if E or H are infinite, then we get an infinite Ramsey set as wanted in Theorem 2. We show this in the following Lemmata 3 and 4. The whole difficulty is reduced to show: if E is finite, then H must be infinite; this will be done in the main Lemma 5. For the convenience of the reader we give now a short informal version of the proof of this lemma; then later on one has to check only the formalization step by step. So let E be finite and let a be the smallest number such that ( V t ) , i E ( t ) . We want to show: for every number b there exists a number c, bY, (XI A

Q (1'1

+

7

5(x, Y ) I .

Lemmata 1 , 3 , 4 together with the following main lemma give the assertion of Theorem 2. LEMMA 5. If E is finite, then H must be infinite: (Vz)

[ r E ( 4 w ( 3 Q ) 23 ( E , Q, 211 A

A

l H ( z ) + + % ( E , H , z)1]

A

(V"?)i E(t)-+(3"t) H ( t ) .

248

D. SIEFKES

Proof. To make better reading we will not follow the rules of our formalism as close as we did in Section 2; but there will be no difficulty to translate the following considerations into a strictly formal proof. So let E and H satisfy the premises of Lemma 5, let the number a be fixed for the whole proof such that ( V t ) , i E ( t )A (Vt); (3x),E(x) (thus a= 0, if ( V t ) i E(t), and a= c', if E(c) A V t ) = ,E(t)). i We assert (Vy) (3x),H(x) and to prove this we show by induction over b (3x),H(x). 1) Induction beginning: b = 0. 1. case: a = 0. Thus we have ( V t ) i E(t), especially i E(O), therefore by (1) (VQ).(Vx) Q'(Q, 0, x>+(v"t>lQ(t>. This implies

(VQ) . (Vx) from which follows (VQ). (Vx)

rQ (x) +-+x= 0 v 1 5(0, x)l

rQ (x) -0

<x

A i

5(0, x)l

(V'"t)

-+

+(

Q ( t ),

Y t ) Q(t)

.

By (COMP) holds (3Q).(Vx) rQ(x)++O < x

A 1&(O,

x)l ,

this gives together (3Q).(Vx) rQ(x)C-'O < x

A i &(O,

x)l

A

(3"t) Q(t)

and therefore (3Q)" (vx) 8-(Q, 0, x). According to (2), this is equivalent to H(0). 2. case: O(V'X)~ ~ Q ( x >A

If i < m - 1, we are ready. So let (3"O D ( t )

A

(VY) (VxX

Q (Y>

+

fii(X,

Y)]

*

i =m - 1, let ff be a predicate with

(XI A D ( Y )

-+

Ssm-

(x, Y ) v

B,,,(x, 4'11 .

ONE-PLACE SECOND ORDER SUCCESSOR ARITHMETIC

253

Similar restrict the recursion for H and the proofs to the predicate D.Then Lemmata 1-5 hold as before and one gets as Theorem 2

rQ(x) A Q ( Y ) D (XI A D ( Y ) A $L-(x, ~ 1 v1 v ( V Y ) (VxX rQ(x> A Q ( Y ) D (XI A D ( Y ) A 1ti,,-(x,~ 1 .1

(I€!)".( V Y ) (VxX

-+

1

-+

we get the desired formula

At last we extend Theorem 2 to the case of arbitrary n-tuples.

THEOREM 4. n-

1

m

-+(3Q)". V (Vx, ,..., x,) i=1

m

1

(vx,, ..., xn) ri= A xi
d V 5i(xl,...,xn)1.

We may get a proof of this theorem by trivial changing of some points in the proof of Theorem 3. Or we use the completeness of SC announced in the introduction (shown with the help of Theorem 3 only) and get Theorem 4 by Theorem 1. We conclude with a consequence of the decidability of SC: If we drop the quantifier (3Q) in Theorem 3 and replace the variable Q by A , we get a formula %(A) which is satisfiable if and only if Theorem 3 is true. Now it follows very easily from a result of Biichi [I] that, if a formula % ( A ) is satisfiable at all, then by an ultimately periodic predicate. This shows that the Ramsey set M in Theorem 1 can always be chosen as ultimately periodic if we start with sets L, definable in SC.

