STUDIES I N LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 61
Editors
A. H E Y T I N G , Amsterdam H. J. K E I S L E R, Madison A. MOSTOWSKI, Warszuwa A. R O B I N S O N , New Haven P. S U P P E S , Stanford
Advisory Editorial Board
Y . B A R  H I L L E L , Jerusalem K. L. D E B O U V ~ R E ,Santa Clara H. H E R M E S , Freiburg i. Br. J. H I N TI K K A, Helsinki J. C . S H E P H E R D S O N , Bristol E. P. S P E C K E R , Ziirich
NORTHHOLLAND PUBLISHING COMPANY AMSTERDAM
LONDON
LOGIC COLLOQUIUM '69 PROCEEDINGS O F THE S U M M E R SCHOOL AND COLLOQUIUM I N MATHEMATICAL LOGIC, MANCHESTER, A U G U S T 1969
Edited by
R. 0. G A N D Y C. M. E. Y A T E S
1971
NORTHHOLLAND PUBLISHING COMPANY AMSTERDAM
LONDON
0 NorthHolland Publishing Company, 1971 All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior permission of the copyright owner.
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PRINTED IN THE NETHERLANDS
PREFACE A summer school and colloquium in mathematical logic, generally known as ‘the 69 Logic Colloquium’, was held at the University of Manchester from 3rd August t o 23rd August, 1969. This volume constitutes its proceedings. Information about the scientific programme and its relation to what is printed in this book may be gleaned from the table of contents and the ‘List of Participants’. The conference was recognised as a meeting of the Association for Symbolic Logic, and abstracts of most of the contributed papers will be included in the report of the meeting which will appear in the Journal of Symbolic Logic. The organising committee for the 69 Logic Colloquium consisted of R.O. Candy (Manchester), H. Hermes (Freiburg), M.H. Lijb (Leeds), D. Scott (Princeton) and C.E.M. Yates (Manchester). The conference was supported financially by the Logic, Methodology and Philosophy of Science division of the International Union for Philosophy and History of Science, and by a generous grant from NATO. It was recognised as a NATO Advanced Study Institute *. On behalf of the organising committee we wish to thank the above mentioned organisations, the ViceChancellor and University of Manchester, Mrs E.C. Connery (Administrative Assistant t o the Department of Mathematics, Manchester), and all the people who by their help made the conference a success. As will be seen from the table of contents the influence and inspiration of Turing may be traced in many of the contributions. Because of this, and because of his association with Manchester University (where he was a Reader from 1948 until his death in 1954), it seemed appropriate to us to dedicate this volume t o his memory. Robin Candy Michael Yates
* We record here that
36 of the participants signed a declaration dissociating themselves from NATO’s aims and expressing their conviction that scientific conferences ‘should not bc linked with organisations of this character’. vii
LIST OF PARTICIPANTS The symbols in parentheses after any name record contributions to the programme of the colloquium according t o the following code: L = Course Lecturer, I = Invited speaker, C = Contributor of a paper read at the conference, T = Contributor of a paper by title, J = Joint contributor of a paper, a = An abstract of the paper will appear in the report of the meeting in the Journal of Symbolic logic, b = The paper is printed in this volume. Ash, C.J. Ash, J.F. Aczel, P. (Ca) Aloisio, P. Bacsich, P.D. (Ca) Barendregt, H.P. Barwise, J. (Ib, Ja) Bezboruah, A. Bud, M.R. Boffa, M. (Ca) Boos, W. Booth, D.V. (C) Bridge, J.E. Biichi, J.R. (I) Bukovsky, L. (C) Carpenter, A.J. Chudacek, J. Clapham, C.R.J. Cleave, J.P. Cooper, S.B. (Ta, Ta) Cramp, G.A. Cusin, R. Cutland, N.J. (Ca) Daguenet, M. Daub, W. Davies, E. Derrick, J. Devlin, K.J. Dickmann, M.A. Drabbe, J. Drake, F.R. Dubinsky, A.
Dymacek, M. Eklof, P.C. (Ja) Elliot, J.C. Engelfriet, J. Feiner, L. (Ca) Felgner, U. (Ca) Fenstad, J.E. (Jab) Fischer, G. Fitting, M. Flum, J. Fredriksson, E. (Ca) Friedrichsdorf, U. Fris, M.A. Gabbay, D. (Cab, Ta) Candy, R.O. Gavalu, M. Gielen, J. Gielen ,W. Gilmore, P.C. (C) Gjone, G. Gloede, K. Glorian, S. Goodman, J.G.B. Gordon, C. Gornemann, S. Gostanian, R. Grant, P.W. GreifMuhlrad, C. Grigorieff, S. Grivet, C. Hajek, P. (Ib) Hamilton, A.G. (Ca) xiii
Hart, J. Hay, L. Hayes, A. Henrard, P. Herman, G.T. (Ca) Hermes, H. (Ib) Hinman, P.G. (Jb) Hirscheimann, A. Hodges, W. Hopkin, D.R. Howson, C. de Iongh, D.R. Isaacson, D.R. Jackson, A.D. Jahr, E. Jech, T. Jensen, R.B. (I) Johnson, D.M. de Jongh, D.H.J. (Ca) Jugasz, I. Jung, J. Karp,C. (L) Keune, F.J. Koppelberg, B.J. (C) Krcalova, M. Kripke, S. (I) Kunen, K. (Ib) Lablanquie, J.C. Lacombe, D. (L) Lassaigne, R. Lolli, G. Lucas, T.
xiv
LIST OF PARTICIPANTS
Lucian, M.L. Lynge, 0. McBeth, C.B.R. Macchi, P. Machover, M. Machtey, M. (Ca) Macintyre, A. (Ca, Ta) Magidor, M. (C) Marek, W. (Ja) Martin, D.A. (1) Meyer, E.R. (Ja) Minsky, M. (1) Monro, G.P. van der Moore, W.A. Morley, M. Moschovakis, Y.N. (Ib, Jb) Moss, B.J. Moss, M. (Cab) Myhill, J. Nyberg, A.M. (Jab) Ore, T. Jr. Pacholski, L. (Ja) Paris, J.B. (Ta) Paterson, M.S. (la) Pelletier, D.H. Perlis, D. (C) Philp, B.J. Pilling, D.L.
Pincus, D. (Ca) Platek, R. (Ib) Potthoff, K. (Ca) Pour  El, M. Prestel, A. Putnam, H. (Ib) Rabin, M. (L, I ) Renc, 2. Richter, M. Richter, W. (Ib) Rodino, G. de Roever, W.P. Rogers, H. Rogers, P. Rose, H.E. Rousseau, G. Rowse, W.G. Sabbagh, G. Sacks, G. (Ib) Sakarovitch, J. Scholten, F.P. Scott, A.B. Shepherdson, J.C. Shoenfield, J.R. (Lb) Siefkes, D. (Ca) Silver, J. (L) Simmons, H. Slomson, A.
Smith, J.P. Sochor, A. Solovay, R.M. (1) Staples, J. Stepanek, P. (Ja) Stomps, H.J. Strauss, A. Suzuki, Y. De Swart, J. Tall, F.D. Telgarsky, R. Thomas, M. Treherne, A.A. Truss, J. Ungu, A.M. Villemin, F.Y. Volger, H. Wainer, S.S. Wdsilewska, A. Waszkiewicz, J. (C) Wcglorz, B. (I) Weiner, M. van Westrhenen, S.C. Whitehouse, R.M. Wilmers, G.M. Worboys, M.F. Worrall, J. Yates, C.E.M.
RECURSION THEORETIC STRUCTURE FOR RELATIONAL SYSTEMS Daniel LACOMBE '1 Facult6 des Sciences de Paris, Dipartement de Mathtmatiques, Paris
Given an algebraic structure E there are many ways in which we can introduce into the study of E notions of effectiveness. We look at some natural niethods of doing this. Section 1 is a short preliminary discussion on types of algebraic object. The remainder falls into two main parts. In Section 2 we define various notions of recursiveness and relative recursiveness over an algebraic structure, and the more interesting notions are singled out and various equivalences given. Maps a : N + E will play an important part. In Section 2 they give a means of importing recursion theory into the structure of E . In Section 3 such maps  or rather the equivalence classes of such maps with respect to recursive interreducibility  become interesting objects in themselves. We develop a complete structure for the set fo these equivalence classes and trace relationships between this structure and the nature of E itself.
l . N will denote the set of nonnegative integers and E will be any set. By means of relations { R i } and functions { p j } we may give E an algebraic structure. We look at some examples. It will be seen that little can be expressed using only monadic predicates. (i) Let A C 9 E (the power set of E) and also: (a) A is closed under the Boolean operations n, C, (b) (VX E E ) (CX 1 EA 1, and (c) E is infinite (denumerable or not). Suppose A' satisfies the same conditions for E . Then the elementary theories of inclusion on A , A' are elementarily equivalent. These notes on Lacombe's lectures were taken and reshapen by Barry Cooper.
3
D.LACOMBE
4

(ii)
SupposeAs’PE,

E=N,, A = N o .
If we put on A the condition ( V X E A ) ( V Y ) ( X = Y(modu1o the finite subsets o f E ) + Y € A ) , 11
then there exist continuously many maps T : E onti, E for which TA = A . This holds because we only consider monadic relations. We obtain a counterexample if we take E = N a n d let A = the set of all twoplace recursive relations. In this case the set of maps T is just the set of recursive permutations and there are only No of these. (iii)
Take A
E ’PE where E, A satisfy the following properties:
 
( a ) E = X = N,, (b) A is closed under the Boolean operations n, C(and so is a ring), (c)(Vx E E ) ( { x ) E A ) , (d)(VXEA)(Xinfinite + ( 3 Y , Z E A ) ( Y , Zinfinite A Y U Z
= x A Y ~ z = ~ ) ) . Let E ‘ , A ‘ satisfy the same conditions. Then we can prove that they are isomorphic: (37 : E o z E ’ ) ( T A = A ’ ) .
We use the fact that members of A are only monadic predicates and that A is a ring. If we only impose closure under the lattice operations fl, U then isomorphism may fail. There are interesting problems involving denumerable lattices of subsets of N. Consider the following lattices: (a) the set of recursively enumerable subsets of N , (b) the n: subsets of N , (c) the subsets of all hyperarithmetic functions. For a long time it was a problem whether these are isomorphic. Owings [S] shows that not only are they not isomorphic, they are not even elementarily equivalent.
2. E will now be a denumerable set except where otherwise stated.
Definition 2.1. An enumeration of E is a map a : N + E which is onto. A numbering of E is a map a : N + E which is onto and oneone.