254

D. SIEFKES

References 1. J. R. BUCHI,On a decision method in restricted second order arithmetic, in: Logic, Methodology and Philosophy of Science, Proc. of the 1960 Intern. Congr. (Stanford, 1962) 1-11. 2. A. CHURCH, Logic, arithmetic and automata, Proc. Intern. Congr. Math. 1962, 23-35. 3. G. HASENJAEGER, Uber o-Unvollstandigkeit in der Peano-Arithmetik, J. Symb. Logic 17 (1952) 81-97. 4. D. HILBERT and P. BERNAYS, Grundlagen der Mathematik I1 (Berlin, 1939). 5. R. MCNAUGHTON, Some formal relative consistency pioofs, J. Symb. Logic 18 (1953) 136-144. 6. G. H. MULLER and D. SIEFKES, Decidability and completeness in restricted second order arithmetic, to appear. 7. M. 0.RABINand D. SCOTT,Finite automata and their decision problems, IBM J. Res. Develop. 3 (1959) 114-125. 8. E. P. RAMSEY, On a problem of formal logic, Proc. London Math. Soc. 30 (f929-30) 264-286. 9. R. M. ROBINSON, Restricted set-theoretical definitions in arithmetic, Proc. Am. Math. SOC.9 (1958) 238-242.

REFLECTION PRINCIPLES OF SUBSYSTEMS OF ANALYSIS *

Dedicated to Professor S. Iyanaga for his 60th birthday G . TAKEUTI lnstitute for Advanced Study, Princeton, New Jersey

and

M. YASUGI University of Bristol Let 6 be one of the subsystems of analysis, SINN and the system with extended inductive definition (called SEID in this article) (cf. [7]). SINN is second order Peano arithmetic with full induction and the II: comprehension axioms. SEID is an extension of SINN, which is obtained by adding to SINN some inductive definitions. Let rD be the system of ordinal diagrams which is used in proving the consistency of 6 and let ‘Ip be first order Peano arithmetic with second order parameters. Then we can prove the following reflection principles.**

THEOREM 1. Ind,(

a),Prov, (‘Vx3yR (a, x,y)’)

--f

V x l y R (a, x,y )

is p-provable, where R ( N ,x,y ) is elementary in a, i.e., all quantifiers in R(a, x, y ) are numerical and bounded, and Ind,(D) is the schema which allows transfinite induction along 3 with respect to Z y formulas (without second order parameters).

THEOREM 2.

Ind,( %), Prov,(‘A (a)’)

-+

A (a)

is p-provable, where A (a) is an arithmetical formula with a parameter a and

*

Part of this work was supported by NSF GP-4616.

** The results here can also apply to the extended system footnote 2 of [71.

255

SIN”

of

SINN

defined in the

256

G . TAKEUTI

and

M. YASUGI

Ind,(%) is the schema which allows transfinite induction along % with respect to the formulas of ’p. By modifying the proofs of Theorems 1 and 2 we can prove the uniform reflection principles (cf. Introduction of [3]), that is the following two theorems are p-provable.

THEOREM 1’. Ind,

(a)+ V i n (ProvG(‘Vx3yR (x, y , a, n (m)>.)b Vx3yR (x,y , LY, i n ) ) ,

where n ( m ) denotes the “m-th numeral” and R(a, b, a, c) is elementary in LY.

THEOREM 2’. Ind, (9) -+ Vm (Prov, (‘A (a, n (m))’) F A (a, m)) , where A (a, m) is arithmetical in LY. We can also prove another form of the uniform reflection principle.

THEOREM 3. I n d ’ ( 3)

--f

Vm (Prov, ( V + A (4, n (m)).)t VdA (4, m))

is 6-provable, where A(a, a) is arithmetical in a and Ind’ is applied to Z: formulas with a second order parameter. For the meanings and consequences of the reflection principles, the reader should refer to [3] in which a list of references concerning those problems is also found. We are concerned with special cases only. Throughout this article, acquaintance with [7] is presupposed. Both authors started their study of logic in Professor S. Iyanaga’s seminar. We should like to take this opportunity to express our thanks to him. Chapter I In this chapter we shall prove Theorem 1 (which has been stated in the introduction) and its corollary.

1. DeJinition of the systems and elementary predicates. Let 6 be one of the systems SINN, G,, GI, SJNN and the system with extended inductive definition (denoted SEID) and let % be the system of ordinal diagrams that is used in proving the consistency of G(% is denoted by S in [7]). Those systems are defined in [l] and are also to be found in 1 of Chapter 2 , 7 of Chapter 3, at the beginning of Section 2 in Chapter 3, at the beginning of Chapter 3, and in 1 of Chapter 4, respectively. Although 6, and 6,are not to be considered after 2 of Chapter I in this article, we have introduced

SUBSYSTEMS OF ANALYSIS

251

them in order to prove Proposition 1 for SJNN. For the elementary notions and the notations, refer to Chapter 1 of [7]. 1.1. We shall restrict the non-logical constants of 6 to the following. Individual constants; 0, 1. Function constants; +, .. Predicate constants; =, < . 1.2. A formula of 6, R(bl,..., b,, pl,. . ., PI), whose only free variables are b,, ..., b,, pl, ..., (including the cases m=O and/or l=O) and which has no quantifiers on f-variable is called elementary iri pl, ..., pl if all quantifiers appearing in R are bounded. 1.3. The beginning sequences of the system 6 are all those of the forms D-tD and s= t, A ( s ) + A ( t ) , where D and A ( a ) are arbitrary formulas, and mathematical beginning sequences. We may restrict the mathematical beginning sequences to the well known quantifier free axioms concerning the constants given in 1.1. 1.4. All other definitions of 6 in [7] are effective here. We shall use the logical symbols v , k and 3, as well as 1,A and V, although they are not formally defined in 6. Remark. The class of the predicates which are elementary in some free f-variables (cf. 1.2) is smaller than the class of the predicates which are primitive recursive in some freef-variables. This does not weaken our result, however, since the classes of the predicates of the form VxR(a,x , a) and 3xR(cc,x, a) with R elementary in a respectively cover the predicates I7: in a and those Zy in a (cf. Theorem 1). 1.5. A cut is called essential if its cut formula is not of the forms= t o r s< t. 2. PROPOSITION 1. Let R(a, &, ..., 0), be a formula of 6 which is elementary in PI,..., p, and assume that +3xR(x, P1,..., P,) is 6-provable. Then there exists a proof-figure of 6 to the above sequence which does not contain any essential cut or any induction. Moreover, this can be proved with the system of o.d.’s 9, i.e., we can prove the above statement by transfinite induction on the o.d.’s of 9 which are assigned to the prooffigures. The treatment of SJNN is slightly different from the other cases. Proof. For simplicity, we shall prove the proposition only for the case m = 1 and denote the formula R(a, cx). Let us first consider certain conditions on the sequences of 6. Let S be a sequence A , ,..., A j - + A j + l,..., A , of 6. S is said to have the property P if it satisfies the following. P.l. S has no free t-variable.

258

G. TAKEUTI

and M. YASUGI

P.2. Each formula which is in the left side of S, i.e. one of A,, ..., A j , is elementary in a. (This implies that none of A,, ...,A contains unbounded quantifiers.) P.3. Each formula which is in the right side of S, i.e., one of A j + l , ..., A,, is either elementary in ct or 3xR’(x, a),R’(0, a) being elementary in 3.

LEMMA. If a sequence S of 6 which has the property P is 6-provable, then it is provable without essential cut or induction, except for the case 6 is SJNN. Obviously the proposition is a trivial corollary of this lemma, except when G is SJNN. The case where 6 is SJNN shall be treated separately. Proof of Lemma. Suppose a proof-figure P to S is given. The proof is carried out with several steps following the consistency proofs of G (cf. [7]). We shall see that, at each reduction step, the end sequence of the resulting proof-figure still satisfies the property P. 2.1. G is SINN. 1) 2 through 8.2 in Chapter 2 of [7] are effective here. 2) We add the following inference schema “bounded-quantification” (abbreviated to bq) to our system.