RECURSION THEORETIC STRUCTURE
5
Given a structure S = { R j }U {pj} on E , a numbering a : N + E will enable us to compare this structure with various wellknown ones on N . In particular we consider what it means for S to be 'recursive' with respect to a. If P is some property on relations, functions on N , we can ask: does there exist a numbering a such that a'S has the property P?  or do we have for all numberings a that alS has this property? Definition 2.2. (i) s is 3recursive (of recursive type, decidable) iff 11
(3a : N o  E ) ( a  ' is ~ recursive). 11
(ii) S is Vrecursive iff ( V a : N .=E)
(CIS is recursive).
Property (i) looks like a Z i relation. If we limit S then it may become arithmetical. Say we take S = {p} where p is a permutation of E. Then (i) is equivalent to the property: {(m,n ) I 3 at least m cycles of cp with n elements} is r.e. Another such case is that in which S = the equivalence relations on E. We may ask: is property (i) properly 2: and nothing less? We could consider a standard numbering of twoplace r.e. predicates on E (= N), and take all indices which correspond to r.e. sets of recursive type. A strong result would be that this set is properly Z i . Example. Let O,(N) = the recursive permutations of N (or we could have taken the primitive recursive inverses).
We have the usual group operation X on members of Ore#). Then we can choose a,fl E Ore, where the subgroup generated by a,fl is not only not 3recursive but not even recursively presentable. All automorphsms of Ore, are inner. Another question is whether all permutations of recursive type are of primitive recursive type. We can show that they are not but the counterexample is not easy to construct since, for instance, such a cp must have no infinite cycles and not infinitely many cycles of any given number of elements. On the other hand, for a wellordering on a denumerable E to be of recursive type is equivalent to it being of primitive recursive type, or of arithmetic type, or even of Z t type: it is to be a recursive ordinal. FraissC [ 2 ] makes two interesting conjectures about necessary and sufficient conditions for 3recursiveness. While these have since turned out not to be true, the most interesting part of his paper is devoted to a notion of relative recursiveness which will be shown to be equivalent to our Vrecursiveness in (Definition 2.3). Since we may replace functions by graphs we will only consider relations p = (PI, ..., P, 1, Q = ..., Qn) on E.
(el,
Li. LACOMBE
6
5
Definition 2.3. (i) P is 3 recursive in Q iff ( 3 a : N E ) (al P is recursive in a  l Q ) . (ii) P is Vrecursive in Q iff ( ~ : Na E ) (a1 P is recursive in a  1 ~ ) .
os
We cannot immediately define a notion of 3!recursive (in) since if a is a numbering for which a  l P is recursive (in aIQ)then recursive permutations of N will yield new such a's. We must divide the numberings into equivalence classes CU under recursive permutation and write ( 3! ti). Consider functions pl, ..., pm on E. Suppose 3 a l , ..., a,, E E such that every element of E is generated by yl, ..., pm from these elements (i.e., E is the Skolem hull of al , ...,a,, under y l , ..., pm) and suppose the structure is 3recursive. Then it is 3 !recursive. And a necessary and sufficient condition for the structure to be 3recursive is that the set of true equations between Skolem terms be recursive.
Examples. (a) Taking E to be 2' (the positive integers) we can arrange a 11 map of N onto E such that =Gis recursive but successor is not. (b) ( Q (the rationals), +) is 3 !recursive (and the only such CU also makes X recursive). (c) ( R A (the real algebraic numbers), t, X ) is 3 !recursive. (d) (CA (complex algebraic numbers), t,X ) is not 3!recursive, since there exist continuously many numberings which make +, X recursive. But we may obtain uniqueness by adding conjugation to the structure. We write
[email protected])E = E E p (i.e., the set of functions EP + E ) , F E = U F(p)E, PEN
[email protected])E = PEP (i.e., the set of pplace relations on E ) ,
We will see later that Vrecursiveness is equivalent to the simple property of being reckonable (Definition 2.4). Corresponding to Vrecursive in there will be two equivalent notions: ( 1 ) reckonable in (Definition 2.1 l ) , and (2) the notions due to FraissC of Forecursiveness and Frecursiveness (Definition 2.12).
Definition 2.4. Let P E R ( p ) E where E may be denumerable or not. Then: (i) P is radical1,v reckonable if P(x , ..., x p ) '3( x i = xi) on E , where
,
I
RECURSION THEORETIC STRUCTURE
‘3 (xi = xi) is a formula built up by Boolean operations from formulae of the kind x i = xi. (Remark: cw3 (xi = x i ) could be any first order formula in =, since we know such a formula is always equivalent to a propositional one.) (ii) P is reckonable if 3 a,, ..., a, E E such that P(xl, ..., x p ) ‘3 (xi = x i , x i = a k ) on E (i.e., we now allow a finite number of constants of E in the definition). Theorem 2.5. If E is denumerable and P E R E , then P is reckonable
*P is W recursive.
The proof uses only two properties of the recursive relations: (a) they contain the reckonable relations, and (b) they are denumerable. So we could write ‘Velementary’, ‘Vprimitive recursive’, etc. instead of ‘Vrecursive’. We now define O(P) C_ R @ ) Fby O(P) = {7(P) 1 7 : E S F } .

Then P i s radically reckonable iff O(P) = 1 (since P is defined by just one formula so is ~(p)). When E is denumerable, P is reckonahle hut not radically
o(p)
= No. And P is not reckonable iff Q(p) = 2%. This is reckonable iff proved using Bairelike methods. An appropriate 7 can be built by steps, each step involving a choice which gives not only different 7 but different ~ ( p ) .
o(p)= 2%
is one gauge of the richness of the nonreckonable relations P. The next result removes any interest from relative 3recursiveness.
Theorem 2.6.(1) Let R,, ..., R, E R E b e reckonable a n d P E R E w h e r e E is denumerable. Then P is 3recursive iff P is 3recursive in (Rl, ..., R,). ( 2 ) if now there is an R i which is not reckonable, then every P E R E is 3recursive in (R,, ..., Rm).
( 2 ) will follow from Theorem 2.8. First we make a definition. Definition 2.7. (i) P E
[email protected])E is reducible to Q E R(q)E by functions q l , ...,qq E
[email protected])Eiff P ( x l , ..., x p ) ZQ(ql(x1, ..., x p ) ,  9
~
~
(
(ii) Let M g FN. Then P i s Mreducible to Q iff
 ,~ x p1 ) ) 3
9
D. LACOMBE
8
3 q 1 , ..., qq E M such that P is reducible to Q by q l , ..., qq.
Theorem 2.8. If Q ER(q)E is not reckonable, then for every PER(P)E there is a numbering a of E for which alP is Mreducible to a'Q for every class M C: FN satisfying the conditions: (i) M contains all constant functions, (ii) V p , q, 3 q l , ..., qq E M for which (a) for each i, pi is oneone from NP to N , (b) cPi(x17 . . . > x p )2 mm(x1, . .  , x p ) (c) i # I + q i ( N P ) n q , ( N p ) = 8, (d) the complement of ( q l ( N P ) U ... U qq(NP) is infinite. We note that the conditions on M are very weak, and that all the following classes satisfy them : the recursive functions, the primitive recursive functions, the elementary functions, and the polynomials with nonnegative integral coefficients. Since very simple M satisfy (i) and (ii), 3 R (where R replaces 'recursive' in Definition 2.3) is not an interesting notion for most natural reducibilities R . This leaves the more important V  R predicate. We now consider the notions: (i) P is Vrecursive in R ...,R , (cf. Definition 2.3), (cf. Definition 2.1 1). (ii) P i s recursively reckonable from R , ...,R , (iii) P i s Fo (or F)recursive in R ...,R , (cf. Definition 2.12). (i) only has meaning for denumerable E and in this case all three notions will turn out to be equivalent (where finitely many constants may be added to the list R 1 , ..., R , in the third case). For the most part we only need denumerability and continuity of the recursive functionals, but to prove (ii) = (iii) we will use a theorem specifically about recursiveness due to Kleene [3] and Craig and Vaught [ 11. We consider what it means for P to be 'computable' from R , , ..., R , (ER('I)E, ..., R('m)E, respectively). (U, X , , ...,X , ,xl, ..., x P ) will be said to be an instruction for a corn utation if U is a finite subset of E and X , , ..., X , E R('I)U, ..., R('m U respectively, a n d x l , ..., x P E E (but not
P
necessarily E U). Twill be the set of all instructions (assuming the numbersr,, ..., r,, p to be given). LetC', C  g Tand define C = (C+, C). We interpret an instruction (U, X , , ..., X,, xl, ..., xp) E C+ (or ) as saying that if Xi= R j r U each
e
RECURSION THEORETIC STRUCTURE
9
i = 1 , ...,m , then P(xl, ...,x p ) (or 1P(xl , ..., x ) respectively). As the defi nition stands, C may give rise to not only well8efined computations P a t points (xl,..., x p ) , but also contradictory computations or perhaps no computations at all. There is an ‘effective’ procedure by which we can ehminate contradictions, but we cannot simultaneously effectively eliminate indeterminacy of computation. C? determines a partial functional @ewhich is a maping into
[email protected])E of the subset of R(‘1)E X ... X R(‘m)E made of those ( R , , ...,R , ) for which there is neither contradiction nor indeterminacy. Thus @e is continuous with respect to the product topologies on the sets
[email protected])E.In the particular case E = N , if we were to impose the extra condition that e+,C  are r.e. (where T is provided with an obvious canonical numbering) then @e becomes a partial recursive functional and our notion of a computation just gives relative recursiveness. Conversely any p.r. functional can be obtained from two appropriate r.e. sets of instructions P,C!. Also, it is wellknown that every partial recursive functional @ p c (where C = C)) isequal to a functional @e’ where = I) and ’ are recursive. To obtain t h s C’ we replace each instruction ( U , X l , ..., X , ,xl, ...,x p ) in C by finitely many instructions (U’,X i , ...,X k , x , , ...,x p ) , taking as U’a ‘large enough’ superset of U and as the ( X i , ...,X k ) all possible extensions of (XI, ...,x,) to u’. Assume now tnat the only structure on E is given by R 1, ..., R , . We will require C to be ‘symmetric’ and could ask C to satisfy: Suppose there exists an isomorphism (in an obvious sense) of (U‘, Xi,..., Xh, x i , ..., xb) onto (U, X , , ...,X,, xl, ..., x p ) . Then it (U, ..., x p ) € C + (or C’) then (U‘, ..., x i ) + (or  respectively). However, tms simple symmetry condition turns out to be too strong and we only need symmetry with respect to a finite set of constants a l , ..., a, E E.
e’ (e”,e
€e
e’+,e
(e+,
e
(e+,
Definition 2.9. The computation C) is said to be (al, ..., a,)symmetric if the following condition is satisfied: If I , I’ E T and there is an isomorphism of I onto I ’ which leaves each ai invariant, then I E C+ (or C ) implies I’ E C+ (e respectively). The (a1, ..., a,)symmetrization of C is the smallest (al, ..., a,)symmetric C ’ such that C C’. Of cource, this symmetrization may introduce new contradictions. We notice immediately that if ae (as a functional) is invariant undel where each permutation of E which leaves each ai invariant then @e = C’ is the (al, ..., a,)symmetrization of C. We will need the following uniformity result: Lemma 2.10. Assume E, F are denumerable and R
,,..., R , ,P E R E . Let us
D.LACOMBE
10
state on F a denumerable set M of (not necessarily symmetric) computations. Then the condition
implies the condition ( 3 n ) ( 3 e l , ..., en E E ) (  J,..., ~ ~f , E F ) ( ~ C E M ) ( W: E ~ F ) (wl=flA
... A T e , = f , + r P = Q e ( ~ R l ,..., TR,)).
To prove this we assume that the latter is false and build up a T step by step (thus using the denumerability of M) which contradicts the former statement. If C? is (al, ..., a,)symmetric we can abstract from all transforms of a given instruction (U, X I , ..., X,, xl, ..., x p ) E C to a configuration ( V , Yl, ..., Y,, y l , ..., y p , b,, ..., b,) independent of the set E. We can define the set of all such configurations (say J) knowing the numbers r l , ..., r,, p , n . J i s denumerable. Given a pair Q=(CD+,Q) where(2,+,(2, C J , we may define a computation C o n E which is ( a l , ..., a,)symmetric by the set of all mappings of members of(2,+,CD into a structure on E where each bi is mapped onto a i . If for this C we have P = Qe ( R 1 ,..., R,) then P i s said to be (al, ..., a,)reckonable from R , , ..., R , by means of (2,. More formally:
Definition 2.11. (i)Pis ( a l , ..., a,)reckonable from R , , ..., R , by means of (2, iff
(3U(finite) C E ) [(U, R l
r U, ..., R , r U, X I , ..., x P , a], ..., an
is isomorphic to ( V , ..., b,)] and
1P(xl, ..., x p ) the
same formula withCD' replaced by 9
(ii) If we can choose n = 0 in definition (i) then P i s said to be radically reckonable from R 1, ..., R , by means of 9. (iii) Assume that we are given a canonical numbering of J . Then P is recur
11
RECURSION THEORETIC STRUCTURE
sively (al, ..., a,)reckonable from R 1, ..., R , if it is (al, ..., a,)reckonable from R 1, ..., R , by means of some (a = ((a’, (23 ), where (a’, (23 are r.e. (iv) Pis recursively radically reckonable from R 1, ..., R , if it is radically reckonable from R 1, ..., R , by means of some r.e. 0. (v) P is recursively reckonable from R i , ..., R , if it is recursively (al, ..., a,)reckonable from R 1, ..., R , for some 41, ..., a, E E . There is a close link between these definitions and Definitions 2.4. In Definition 2.1 1 (i) we can write (a f. = and replace the conditions given there by the denumerable set {(C,’)},EN where (C,)’ is a condition corresponding to the configuration 1:. The condition (C,’) could be written in the form:
...,u k ) g i ( R 1 ,..., R,,
(3U1,
P(x1,
u l , ..., uk,X I , ..., x p . 0 1 , ..., a,)
...J
where k depends on n and is the cardinal of the finite set V in I;, and where ‘3’ is a Boolean combination of atomic formulas in R 1, ..., R , and =, with u1, ..., uk,x l , ..., x p , a l , ..., a, as individuals. There is a similar condition for (Ci). Recursive reckonability and recursive radical reckonability are transitive relations. We can show, as we did above for the p.r. functionals, that the condition that (i3 be r.e. in the above definitions could be replaced by the condition of being recursive or primitive recursive with the same results. We also note that reckonability by means of finite 0 is not a trivial notion. We now move on to Fraisse’s notions of F, Forecursiveness. We need to introduce a formal language. Our predicate symbols will be pi, ..., p , , T , 01, ...,u,, =. The p’s will be interpreted as R’s, T as P,and the U’S will be additional relational symbols. Let CP be a closed firstorder formula in the language. ( E ’ ,R ..., R ‘ , P’, S;,..., SA)will denote a realisation of this language. We write C2R1t...’m for the class of all models of CP w h c h are extensions (enriched by new relations) of the system (E, R , , ..., R , ) . oC2R1,...*Rmis those realisations of CEiR1,tRm cp for which E = E ’ (we may omit the superscripts for convenience).
;,
Definition 2.12. Assume that there exists a CP in our language satisfying: (1)
(V(E’,R ; , ..., R h , P’, S‘, ..., 5’;) E Ccp)(P”t E = P ) ,
D. LACOMBE
12
Then we say that P i s Frecursive in R , , ..., R,. We notice that (2) excludes the case, for example, in which @ is inconsistent. If there is a @ satisfying (1) and the (original Fraisse) condition:
then we say that P is Forecursive in R ..., R , . For each x E E let us introduce a formal constant 2. The diagram of R i is defined by
and A ( R l , ..., R , ) = A ( R 1 )U ... U A(R,). Then ( 2 ) implies that A ( R 1 ,..., R , ) U {@}is consistent. So if (2) or (20) is satisfied, (1) means that
P(xl, ..., x p ) ++ A(R 1, ..., R , )
U {@}
n(21, ..., 2p)
and
We obtain the notions of Fo, Frecursiveness when we take m = 0. Theorem 2.13. The following statements are equivalent: (a) P is Frecursive, (b) P i s Forecursive, (c) P is recursively radically reckonable. (b) +(a) follows immediately from the definitions. (a) + (c) is also easy. We use the fact that we only make use of finitely many formulae as premises for a logical inference, and so the logical inferences are r.e. (c) + (b) makes use of the strong semantical form of the KleeneCraigVaught theorem [ 3 , 1 ] : For each recursively axiomatizable theory T there exists a finitely axiomatizable extension T' of T with additional predicates such that every infinite model of T can be enriched, without enlarging the domain of the model, to form a model of T ' . Note: the Cp's used in (a) and (b) will not necessarily be the same. It may be that the Cp in (a) cannot be realized for any possible enrichment of E alone.
RECURSION THEORETIC STRUCTURE
13
Corollary 2.14. The relativized notions are also equivalent. (In F , and Frecursiveness the constants a l , ..., a, are added to R 1, ..., R , as monadic singletons {al}, ..., {a,} or as an nadic singleton {(al, ..., a,)}). Examples. (1) Take E = N with the oneplace relation ( 0 )and the successor relation { (x, y ) I y = x t 1 }. Then to be recursively radically reckonable in this system is just to be recursive  but, for example, t is not firstorder explicitly definable on W, 0, successor). On the other hand, to be recursively reckonable from (+, X ) on N is also the same as being recursive, but we can get all arithmetic predicates from explicit definition over (N, t,X ). As for implicit definability, we know for instance that some hyperarithmetic nonarithmetic predicates are implicitly firstorder definable (with no extra predicates) over (N, 0, successor). This illustrates the difference between Frecursiveness and other notions of definability. Actually, Frecursiveness is an instance of Kreisel’s general notion of invariant definability [4]. (2) Take the system (reals, +, X ). Then in the particular case in which P C R (i.e., P i s a monadic predicate on the reals) we get that P is recursively radically reckonable in ( R , +, X ) if P and its complement are each the union of a recursive sequence of algebraic (closed or open) intervals. The situation for nonmonadic P i s too complicated t o state briefly. (3) Let E = N N and let the set of operations on E be given by S = (successor (i.e., a monadic predicate on N N ) , K , L (the inverses of the recursive pairing function), 0 (where ( l o q)(x) = ~(l(x)))}. Then the recursively radically reckonable predicates for this system are the hyperarithmetic predicates (on N N ) . We obtain a similar result if we replace N N by the set H of all hyperarithmetic elements of N N (we use Spector’s result [7] on quantification over H): on H, the predicates recursively reckonable in the product 0 are the predicates hyperarithmetic (on H). But since H i s denumerable the hyperarithmetic predicates on H are also the predicates Vrecursive in 0 . 11
3. Numbering a! : N + E have been considered so far. The objects we are really interested in are the equivalence classes of ‘recursively indistinguishable’ numberings. Definition 3.I . A numerotage is an equivalence class of numberings under the equivalence relation: a! E a!’ iff (~101’ is recursive. We can extend our notions to the case where E is finite by considering any function defined on a finite set to be recursive. Thus, on a finite E there will be just one numerotage. More generally, we have:
D. LACOMBE
14
Definition 3.2. Assume a,a’are enumerations N + E of a denumerable set E . We say: (1) a is (recursively) reducible to a’iff there is a recursive function 8 such that a = a‘8, ( 2 ) a, a’are interreducible if a is reducible to a’and a’is reducible to a. An equivalence class determined by interreducibility of enumerations is an enumerage, ( 3 ) a is isogenic to a’ iff there is a recursive numbering (i.e. a permutation) of N , 8 say, such that a = 01’0. There are cases in which notions (1) and ( 2 ) are equivalent, but this is not true in general. Definition 3.3. Consider the following diagram:
N X “1
....... X N  N
1 x ....... XaPlE
El
P
7
PF1
(1) We say cp represents @ with respect to a l ,.... a p ,p iff
(i.e., if the diagram commutes). We say Q is (recursively) representable with respect to a1 ..... a p ,0. (The cp need not be unique since P need not be oneone.) ( 2 ) cp is intrinsic iff
( V x l , .... x p , x i , .... x b E N ) ( a l x l = a l x ; A
... A aP x P
=
So cp is intrinsic iff cp represents some @. We see immediately that the notion of recursively representable with respect to a l ,.... ap, is invariant under interreducibility of these enumerations  and so is a property of enumerages rather than enumerations. When E l = ... = Ep = F = E and a 1 = ... = aP = 0 = a,we can say that @ is representable with respect to a (or with respect to the enumerage (Y). We write Rep(a) (or
[email protected])) for all such @ representable with respect to a (or Z).
RECURSION THEORETIC STRUCTURE
15
Theorem 3.4. A set of functions M =
[email protected]) for some CU iff M satisfies: (1)Mis countable, (2) all constant functions and projection functions are in M, and (3) M is closed under substitution. It follows immediately from M =
[email protected]) that M satisfies (1)(3). And given an M satisfying (1)(3), it is easy to show that M C
[email protected]). The last part is more difficult. Given the denumerable M we build up CU (or rather the enumeration a) in such a way that all members of M are representable with respect to a but only members of M are representable with respect to a . The construction is Bairelike. At the nth step we work with the nth recursive function (of any number of places). It will follows from the way in which a is constructed that the nth recursive function either is not intrinsic or recursively represents something obtained by superposition from M.(Contrast the 11 case where nor all structures on E were 3recursive.) This theorem suggests that we might refine the problem by imposing new conditions on 5. We will say that a relation R C E P is recursive with respect to a iff a  l ( R ) = { ( x l , ..., x p ) I R ( a x ...,a x , ) } is recursive. Similarly R is r.e. iff a  ’ ( ~ )is r.e. Even the simplest relations may not be recursive with respect to a given a. In particular the equality relation is not in general recursive with respect to a. Definition 3.5.(1) CU is pure iff = is recursive with respect to a. (2) ii is semipure iff = is r.e. with respect to a. The definitions make sense since both notions are invariant under interreducibility. If E is infinite, there is a correspondence between pure enumerages on E and numerotages on E. This is because CU is pure iff CU = (Y’ for some numbering a’.If E is finite then there is just one semipure enumerage on E and this is also pure. All the general definitions have meaning and interest for finite E. For in
E
stance if = 2 there is still a richness of enumerages. An enumeration of E becones a Iplace predicate on N a n d our reducibility is the classical manyone reducibility. Given E (where 2 < < No) write Enum(E) = the set of enumerages on E.