where Vy‘yq-r'+A', V y < n R b ( y , a ) ~

~~

~

~-

~~

I"

+

~~~

-

~

~-

-~~

A 1%VY < sRb ( Y , a), A , ,

where A' is A , , V y < s R & ( y ,a), A , .

~

~

-~

G. TAKEUTI and M. YASUGI

260

4) Reduction in case there is no explicit logical inference or induction in the end piece. We can follow the proofs in 8.4 through 10 of Chapter 2, [7]. We shall only mention the cases where some extra considerations are needed. 4.1) The end piece of P contains a beginning sequence of the form D-tD. Let P be of the form

D-D

..

r-; A , B r, n

.. B, n-;A ~ b,, A , A, A,,

.. 3+@.

b, A ,

up to term-replacements, the reduction in 8.5.1 in If 6 is identical with Chapter 2, [7], works. If not, then b is of the form

and fi is of the form Vy < tS‘(y), where s= n and t = n for some numeral n, so=O, ..., ~ , , - ~ = =1n, -and S’(0) is either S(0) itself or obtained from S(0) by some term replacements, since there is no explicit logical inference or induction in the end piece. Hence P is reduced to the following proof-figure. I

4.2) Elimination of weakenings in the end place of P (cf. 8.6 of Chapter 2, [7]). If the last inference of the proof-figure Q is a bq, say

A , (0 < kl- S(0)) A r -+ A , Vy < k S ( y )

then the proof goes as follows.

... A ( k - 1 < k t S ( k - 1)) ~

SUBSYSTEMS OF ANALYSIS

261

..

If Q: is T * A A * , then Q* is Q:. If Q: is

..

then

r* + V 4 3 x V y R (4, x, Y >

by adding to G all true 1;sentences as axioms. (The idea of the proof is that we cannot obtain a new provable well-ordering by this addition by a result of [ 2 ] ; on the other hand, (I) provides a new provable well-ordering whose order-type is the supremum of all the provable well-ordering in G.) This implies that we cannot prove Prov, ( W A (4)')

-+

V#A (4)

2

where A ( a ) is arithmetical in a, by adding to G Ind(53) which applies to arithmetical formulas without second order parameters. 14. Appendix Here we wish to state an application of the proof of the theorem in Chapter 5 of [7] (also cf. [6]), although this has no direct bearing on the reflection principles :

THEOREM. Let 6 be SINN or

(which may contain recursive functions as functional constants) and 33 the system of ordinal diagrams used to prove the consistency of (5;and let Q ( a ) be a recursive predicate (containing no logical symbols) and a < .b a recursive linear ordering of the set Q (= {a[Q (u)}). Then, if 1 &($(Cl(m)) > 1 + +

A (a?$ (i( m ) )- 1))) *

$ is a function defined on finite sequences of natural numbers. (Therefore the requirement that A does not contain $ free is superfluous here, since A contains variables for functions from N into N only.) A weak consequence of (11) is the enumeration principle: (111)

Va3nA(a, n ) -+ 3y(Va3nA(a, y ( n ) )&Vn3crA(cc, y ( n ) ) ) .

The strongest form of an intuitionistic continuity postulate we shall use is Brouwer’s principle for functions ( [ 2 ] ,p. 73)

(IV) Va3PA(a,P)-+3$Va(Vm3!n$(m,a(n)) > I & & VP [ V m h $ ( m , E ( n ) ) = P ( m ) + 1

A ( a , B11).

-+

Without essentially strengthening (1V) we can impose the following condition on the II/ in (IV):

(*I

VrnVn3k($(nz,&(n))< l & $ ( r n + l , E ( k ) ) > l - + n < I c ) .

The form of IV with (*) included is denoted by (IVA). In (IV) and (IVA) a mapping $ is defined for pairs ( m , a), mEN, CEO,. The principles (I)-(IVA) remain valid if the range of the variables a,p is restricted t o D(0). These restricted forms we denote by (I*)-(IVA*). Further we need the axiom of choice in the following form (a not necessarily in D(0,)).