E
So G ( E ) = 2No. The quotient relation of reducibility by interreducibility given an ordering on Enum(E). This ordering is analogous to that on Turing degrees. For example we can prove that it is an upper semilattice which is not
16
D.LACOMBF
a lattice, and that (following Spector [8] ) an increasing denumerable sequence of elements has no least upper bound. But if E is infinite Enum(E) has many minimal elements  unlike the luring degrees for which 0 is the unique minimum. For example any semipure enumerage is minimal, since if a is semipure and a‘ is reducible to a,then a is reducible to a’.The semipure enumerages do not exhaust the minimal elements since there are at most countably many of these, and using Bairelike arguments we can obtain 2’0 minimal elements. There are other interesting kinds of elements of Enum(E). For instance: Definition 3.6.5 is homogeneous iff a l , a2 E ti implies that a1 is isogenic to a2.
Examples. (a) H. Rogers Jr. [6] showed that on the set of partial recursive functions the enumerage consisting of all standard ‘numberings’ is homogeneous. (b) It is easy to show that for a group G to possess a semipure enumerage compatible with the group operation is equivalent to it being ‘recursively presentable’  that is, is the quotient of a free group by the normal subgroup generated by an r.e. subset. If G has a finite set of generators then there exists at most one semipure ii compatible with the group operation. If C is recursively presentable but not 3 recursive, then the semipure Z compatible with the group operation is homogeneous. We can form a category in which the objects are the enumerages on all finite or denumerable sets and the morphisms (E, Z) +(E’,E’) are those mappings 0 : E + E’ which are representable with respect to Z,Y’. We use N with the structure given by the recursive functions as a fixed auxiliary set, in the same way, for instance, as we use the field K in the category of vector spaces over K . There are several types of problem arising from the category of enumerages. There are general category problems. For example, it is easy to define the following operations: direct product of finitely many enumerages, homomorphic image of an enumerage under a mapping, quotient of an enumerage by an equivalence relation, and induced enumerage  that is, the restriction of an enumerage CU on E to some subset E‘ of E. This last notion is not always defined. One can show that a necessary (but not sufficient) condition for it to be defined is that E’ be r.e. with respect to ii, in the weak sense that E’ = cp(N) for some cp : N + E recursive with respect to E . And a sufficient (but not necessary) condition is that E’ be r.e. with respect to 5,in the strong sense that CU~(E’> is r.e. Another approach is to look at interesting subcategories. We say that an
17
RECURSION THEORETIC STRUCTURE
enumerage 5 on E is autogenic if the set of all 5recursive mappings of N into E can be enumerated by an 5recursive mapping of N2 into E (where by %recursive we mean recursivelyrepresentable with respect to (T, Then, for instance, most classical results on standard numberings of partial recursive functions can be extended to the class of autogenic enumerages.
a)).
References [ 11 W. Craig and R.L. Vaught, Finite axiomatizability using additional predicates, J. Symbolic Logic 23 (1958) 289308. [ 21 R. FraissB, Une notion de rkcursiviti relative, in: Infinitistic methods, Proceedings of the symposium on foundations of mathematics, Warsaw, 1959 (Pergamon Press, 1961) pp. 323328. [ 31 S.C. Kleene, Finite axiomatizability of theories in the predicate calculus using additional predicate symbols. Two papers on the predicate calculus, Mem. Am. Math. SOC. 10 (1952) 2768. [ 41 G. Kreisel, Model theoretic invariants: applications to recursive and hyperarithmetic operations, in: The theory of models, Proceedings of the 1963 international symposium at Berkeley (NorthHolland Publ. Co., 1965) pp. 190205. [5] H. Rogers Jr., Godel numberings of partial recursive functions, J. Symbolic Logic
23 (1958) 331341. 161 C. Spector, Recursive wellorderings, J. Symbolic Logic 20 (1955) 151163. [7] C. Spector, On degrees of recursive unsolvability, Ann. Math. 6 4 (1956) 581592.
MEASURABLE CARDINALS J.R. SHOENFIELD Duke University, Durham, N.C., USA
These lectures are an introduction to the role of large cardinals in the solution of problems in set theory not solvable by conventional methods. The reader is assumed familiar with the development of ZFC (ZermeloFraenkel set theory with the Axiom of Choice) through elementary cardinal arithmetic and with the elements of model theory. Some notation: u, v and T are ordinals; K , A, and p are infinite cardinals; PS(x) is the power set of x ; Ix I is the cardinal of x;yx is the set of mappings from x t o y . Why can large cardinals solve problems which cannot be solved in ZFC? We recall that every set belongs to some R (u), where R ( 0 ) is defined (using transfinite induction) by
R(O)=0 , R(o+l)=PS(R(a)),
R(o) =
u
R(T)
for u a limit number.
7
KK
& f i s not almost constant] .
Let i be the identity mapping of K into K . Since singletons have measure 0, i is not almost constant. Thus K *  K # 0. From this and K < K * :
Lemma 3. K
w and I B I > 2 , then no generic mapping is in M.(Hint: For every f, [p Ip f ] is dense.) Existence theorem. For every p , there is a generic f such that p C f. Outline of proof. Let D o , D , , ... be the dense sets in M ,and extend p step by step to include a member of each D i . Now fuc a generic f. Define a relation Ef on M by a Ef b f, 3p(p
c f & (a, p) E 6 ) .
36
J.R. SHOENFIELD
Then Ef is wellfounded ;so we may define a mapping K f on M by
We then set
It is easy to show that for each a there is an 2 such that K f ( 2 ) = a for all f. Moreover, the operation  is definable in M. It follows that M C M [ f ] . It is also easy to define an F E M s u c h that K f ( F ) = f f o r allf; s o f € M [ f ] .
Fundamental theorem. If f i s generic, then M [ f ] is a countable transitive model of ZFC including M and containingf. We shall not give the proof of the fundamental theorem; but we will sketch some notions used in the proof which are essential for further developments. We add to the language of ZFC a constant a for each a E M . The resulting language is called the forcing language. It is used to describe M [ f ] , with the variables ranging through M [f]and the constant a designating K f ( a ) . Thus the language is independent o f f ; but the meaning of the language depends upon f. Let @ be a sentence in the forcing language. We write k M [ f l @ if @ is for true in M [ f ] . We say that p forces @, and write p I+ @, if FMIfl every generic f such that p C f. There are three basic lemmas on forcing.
Extension lemma. If p H @ and p C q , then q H @.
Truth lemma. Iff is generic, then
[email protected] iff 3 p ( p C f a n d p H
a).
Definability lemma. Let @ ( x l ..., , x n ) be a formula of ZFC. Then there is a formula *(j, z , x 1,..., x n ) of ZFC such that
The definability lemma shows that in defining sets in M we may make use of forcing. We often use this fact tacitly. The fundamental theorem is useful only in conjunction with other results
MEASURABLE CARDINALS
37
connecting M a n d M [ f ]. To illustrate, we sketch the construction of a model in which 2tso 2 Hz.Take A = w X B = (0, 1 }. Choose a generic f. For u < Hy,
#,
x , = [ i l i E w &f(i, u ) = 01
is a subset of w in M [f]. It is easy to show (using the genericity o f n that these sets are distinct. Thus in M [ f ] there are > H y subsets of w . To complete the proof we need to know that H Y = Hylfl . This is true, but the proof is nontrivial. Now suppose our basic set theory is ZFM. Then M will be a model of ZFM. We need to know that M [ f ] is also a model of ZFM. This will not always be true; but it will be ifA and B are small.
Theorem 9 (EvySolovay [ 141). If K is measurablp and IA IM IB IM < K , then K is measurabl& [fl .
< K and
Proof. A simple calculation (in M) shows that I CIM < K . Suppose that in M, p is a measure on K . For x C K and x E M [ f ] , we define p’(x)=O
if 3 a ( x C a a n d p ( a ) = O ) ,
p’(x)= 1
if 3 a ( a C x a n d p ( a ) = 1 ) .
It is clear that p’ E M [ f ] . We claim that in M [ f ] ,p’ is a measure on K . We shall show that: (a) if x C K and x E M [ f ],then p’(x) is defined; (b) in M [f],the union of < K sets of p’measure 0 has p‘measure 0. The desired conclusion then follows easily. For (a), let a designate x (i.e., let K f ( a ) = x ) . For p E C define
d p = [ U l U < K & p tt6Eal . Then d, E M , and the function mapping p into d, is in M ,so that u d, is PC, in M. We have
For if u E d,, then p it 6 E a and hence kMIfl6 E a by the truth lemma. But this means that u E x . If u E x , then b M [6f E l a ; so by the truth lemma,
38
J.R. SHOENFIELD
there is a q C fwhich forces 13 E a. We may suppose p C q ; otherwise we replace q by p U q and use the extension lemma. Hence u E dq C u dq. In view of (S), it will suffice to prove 3 p ( p C f a n d p E D ) p c q where
Sincefis generic, it suffices to prove that D is dense and inM. Clearly D E M . Let p E C;we must findq E D w i t h p C q . I f p ( u d q ) = 0, Pcq
pED;sowemay takeq=p.Nowletp(
u
d q ) = 1.Since I C I < K a n d p i s
Pcq
Kadditive in M , p ( d q ) = 1 for some q with p C q . Then q ED. For (b), let g be, in M [ f ], a mapping from some u < K into PS(K). We must show that either p ’ ( g ( 7 ) )= 1 for some r or p’( u g(7)) = 0. Let dP,,be T K + (where K is measurable); for to obtain this i n M [ f ] we must take Cso that lCl* > K . Theorem 9 (in the generalized form) is sometimes stated: measurability is
MEASURABLE CARDINALS
39
preserved by mild Cohen extensions. It has been shown that almost all the large cardinal properties which have been considered are preserved by mild Cohen extensions; so these large cardinals do not suffice to settle CH. It has been known for a long time that the existence of a measurable cardinal settles many mathematical problems; see [ 101 for examples. In all of these cases, the existence of a measurable cardinal makes the situation more complicated. Mathematicians would be more inclined to accept measurable cardinals if these cardinals had 'nice' mathematical consequences. We shall now examine some such consequences. Since ww is the product of w copies of w , we can give it a topology as follows: give w the discrete topology and then give owthe product topology. (With this topology, ww is homeomorphic to the irrationals.) We use a , 0, y for elements of o w . We identify ( w " ) ~ with ww by identifying (a,0)with the y defined by y(2n) =
[email protected]), y(2n+l) = P(n). For X C o wProj , ( X ) is the projection of X , on its first coordinate. considered as a subset of We say a subset X of ww is: (a)
Z; if it is open;
(b)
Zk+l if it is Proj(Y) for some Q1 set Y;
(d)
A: if it is both Z i and Hi.
N te that this is a proper definition by induction on n . A set which is 2; or for some n is called projective. The classification of projective sets given by (1)  (4) is called the projective hierarchy. For orientation, we remark that, by a theorem of Souslin, A; sets are identical with Bore1 sets. It is known that sets low in the projective hierarchy have certain nice properties. Solovay has shown that in ZFM we can prove sets a little higher in the hierarchy have these nice properties. The nice properties are: (1) being measurable for all reasonable measures; (2) being countable or including a perfect subset; (3) having the property of Baire. We shall consider only (3). A set X in o w has the property of Baire if there is an open set Y such that X A Y (which is by definition ( X  Y ) U (YX)) is of the first category. This property is similar in many ways to being measurable. (See [ 131 for this and much further information about pro'ective sets.) Lusin proved that every z: or set has the property of Baire (see [ 131).
n,P
nl1
J.R. SHOENFIELD
40
This is the best possible result in ZFC; for results of Godel show that if V=L,then there is a A: set which does not have the property of Baire. We prove in ZFM that every C i or 11; set has the property of Baire. This is again best possible, since results of Silver show that if V=L,, then there is a A$ set which does not have the property of Baire. We need the following result, whose proof is like that of Theorem 7.
7'heorem 7u (Rowbottom [21] ). If K is a measurable cardinal, w a n d o l E w W , t h e nI R ( a ) n L a I G l a l . Corollug. If countable.
K
0 there is a total S," E Fm+lsuch that for all x j l ,...,x m we have
Definition 1.3. An enumerative system (ES) is a pair (D,F)such that (1) D has at least two elements, (2) F is a collection of Q 2ary partial functions on D , ( 3 ) F is closed under generalized composition, (4) Xx(x) E F, and there is a pairong mechanism (P,fl,f2); i.e., P E F 2 , P 11 and total,fl, f2 E F 1 , a n d for a l l x , y E D , f l ( P ( x , y ) ) = x , f 2 ( P ( x , y ) ) = y , (5) each Cj, x E D , is in F ,
116
H.M.FRIEDMAN
( 6 ) for each z, w E D , z f w, E, E F2, where Ez,w = Axy ( z if x = y ; w if x # y ) , ( 7 ) there is a @ E F2 such that F l = {hxl(@(a,xl)) :a ED}. We now prove a kind of mutual interpretability between ES and BRFT, the first half of which is due to Strong and Wagner, the second half due to us. This theorem is of independent interest since ES is a very elegant functorial condition on monadic and binary partial functions.
Theorem 1.1. Let (D,F,@,,)be a BRFT. Then (D,F1 U F 2 ) is an E S . Let (D,G) be an ES. Then there is a unique adequate (D,H) with H l = G l ;furthermore, for this unique (D,H) there exists a BRFT (D,H,JI,), and H 2 = G 2 . Proof. Let (D,F,@,) be a BRFT. Choose e E D such that for all b,c,a,x, = fix y f e. Then (Xbc($(e,b,c)), is a pairing mechanism for (D,F). Clearly Ez,w = Xz(@2(z,e,e));Xz(@2(~,e,y))) hxy(a(x,z,w,y)).So (D,F1 U F2) is an E S . Let (D,G)be an E S . Let (P,fl,f2) be a pairing mechanism for (D,G),@ as in (7). Then the following notation will be used throughout this paper: DefineP1 = W X ) , ~ ; = ~ x ( x ) , ~ k =+~l 1 .  x k + l ( ~ ( xP1 (, x 2 , .  , ~ k + l ) ) ) , =f1, = Xxyl"Q.(x))), 1 < i < k . Then eachfiitE G l ,and fi (Pk(xl ,..., x k ) ) = x i . Furthermore, eachPk is total and 1  1 . Define H , = {hx,...xn(g(Pn(xl,..., x,)): g E G, }. We now check the adequacy of ( 0 , H). To check closure under generalized composition, let h E H , ,fl1, ...,fl, E Hm Set h ( x 1 ,..., x , ) ~ ~ P , ( x ..., l , x,)),P~(xl,. . . , x m ) ~ g ~ ( P m ( x l , . . . , x mThen )). h(Pl(x1,..rxm),.., Pm(X1 ,*.)x m ) ) g(P,(gl(Pm(xl~***~ Xm)),..., gn(Pm(xl,...,x,))). Set f = Ax('g(P,kl(x), ...,g,(x )))). Clearly f i s obtained by generalized composition, from P,g1 ,..., g,, g, and so is in G 1 . So h1 ,.,xm(hCOl(xl,.*.,xm),.,P,(xl ), Xm))) E H m . To check that H contains the projection functions, note that u,"(xl,..,xm) =f,"(Pm(xl ,**,xm))To check that H contains each C," ,note that G l contains each C i . To check that OL E H , we need to use @. Choose z , w such that ha(@(z,a)) = f l , Xa(@,(w,a)) = f 2. Then z f w. So Ez,w E G 2 .We must produce (4(y)), y), such that a = Xabcx(g(P4(a, b, c, x ) ) ) .Choose g = h ~ ( @ ( E ~ , ~ @ffEG1 &A))). Clearly g E G, Obviously H I = G,, H2 C G2. To show H 2 = G2, let h E G,. Then h ( x l , x 2 ) g(P(xl, x z ) ) ,w h e r e g = W W 1 ( x ) , f 2 ( x ) > )Clearly . g E G,. @ ~ ( ~ , ~ , C , L I , . Xa(a,b,c,x). ) Now
frl
WCv),
AXIOMATIC RECURSIVE FUNCTION THEORY
117
Uniqueness of (D, H): In fact, we show that if (D, F) and (D, C ) satisfy (l), (2), (3), (4) of Definition 1.1, and have a pairing mechanism, then F , = GI + F = C . To see this assume Fl = G l and let (P,f l , f 2 ) be a pairing mechanism for (D, F).Let g E G, . Let h = h X @ ( r ( x ) ,...,f/(x))). Then since each EF, we have that h E G . Hence h E F. Note that Axl ...xm(h(P,(xl ,...,x,))) = g. However, it is obtained by generalized composition from h, P, and projection functions, and so lies in F. So G C F. By symmetry, F = G . This completes the proof of uniqueness. For the existence of a BRFT, (D, H ,,),)I define @,(x,x 1,..., x,) = @(x,Pn(xl,..., x,)). To see that @ enumerates H , , let h E H , . Let g E G l have h(xl ,..., x,) g(P,(xl ,...,x,)). Let a be such that Ay(@(a,y ) ) = g. Then clearly @,(a, x1,..., x,) N h(xl ,..., xn). However, we cannot conclude the existence of S r functions for these enumerations. So we will define new enumerations $, in terms of the old ones by IJ~(x, xl,..., x,) = @,+l(fl(x), f2(x),x1, ..., x,). To see that the J / are,still enumeration functions for (D, H), let h EH,. Then choose b such that @n+l(b, x , x1,..., x,) N _ h(xl ,..., x,). Then clearly $,(P(b, b), x1,..., xn) = h(xl ,..., xn). To complete the existence, we have to find S," functions for the $, . Let x, x1,..., x, be fmed elements of D. We wish to define S,"(x,..xl ,..., x,). That is, we wish to find a in terms of x, x1,..., x, such that $,(a, y l ,..., y,) = $,+,(x, x1,..., x,, y 1,..., y,). In other words, we wish to find 6, c such that @,+1(b,c, y1,..., y,) = $n+m(X, X I ,.... x,, y1,..., y,) and then set a = P(b, c). The first step is to associate an element of G,+l with , ,$ by considering h = Ayy l... yn(J/,+,(fYf1Cy) ,..., fgz:Cy),yl ,..., y,)). Choose d such that h = Ay(@,+l(d, y , y1,..., y,)). Then clearly @,+l(d,P,+l(x,xl,...,Xm), y1*.*, Y,)? $n+m(Xp x 1 ~ . * * ~ylv..., x m ~Y,) SOwe set ~ = P ~ ( d , P , + ~ ( x,..., , x ~ ~ ) ) = P , + ~ ( d , ,..., x , x,).Thuswe x~ defines," = X~x~...x,(P,+~(d, x , x l , ..., x,)). The choice of d was independent of x, x,, ..., x, . In connection with Theorem 1.1 and its proof, we remark that the Smn condition on BRFT cannot be derived from the rest of the conditions. For by the recursion theorem for BRFT (see Theorem 1.6) it is easily seen that for ), there is an a f b such that Ax(@1(a, x)) = Ax(@,(b, x)). any BRFT (D, F, @ So if D = a,F is the class of partial recursive functions of many arguments on a,@, are enumeration functions for (w,F),and @1 is an enumeration without repitition (see Friedberg [2]), then (D, F, Gn)is not a BRFT.