Vn3xB(n, x)

(V)

-+

3aVnB(n, a ( n ) ) .

DEFINITION 1. A set X i s said to be represented by (0, 9, -) if a) (0, 9) is a spread, S, b) S*, the set of equivalence classes of S with respect to can be mapped bi-uniquely onto X . DEFINITION 2. We define a mapping (the standard mapping) x from D(0,) onto D(0),as follows:

-

-

E ( n )E 0 + xa ( n ) = i ( n ) ,

E(n)+O

-+

-

x a ( n >= x a ( n - I > * ( p , ( G ( n - I>*( m >E 0 ) ) .

292

A. S. TROELSTRA

DEFINITION 3. A spread ( 0 ,9) is called homogeneous, if

~atl/?trm (92 (m)

= 9p(m)

--f

37 ( E ( m )= ?; ( m )& 9/? = $7))

.

THEOREM 1. If ( 0 , 9 ) is homogeneous, then

k ’ a ~ D ( 03nA(9a, ) n ) + V a E D ( @ ) 3 m 3 n ’ d P ~ D ( @ ) ( Q ~= ( m9p(m)--t ) A (SB, a ) ) . +

Proof straightforward. Standard construction. (x,), is a given sequence. Let X be the set of finite sequences of elements of (x,),, and let R be a relation defined for pairs ( a , x), O E X ,XE(X,),. A sequence a=(y,, ..., y,) is called admissible if R(O,y , ) & R ( ( y , ) , y 2 ) &...&R((YI,..., Y n - I ) , Y 3 . A sequence a is called admissible if E(n) is admissible for every n. If { x : R ( a ,x)} is enumerable for every g, then a spread (0,,, 9) can be constructed which contains all admissible sequences and which is homogeneous. This is done as follows: Let (a,), be an enumeration of X , g1=O. We suppose (x: R(o,, x)} to be enumerated as ( x ( n , m)),. If to every sequence (mi,. .., m,) of length t a number y2 has been determined such that

9 ( m , , ..., In,) then we put for every m.

= 6,

9(ml,..., m,,m) =a,*(x(n,m))

3. Applications In intuitionistic mathematics all applications of Brouwer’s principle were hitherto in fact applications of the fan theorem, which is a combination of the bar theorem and Brouwer’s principle for a special case, the case of finitary spreads ( [ 2 ] , p. 59). The following examples show that the general form of Brouwer’s principle has strong consequences too. THEOREM 2 . I c N , Wi subsets of a space r.

U {Wi: i ~

l3}T - t

U {Interior Wi: ie1) 3 T

([4], Theorem 6).

ProoJ In this as well as in the other applications treated, the essential idea of the proof is always the construction of a suitable spread to which Brouwer’s principle can be applied.

293

"BROUWER'S PRINCIPLE"

r is separable, therefore we may suppose (p,,),, to be a sequence dense in the space. p is a metric for r. We put Ui,j=U(i-l, p j ) and we define a relation T on ordered pairs by T(Uk,l, Ui,j)tSk-l > i - l + P ( P j 9 P I ) . We remark that T(U,,,, Ui,j ) implies Ui,c Uk,1. The Ui,constitute a basis for r. T is an enumerable relation, in other words the pairs ( Uk,1,Ui, j ) for which T holds can be effectively enumerated. For let q ( n , m, k ) be a rational-valued function such that then



=

ui,j> 7

(ukt,lt?

we can carry out the standard construction of a spread S = ( B o ,9) such that YES-VnT(Y(n), r ( n

+ 1)).

In the sequel of this proof we suppose a, PES. If diameter a(n)n.

We remark that (3)

Vt3P E D (0)( t

> m & qt E u (2-k(t),p ) n v

--t

01" ( m ) = = p(m)&Iim$P = q t ) .

+