Definition 1.4. Let (D, F, @),
be a BRFT. Then we set
[XI
= Ax l... xk
H.M. FRIEDMAN
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Definition 1.5. Let (D, F, 4,) be a BRFT, 0 < r , 0 <mi, 0 < k , G a partial function from Fml x... XFmk into F,. Such G are called functionals. Then we write PRep ((D,F, @,), e, g ) if and only if for all x 1,..., x,, if G ( [ X l ] m ,..., l [ X k l r n k ) is defined, then its value must be [@k(e, X I , ..., Xk)],. We write TRep ((D, F, @,) e, g ) if and only if PRep ( ( 0 , F, @,), e, G ) & e E Tk. We say G is partial representable if and only if ( 3 e ) (PRep ((A,F, @,),e,G)). The idea behind the notation is that PRep mean “partial represents”, TRep mean “total represents”. Theorem 1.2. Let (D, F, @), be a BRFT, G: Fml X ... X Fmk Fr, 1 < r, PRep ((D,F, @,), e, G). Then there is a y such that TRep ((D, F, @, Jy, G). +
Proof. Consider Axl ...x k y l ... yr( [ [e ]k (x l,...,x k ) ] r ( y 1 ,..., vr))[x]k+r.Define XI ,...,X k ) = S,k ( X , X I ,..., x k ) . Choose b ] k =f. Then TRep ((D,F, @,), J’, G). Definition 1.6. Let (D, F, 4), be a BRFT. Let h EF,, 81,...,g, E F , ,0 < n , 0 < m. Then by h(gl ,...,g,) we mean that element of F, obtained by generalized composition, remembering the convention Fo = D. Theorem 1.3. Let (D, F, @), be a BRFT. Then (a) each functionalH: Fml X ... X Fmk + Fr given by H ( x l ,..., x k ) = x i is partial representable, (b) each functional H : Fml X ... X Fmk + F,. given by H ( x l ,..., x k ) = y is partial representable, (c) ifH, G l , ..., G, are partial representable functionals,H: F m lX ... X Fmk + F,., Gi: F,, X ... X F,, + Fmi, then C: Fnl X ... X F,, + F,. is partial representable, where G(x l ,..., x,) = H(Gl (x l,..., x,) ,...,Gk(x1,..., x,)), (d) ifH, Gl, ..., G,. are partial representable functionals,H: Fml X ... X Fmk F,., Gi: F1 X ... X Fmk + F p , then G : F, X ... X Fmk F, is partial representable, where C ( x l,..., x k ) = H ( x l,...,x k ) ( G l ( x l,...,x k ),..., G, (x l,...,x k ) ) in the Sense of Definition 1.5. +
+
Proof: (a), (b) are obvious. For (c), let PRep ( 0 , F, @,), e, H), PRep ((D, F, @,), qc Gi). Then if [ a ] , = [e]k([ql]p,...,[qk],) in the sense of Definition 1.6., then PRep ((D, F, @,),a, C). For (d) let PRep ((D,F, @,), e, H), PRep ((D,F, @,), qi, Ci). It suffices to find a n f E F H 1 such that f i s total and [ f ( X 1 ,...,X,, X ~ l ) ] =k [ X H 1 ] , . ( [ X l ] k , . . . , [X,.]k). Towards this a h , choose a
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We will make use of the following result later. Theorem 1.4. If (D, F, @,), (D, F, I),) are BRFT, then for each 0 < m there is a n f E F l such that f i s total and for all x , x l ,..., x , E D we have @ m ( X , x l , . * * , X , ) eI ) , ( f ( x ) , x 1 , *   , x m ) .
Proof: Choose u E D with
= .$ ,
T a k e f = hx(SA(u, x ) ) in (D, F, I),).
Definition 1.7. Let D , E be sets, G a total 11 monadic function from D onto E . Then we take T(G, f),for f a partial nary function on D , to be given by G ( f ( x l,...,x , ) ) = T(G,f)( G ( x l ),..., G(x,)). T(G, f ) is an abbreviation for “the translate” o f f b y G. When G is onto and total,f(T(G, f)) is a 11 total function from uD P ~ onto ) uE(ESpreserving application. k
k
Definition 1.8. Let (D, F, $), and (E, G, I),) be BRFT’s. They are called piecewise isomorphic if and only if for each m there is a 11 onto total function H : D +.E such that for all x E D , T(H,[x]m ) = [H(x)] ,where the second brackets refers to (E, G, I),). If we wish to use the [ notation for two BRFT’s at once, we use [ for the first and { instead of [ for the second, from now on.
,
After Rogers [7] we take Definition 1.9. Let (D, F, I),), (D, F, I),) be BRFT’s. Then they are called piecewise Rogers isomorphic if and only if for each u there is a 1 1 onto function G: D +. D such that for all x E D , [ X I , = { G ( x ) } , , and C E F1. (D, F, 4), is piecewise Rogers imbedduble in (D, C , I),) if we require G to be a 11 function, but not necessarily onto. Definition 1.10. We write 1isomorphic, Rogers 1isomorphic,Rogers limbeddable, just in case the respective definitions above hold for n = 1. The notion in Definition 1.7, in contrast to those in Definition 1.8, is connected with the usual notion of modeltheoretic isomorphism. For this reason, if two BRFT are piecewise isomorphic then they are elementarily equivalent with respect to an appropriate rich language, described below. We may think of a BRFT (D, F, @ ), as a relational structure
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(D, F1, F2 ,... ; A 1, A 2 ,...; $1,$2 ,...),where there are infinitely many disjoint domains D, F1, F2,..., and infinitely many it2ary relationsAi, and infinitely many constants $i E F i + l ,with the stipulation that AiU,xl, ..., xi, y ) is wellformed just in case f € Fi, xl, ..., x i , y € D. The intention is that A i ( J x1,...,xi, y ) if and only iff(x1, ..., x i ) = y . For each n let I.(,, be the similarity type of manysorted structures (D, F l , F2,...;A A 2 ,...;G,,). Let L,, be the language LmF based on p,, only. Finally let L be the least language closed under arbitrary conjunction and disjunction, and negation, and containing each sentence in each L, .
,,
Theorem 1.5. If (D, F, $,,), (E, G, G,,) are piecewise isomorphic, then (0,F1,...;A1, ...;$i,...)EX(F, G I ,...;A 1 ,...; GI,...). Proof. One merely verifies that for each m ,and total 11 onto H : D += E with for all x E D , T ( H , [XI), = { H ( x ) } , ,there is an extension of H to a modeltheoretic isomorphism from (D, Fl ,...;A ,...;4,) onto (E, G l ,...; A 1,...; G,). To see this, takeH*(f)= T(H,f) f o r f E F . Clearlyf is total and ll,H*($,)= $ , , H * extendsH. It suffices to prove Rng ( H * ) = G U E . To see this, note that (E, Rng (H* r F ) , H*(@,)) is a BRFT. By the part of Theorem 1.1 concerning uniqueness, it now suffices to prove that Rng (H* r F 1 )= G l . Clearly if m = 1 we are done. If 1 < m then l e t f € F 1 and s e t g = A x l... x,(f(xl)).ThenH*(g)EG,. ButH*(f)= h x l ( H * ( g ) ( x l , x l , ..., x l ) , a n d s o H * ( f ) E G . I f f E G 1 then s e t g = Ax ,... x, (f(xl)). Then (H*)l(g) E F,. But ( H * )  ' ( f ) = Ax, ( ( H * )  l ( g ) (xl,xl, ..., xl)), and s o f E Rng (H* F 1 ) .We are done.
We d o not know whether 1isomorphic, Rogers 1isomorphic, Rogers 1imbeddable are equivalent to, respectively, piecewise isomorphic, piecewise Rogers isomorphc, Rogers imbeddable. The recursion theorem, in the language of BRFT, says that for each 0 < k and each total f E Fl there is an x E D with [ x ] ~= If(x)]k. Not only is this true in all BRFT, but the stronger effective version is true. Theorem 1.6. Let (A, F, $,,)be a BRFT, 0 < k . Then there is an R, E F , such that for allx E T l we have [Rk(x)lk = [Gl(x,Rk(x))]k.
Proof. Fix e E D such that for all x, y , x1,..., Xk E D we have 4,+2(e, y , x, XI,.. .,x k ) N 4k (41 (y, 1 (x. x)), x1 ,...,xk). Hence for all y , x , x l , . . . , x k E D w e h a v e 4 , + l ( S k + l ( e , y ) , ~ ,,..., ~ 1 xk)= 4, (41 (y, S,1 (x, x)), x ,...,x,). Clearly for all y , x ,...,x k E D we have
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Proof. Upon examination of the proof of Theorem 1.2, we see that it can be made “effective”. Thus, let HI : Fl + F,+l by H1 (g) = h x l y l ... y, XI)], 011,..., y,)). Then by Theorem 1.3,Hl is partial representable. Hence by Theorem 1.2, ( 3 q ) (TRep (0,F, q , H1). Fix this 4. Fix a with [a] 2 = hxy(Sk (x, y)). Then whenever PRep ((0,F, Q,,), e, H ) , we have TRep ((D, F, @,,I,Si (a,[crl1 (el), HI. Hence whenever PRep ((D, F, Qn), e, H I , we have that H( [ R , ( S l (a,[q] (e)))],) is either undefined or is [R, (Sl (a,[q] (e)))], by Theorem 1.6. Hence set FP, =
[email protected], (S: (a. @ 1 ( 4 9 e)))). FP stands for “fixed point”.
en),
Theorem 1.8. Let (D, F, @,) be a BRFT, and let 0 < m , 0 E D . Then H: F, X F, F, given by H(f, g) = h x ,... x, (f(xl, ...,x,) if x, = 0; g(xl, ..., x,) if x, # 0) is everywhere defined and partial representable. +
A word of caution about H above is in order. Note that H( f, g) is not, in general, a ( C r , f, g, U,“), since the latter will always be undefined at any arguments at whichfis undefined.
Section 2
In this section we first present a new minimality theorem for the partial recursive functions which we derive immediately from Wagner and Strong’s original minimality theorem via Theorem 1 . l . We will exposit a proof of Wagner and Strong’s result. In so doing we will establish key lemmas necessary for the second part of this section, which concerns relative categoricity properties of BRFT; roughly speaking, any two BRFT on w which contain
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Xx(x + 1) (aBRFT's) and which have the same partial functions are piecewise isomorphic. However, rather than proceeding directly in the second part with the relative categoricity result, we first will develop a general sufficient condition for piecewise isomorphism, and use this to derive the relative categoricity result. In the third part we classify those sets of total functions on w which are the total functions present in some wBRFT. Definition 2.1. An wBRFT (wES) is a BRFT (ES), (a,F, 4), ((w, F ) ) , such that Xx(x + 1) E F,. We will use the convention n  m = 0 if n < m . Theorem 2. I . (Wagner and Strong). If ( a , F, @), is an wBRFT then PR (w) C F , where PR (w) is the class of partial recursive functions of many arguments on w.
Proof. Fix (a, F, @,) as an oBRFT. It suffices to prove (a) wheneverfEF,+l, O < n , f t o t a l , t h e n p f = Ax l... x,(py(fb,x l,...,x,) = 0) if ( 3 y ) (fb, xl ,..., x,) = 0 ) ; undefined otherwise) E F, . (b) whenever g E F,, h E Fm+2,0 < m, h, g total, and f = Ax1... x,+l('g(xl ,...,x,) ifx,+l = 0; h(xl ,..., x,, x,+l,f(xl ,..., xm+1 1)) if x,+~ # 0), then f = PRIM ('g, h) E Fm+l. The partial functionals PRIM, p are defined only on total arguments. Towards (a), we need to consider H : F,+l + F,+l given by H ( g ) = Xyx1... X, ( O i f f O , x l ,..., x , ) = O ; S ( g ( S b ) , x l ,...,x , ) ) i f f O , x l ,..., x,)#O)). By Theorem 1.3 and Theorem 1.7 and Theorem 1.8 we have that H has a fixed point, h E F,+l. We wish to prove Ax ,... x, (h(O,xl ,..., x,)) = pf. If p f x ,..., x,) = z then clearly g(z, x ,..., x,) = 0, and hence g(0, x ,..., x,) = z, sincefis total. If pf(xl, ..., x,) is undefined then clearly g is nowhere defined. Towards (b), we first note that hx(x  1) E F since Xx(x  1) = p i where f(x, y ) = 0 if x = 0,y = 0; 0 if Sb)= x; 1 otherwise. Then define H : F,+1 + F m + 1
[email protected])=Xxl... x,+l(g(xl ,..., x r n ) i f x m + 1=O;h(x1,...,x,, Xm+l, p(xl...., x,+~  1)) if x,+~ # 0). Then by Theorem 1.3 and Theorem 1.7 and Theorem 1.8, there is a fixed point for H . But any fixed point for H is PRIM ('g, h) and lies in F . Upon examination of this proof, we can obtain a stronger theorem of this kind which will be useful later. Theorem 2.2. Let (w, F, @,)be an wBRFT, and let 0 < n , 0 < m. Then the partial functionals PRIM: F, X Fm+2 + Fm+2 + Frn+l and p: F,+l + F, are defined on and only on all total arguments and are partial representable in (0,F, @,).