If a* (m)= a* (m- l), then U ( 2 - k ( m )p, ) n V= 0 ;if a* (m)= a* (m- 1) 1, then q , ~ U ( 2 - ~ ( ~ ) , pV.) nIn the second case we can take a P E D ( O ) such that E*(m)=P(m),and lim9p=ym (an application of (3)); then A ( p , n) holds i.e. U(2-", lim9p) = U(2-", )4, c W , p ( p , 4,) < 2-k(m)+2-k(rn) - p ( p , 4,) = 6 > 0. We remark that k ( m )2 m>n, hence 2-" 3 2-k(m).Therefore U ( 2 - k ( m )9,") , c W, hence U ( 6 ,p ) c W. So we have proved for an arbitrary p : ~E(U(p E ), n V = 0 v U ( E ,p ) c W )

and this proves V C W. THEOREM 8. Every open covering of a space r contains an enumerable sub-covering. (Compare [ 5 ] , 2.2.6.) Proof. Let (O,, 9, ) be the representation of r as described in the proof of Theorem 2, and let { W,: X E X )be an open covering of r ; ( p , ) , is a sequence, dense in r. We introduce a predicate :

-

A(a, k, m ) t , 3 p E r 3 x E X ( n a ( n ) We see that

=

{ p } & p ~ U ( 2 -p,) ~ , c W,).

fl

Va3k3mA (a,k , m ) .

291

“BROUWER’S PRINCIPLE”

From (111) we can derive (using the well-known pairing functions for natural numbers) (a, P, y ~ D ( 0 , ) ) 3P3Y (Va3nA(a, P (n), Y ( H I )

V’n3aA(a, P ( n ) , y (n”

We remark that ( U(2-p(”’,p , ( , , ) ) , covers r ; further we have Vn3x(U(2-P(”),P,(,,)c W,).

Applying (V) we obtain therefore a sequence ( Wxcn,>,which covers r. Remark 2. More generally, this theorem can be proved for topological spaces which can be represented by a spread. THEOREM 9. Let V be a set of sequences of elements of (x,),, and let be an equivalence relation defined for pairs of elements of V ; the corresponding set of equivalence classes is denoted by V*. We suppose V* to be represented by (0, 9, -); (0, 9)=Sc V. Let T be a predicate such that v a E V l p E v ( ( a - P&T(P)).

-

-

Then there exists a representation (O,, 9’, ), (O,, 9‘) = S,, such that v a E s, (T(a)). Proof. Let x be the standard mapping from D ( 0 , ) onto D ( 0 ) according to Definition 2. For a, PED(O,) we introduce a predicate A We see that

A (a, PI + + ~ X N

-

<Xp(n))n

& T ((Xpcn)),)

*

VO1VA(a, P)

Hence we can apply (IVA”), and obtain a function II/ as required there. We define a finite sequence of natural numbers ~ r n l , m z , , , , , r nfor n every (ml,..., mn)EOo,such that

298

A. S . TROELSTRA

Let Y E V . Then 3 a ~ D ( 0 , )( 9 p - y j . We can find a

Y

N

~ X N

N

(X~cn,)m

T ((x,n>

p such that

A ( a , p) i.e.

*

This proves our theorem. A useful topological application of this theorem is the following theorem (cf [ 5 ] , 3.2.20). 10. Let ( W,), be a sequence of closed pointsets of r, and let r THEOREM be represented by ( 0 , 9, -), ( 0 , 9) = S , such that Va E SVn ( a ( n ) E

< V n > m & 3~ E r ( { P } = na (n>>>.

nL

I1

If to every sequence (W,ci,)i, W n C i , = { p >a, sequence (Wn,(i,)i can be found such that F n ( i ) = { pVi(Wm(,+l) }, C W , , i J then there exists a representation (O,, 9’, -), (0,, 9’)=S’, such that V a ~ S ’ V m ( a ( m +1) Ca(m)j. Proof. lmmediate by an application of Theorem 9.

ni

References 1. A. HEYTINC,Intuitionism, an introduction (second revised edition, Amsterdam, NorthHolland Publ. Co., 1966). 2. S. C . KLEENEand R. VESLEY,The foundations of intuitionistic mathematics (Amsterdam, North-Holland Publ. Co., 1965). 3. G. KREISEL,Reports of the seminar on Foundations of Analysis, Stanford University, Summer 1963 (Mimeographed), Section IV: Theory of free choice sequences of natural numbers. 4. A. S. TROELSTRA, Intuitionistic continuity, Nieuw Archief voor Wiskunde (3), 15 (1967) 2-6. 5. A. S. TROELSTRA, Intuitionistic general topology, Thesis, Amsterdam 1966.