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Proof. ConsiderH*(f,g)= hyx l...x, (O.iff(y, x1,..., x,)= 0; S(g(Sb),xl,..., x,)) if fb, x1,..., x,) = q for some q # 0; undefined otherF, @,), e, H * ) by Theorem 1.3. If a € Tn+l then wise). Then let PRep ((a, d [ a I n + 1 ) = [S,l(
[email protected],l(e, Q)),O)ln. Consider H**(g, h, 0) = h x l...x,+~ (g(xl ,...,x,) ifx,+l = 0; h(X1 ,...,X,,x,+l,P(xl,...,~,+l  I))ifx,+i ZO). Let PRep ((a, F, @,), e, H * ) by Theorem 1.3. If a E T,, b E Tm+2 then PRIM ([a],, [b1m+2)= [FP(s:(~, Q, b))1,+1. U PR2 (a) C F. Theorem 2.3. If (w, F)is an wES then PRl (a)
Proof. Let (0, F ) be an wES. Then by Theorem 1.I there is a BRFT (w,G,@,) such that F l = G l , F2 = G2. It is well known that Theorem 2.1 and Theorem 2.3 are best possible in the sense that there is a BRFT (w, PR(w), 4,). Theorem 2.4. Let (0,F, @), and (D, F, $,) be two BRFT satisfying (a) the inverse of every permutation of D that lies in F , lies in F l (b) the partial inverse functional I : Fl + F l given by Z(g) = gl if g is a permutation of D ; undefined otherwise, is total representable in (0,F, &). (c) for each 0 < m there is a total function fi E F2 such that whenever a, b E T I we have that [P(Q, b)] is a permutation of D , and for all x either i [ P ( a , b)ll(x)), = t[al1(x)}m or [XI, = [[bll([P(a, b)11(~))1m* Then (D, F, 4), and (D, F, $,) are piecewise isomorphic.
Proof. Let (D, F, @,), (D, F, $,) be as in the hypotheses, and fix 0 < m,0. We first note the existence of a permutationfE Fl such that [XI, = cf(x)}, . This is clear from Theorem 1.4, and (c). We fix thisf. Next, note that T * :F l X F, +F, by T * ( G , g ) = X x l...x,(G(g(Z(G)(xl),..., I ( G ) (x,)))) is partial representable by (b). Hence let TRep ((D,F,@,),p,T*). Similarly let T**(G, g ) = T*(I(G),g), and let TRep ((D, F, #,),q, T**). Then let r be such that [I] = f( [ p ] 2). Let s be such that [s] = hxy([q]2(x, flb))). Now let b(x) = P(S: (r, x),S: (s, x)). Then by Theorem 1.7, there is an e such that [el = [h(e)] 1.Then [el = [p(S,' (r, e),S: (s, e))]1. Since Si (r, e), S; (s, e ) E T ,we have that [el 1 is a permutation. Furthermore, either { [el (x)}, = { [S1 '1(r, e)] 1(x)}, = { [rI 2 (e, XI}, = {f([PI 2(e, X))>m = [[PI 2(e, x)l, = T ( [ e l l ?[XI, 1, or [XI,,*= " ~ \ ( s , ~ ) I I ( [ ~ I ~ ( x ) )=I ,~ [ ~ 1 2 ( e , [ e l l ( ~ ) ) l m = [ k I 2 ( e , f  ([ell(x>)>lm= T(I([eI1), [f'(Ie11(x)>l,>= W ( [ e l l ) ? { [el 1 (x)},). Hence, in either case, { [el (x)Im = T([e] 1, [XI).
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We wish to use Theorem 2.4 to prove that any two oBRFT are piecewise isomorphic.
Lemma 2.5.1. Let (a, F , @ n ) be an oBRFT. Then the inverse of every permutation of o that lies in F l lies in F , ,and the partial inverse functional I : F , + F , given by I ( g ) =gl if g is a permutation of o ;undefined otherwise, is total representable. Proof: Iff is a permutation of o thenfl = p(hxy(a(x, 0 , l ,f(y)))). So by Theorem 2.2., I is partial representable. Hence I is total representable.
Lemma 2.5.2. Let (o, F , G n ) be an wBRFT, 0 < m. Then there is an f E F 2 , ftotal, such that [ f ( x ,y ) ] , = b ] ,, and for e a c h y , Rng ( h x ( f ( x ,y))) is infinite.
Proof. First we show that there is a total g E F2 such that whenever [ X I 2 is (Va if ( 3 a Z ) = R m( [ e l3 (x, Y , z))); [Rm([el3 (x, Y , z))l ,(a 1...., a,) otherwise), if h is total; undefined otherwise. Then by Theorems 1.2, 1.3, 1.8, and 2.2, let TRep ((0, F , gn), q, H ) . Then finally set g ( x , y , z ) =
, ,
,
[414 (x, X, Y , Z ) . Next let g = [ k ]3 , and let h ( x ) = Si ( k , x), Then there is a p such that [ p ] 2 = [h( p ) ]2 . Clearly [h( p ) ] is total since k E T 3 . Now [h( p ) ]2 (y, z ) = (kp)12 01.z ) = [ k ]3 ( p , y , z ) = g ( p , y , z ) = [PI (y, z). But since p E T2, we must have, for ally, z , if ( V a < y ) ([ [pI2(a, z ) ] , = [z],) then
[Si
k(p*~,z)l= m[b12(y,~)lm = [zl, andeither(~a &,) omitted some of the most useful infinitary languages. Evidently, for further progress one has to look for structures where Montague’s g.r.t. differs from others (and to apply it). Specifically it might be interesting to describe in familiar terms the class of X subsets C N , where Z 1 ranges over the type structure hereditarily of cardinal < 9 0 , and the basic ‘operations’ on NN are )uz (n + l), inverse pairing functions, composition, and =. (Montague always includes =.) Do not assume the continuum hypothesis.
.
A
xr
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(ii) In contrast to (i), Moschovakis 1371 studies two notions of ‘abstract computability’ for which, as far as I know, there are in general no equally familiar equivalent descriptions. Curiously, though the word ‘computability’ occurs in the title the computability notions of [37] ,prime and search computability, are not studied in much detail in [37] ;in fact, there is no description of the class of prime computable relations in any familiar structure! One would expect these classes to be really interesting for very simple algebraic structures and, at the other extreme, in structures without a (prime computable) well ordering. The difference between the two computabilities corresponds pretty well to that between deterministic rules (of the kind one has in computations) and nondeterministic ‘rules’ (as in the usual systems of inference with detachment) where nothing is said about the order in which the principles are to be applied. More logical uses of these notions will be taken up in 4(c). Here it is to be remarked that in Part I1 of his paper [37] ,Moschovakis develops his theory of hyper projective sets (not mentioned in the title of [37] ) which gives a very satisfactory generalization of Kleene’s theory of hyperarithmetic sets. Though [37] does not present the theory of hyperprojective sets as a g.r.t., this can be done straightforwardly by means of Feferman’s [8] described in 2(a)(ii) above. In any case, Moschovakis, jointly with Barwise and Candy [4] ,have already related hyperprojectivity to Barwise’ g.r.t. using strict n,sets 1 in [3]. (d) Enumerated structures and an axiomatic theoty Trivially, whenever we have a general definition (here: of a recursion theory) on a class of structures as in (a) or (b), we have an induced axiomatic ‘theory’ (but, of course, not necessarily recursively axiomatized) modulo a given language corresponding to the basic definitions: the set of those statements in the language which are true in all the structures considered. (Obvious languages were considered in [23, pp. 2022041 or [ 2 6 , pp. 319,3201 .) It is sometimes said that the axiomatization problem is to generate the set of valid statements. But this is a logician’s parody of the role in mathematics of genuine axiomatic theories! Of course the problem is attractive because it is clear cut. Now, first of all, quite often the set of valid statements is not r.e., when there is no question of a literal axiomatization; but even when the set is r.e., a complete axiomatization is not necessarily useful; for example, if very few axioms yield the bulk of interesting results while, perhaps demonstrably, a complete axiomatization is bound to be complicated. In such a case the law of diminishng returns applies; concerning objectivity in judging partial axiomatizations, see l(b). Naturally, and this comes up in (ii) below, once
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such ‘partial’ axiomatic theories have been discovered they may present interesting material for metamathematical study. The enumerated structures and definitions of essential concepts are given in Malcev [35] and Lacombe’s thesis [31] ’). The axiomatic theories were introduced by Wagner [57] ,developed and studied by Strong [55] and, more recently, Friedman [ 1 11 .(His work and my knowledge of Strong’s go back to an exposition by Mr. Richard Thompson in my seminar at Stanford, in autumn 1968.) The basic notions of recursion theory (listed at the beginning of Section 2) all appear explicitly in these axiomatic theories except for: finite. There is an important additional relation, between the universe and its s.r. subsets, familiar from the s.r. enumeration of s.r. sets. (i) The works of Malcev and Lacombe constitute a g.r.t. on countable structures, defined in terms of oneone numberings (numerotage) or manyone enumerations (BnumCrage). Any enumeration together with given relations of the structure obviously induces number theoretic relations. The enumerated structure is then called recursic if the number theoretic relations are recursive. Evidently since only the (extensional) notions of ordinary recursion theory are used, the theory is free of details about Godel numberings and the like. Both Malcev and Lacombe emphasized a particular application, namely where the enumerated structure consists of the partial recursive functions themselves with composition and some other simple operations (but not: extensional equality). From this point of view, it was natural to analyze properties of the enumeration function which implied the principal theorems of what has been called: preFriedberg recursion theory. Though names were given to such properties (e.g. Lacombe’s ‘autogenic’ enumerations, obviously connected with a partial recursive enumeration of partial recursive functions) no attempt seems to have been made to set out these properties in the form of an ordinary axiomatic theory. It seems fair to say that, because of this, it was not realized just how abstract the analysis was! Specifically, the fact that one was thinking of an enumeration of the structure by means of natural numbers was not used in the theory of partial recursive functions at all. One could just as well think of ’)
I know Malcev’s work only from reviews. I have the impression that Lacombe shows more finesse (but it often happens that reviewers do not appreciate and hence do not communicate the finer points of a theory). This compensates for the distressingly affected style of Lacombe’s thesis.
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an enumeration by ‘themselves’ provided such a function is identified, not with its graph, but a presentation. (ii) Wagner [57] also analyses the usual operations on partial recursive functions, but in a proper axiomatic style, also emphasizing the ‘enumeration’ relation. The language is very simple, the axioms are equations. In h s original version he introduced explicitly an undefined element; consequently, when specialized to the classical case, his basic relation = cannot be realized by an r.e. predicate. This would be interesting to modify this. Wagner and Strong [ 551 developed what might be called Postrecursion theory from these axioms. In view of the work under (i) (or, indeed, from inspection of the older literature), it was not surprising that something like this could be done; but (at least to me) it is remarkable just how much these few axioms yield. Recently, Friedman [ 111 has developed this theory some steps further, in (at least, to me) most satisfactory ways. First of all, he establishes an isomorphism result which again is not unexpected from (i) but was not previously established: if we take the partial recursive functions and consider any two enumerations satisfying the WagnerStrong conditions the resulting structures are isomorphic. He describes it as a relative categoricity result, relative to keeping our class of objects fixed, varying only the enumeration. Naturally this result does not explain why SO much recursion theory can be developed by logical deduction from the few axioms. Indeed, logically quite unrelated axioms in the language of WagnerStrong might also be relatively categorical! This raises the (probably delicate) question : In what way, if at all, are the WagnerStrong axioms distinguished among relatively categorical ones in the language considered? The other novel element which, of course, depends on the existence of g.r.t. described in (a)  (c) above, is that the WagnerStrong axioms hold for a large class of them. And, I think equally important, Friedman [ 1 11 shows that his isomorphism result does not hold for all of them, e.g. not for all admissible La: satisfyingly enough it holds for the subclass of ‘nonprojectible’ La which have already played an important role in the subject. So things fit together quite well. Possibly the WagnerStrong axioms are necessary in the precise sense that they are satisfied by g.r.t. which have the kind of uses described in Section 1.
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(But cf. Friedman’s remark at the end of [ 111 concerning countable admissible A without a definable well ordering.) But it cannot be expected that they are sufficient. It will be interesting to see whether a few additional axioms in the WagnerStrong language are sufficient for the bulk of, say, postFriedberg recursion theory.
3. Metarecursion theory (a g.r.t. on the recursive ordinals) It is useful to begin with an uptodate summary by reference to the list of basic notions at the beginning of Section 2. The generalizations of these notions were defined in [26, p. 3221 in terms of model theoretic invariance discussed in 2a, but the technical development used principally one of two equivalent descriptions, in terms of Kripke’s equation calculus [28] (the standard equivalence proof, which belongs in para. 3 of [26, p. 3241 , somehow having been omitted); and, to a lesser extent, notations and concepts of Kleene’s hierarchy (the logical status of the notations being made clear in [26, p. 319, 1. 2131], but not in [49]). The equivalences with familiar concepts of Kleene’s hierarchy are of importance not only for the proofs, but also for results, because some of the more important applications of metarecursion theory concern precisely these concepts. An important element of metarecursion theory is the notion of metafinite, the generalization offinite in the sense of definability. This notion is taken as primitive in [26, p. 3221 , but turns out to be extensionally equivalent to: metarecursive and included in a proper initial segment of the recursive ordinals. (Thus this use of ‘metafinite’ is not essential to stating results.) But thinking of this conjunction as generalizing ‘finite’ was essential in the development of the subject, to guide (1) conjectures, (2) proofs and (3) the choice of notions. The following examples should be kept in mind.
(1) For reasonable conjectures in applications of the theory to n i sets, that is to meta r.e. sets of natural numbers, one must remember this (where metafiniteness is relevant): these sets are generalizations of bounded r.e. sets (like r.e. sets whose elements are 0 or l ) , since they are included in o,which is a metafinite subset of the recursive ordinals. The distinction between finite sets, given by a standard presentation, and bounded r.e. sets is familiar not only from the intuitionistic literature (cf. [26, p. 3231 ) but from Post’s notion of hyper  and hyper hyper simple sets.
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But the distinction is not always developed in ordinary recursion theory when it could be introduced in a natural and perhaps fruitful way.
( 2 ) The notion of metafinite is used in proofs of theorems which generalize classical results where ‘finite’ is not even mentioned such as enumeration of r.e. sets without repetition or the splitting of nonrecursive r.e. sets into disjoint ones; or generalizations which retain the literal meaning of ‘finite’, as in (the strongest version, in (a) below of) the maximal simple set; cf. Friedberg [lo] . (3) In a more subtle way, one of the reducibility relations, namely <M (‘M’for metarecursiveness), is tied to our emphasis on metafiniteness.
To explain this let me go over the subject of metareducibility; its discussion in [ 2 6 ] and [49] was unnecessarily complicated because of our ignorance of simple mathematical facts below. Nowadays it can be put as follows. The various notions which are equivalent to Turing reducibility in classical recursion theory fall naturally into two groups: those concerned with computation and those concerned with definability. Now if one things of computations as operations from (and to) finite objects, and if the notion of metafiniteness is to be used then <M is the proper generalization: it involves a uniform metarecursive process from and to metafinite neighbourhoods. AN the definability notions that arise turn out to be equivalent. They are denoted by Q [Q I 91 . T is generic if for every sentence 9
[email protected]) there is a P such that T E P , and either P l 9 or P 9 . Clause (1) of the definition of is not orthodox; consequently, the existence of generic T’s is not obvious, but instead requires 2.6. Let 2 be a subset of La. 2 is said to be &definable if there is a formula 9 ( x ) of ZF with parameters in La such that
a&$?
2 = Cx I x E L , and La

(x)}
Kripke [3] and Platek [4] call a function f (from a into La) arecursive if its graph is &definable by means of a C1 formula. A set is arecursively enumerable if it is the range of an arecursive function. A set is afinite if it belongs to La.
a’s, is are9, restricted to
Proposition 2.4 (i). The relation, P l9, restricted to ranked cursively enumerable. (ii) for each n > 0: the relation P t9 E C,, is 2, over La.
Proposition 2.4 (i) is, in the main, a coaequence of the following fact: if R ( X ) is a ni predicate and a is admissible, then the set { X I X E La & R ( X ) } is arecursively enumerable.
+
every set is Don’t forget that 2.5 assumes that a is such that La countable. The proof of 2.5 is similar to that of 3.2 of [5]. The next lemma, the socalled “sequential” lemma (cf. [ 101 ) or “fusion” lemma, is designed to capture the essential properties of C,pointed, perfect closed set forcing. The only worrisome points in the proof arise from the C,pointedness requirement. The proof for C1pointedness [2] is more direct, and if all pointedness requirements are dropped as in [5, 101, the proof is almost immediate.
FRECURSIVENESS
{a
Lemma 2.6. Let Let P be such that
293
I i < o}be an afinite set of
C, sentences of la(9).
(4 (Q)P>Q(ER>Q>R [ R I ail Then there exists an R C_ P such that (i) [ R k
ail.
Roof. Let Lp 6 , then X is constructible in the sense of Giidel.
298
G.E. SACKS
o and let a be We indicate a proof of 3.2 for the case of Zm. Let X a countable, Z:,admissible ordinal. Suppose X 4 La. It suffices to find a T such that = a and X 4 L a ( T ) . T is constructed as in Section 2 save for some additional steps designed to omit X from L,(T). Let c be a term of &(9) which denotes a subset of w. Fix P. We define a Q C P such that for every generic T E Q, X # c ( T ) . First suppose
WL
Let Q = Qo if n 4 X ; otherwise let Q = Ql. Now suppose otherwise:
Let Q = P. If T E Q and T is generic, then
n
4 c (T) * (ER)Q>R [ R I fi 4 C I
for all n ; but then c ( T ) E La, since the forcing relation restricted to ranked sentences is arecursive. The idea underlying the above argument goes back to Kleene and Post [Ill. 4. Secondorder absoluteness of F Let 9 be a subset of 2 w , and let a be a countable, admissible ordinal. We say 9 ' 3 is acomputable if there exists a ranked sentence 9 of &(8) such that
3,
This is not the definition of atcomputable subset of 2'" given in [ 21, but it is equivalent to it. The definition used in [ 21 is longer but more to the point, since it is given solely in terms of computations whose heights range over the ordinals less than
a.
FRECURSIVENESS
299
(The notion of acomputable subset of 2w is discussed in more detail in [2] .) Since 9 is ranked, it has rank equal to some 6 < a. Thus the question of whether or not an arbitrary S belongs to CIO can be decided by means of a computation that belongs to L a ( q and whose height is at most 6. Although the computations associated with 33 are bounded in height, they are not bounded in complexity, since S is arbitrary. If 33 is acomputable, then CM is Borel. It is wellknown that is lightface A: if and only if CZ3 is olcomputable, where o1is the least nonrecursive ordinal. ‘93 is said to be Z, over a [ 2 ] if there is a Zl sentence 9 of L a ( 3 ) such that
Proposition 4.1 is an immediate consequence of Barwise’s compactness theorem [7] .
C 2w and let a be a countable, admissible ordinal. Proposition 4.1. Let over is acomputable if and only if both 93 and 2w  c)3 are Then a. Let 33 is said to be Bqrel in T if there exists 2 ; ^formulas P ( X , Y ) and = SP(S, T) and 2w  33 = SQ(S, T). A set is Q ( X , Y) such that Borel if and only if it is Borel in some T , A set ?e is Borel in T if and only if there is a ranked sentence 9 of d: T ( 9 $) , such that W1
We say 33 is Borel in a if B is Borel in every T such that oT = a. We say 33 is uniformly Borel in a if there exist Zi formulas P(X, Y ) and Q ( X , Y ) such that = SP(S, T )
and 2w

33 = SQ(S, T )
for every T such that o? = a.
fieorem 4.2 ( [ 2 ] ) .Let 3 C 2w and let a be a countable, admissible ordinal. Then (i), (ii) and (iii) are equivalent. (i) 33 is Borel in a. (ii) 33 is unformly Borel in a. (iii) 33 is acomputable.
G.E. SACKS
300
It seems wise to discuss the proof of 4.2 before proceeding to a generalization of 4.2. First we draw the reader’s attention to the uniformity expressed by (i) + (ii). We do not know a direct proof of (i) + (ii), although we tend to thmk there is one; (iii) + (ii) is almost immediate, so we confine our efforts to (i) + (iii). Suppose 93 ’ is not acomputable. We sketch a T such that a; = a and CM is not Borel in T . The construction of Section 2 is employed to insure that = a. Some additional steps are woven in so that 33 will not be Borel in T . Let 9 be a ranked sentence of
[email protected], 3).Fix P. We show there is a Q g P such that for every generic T E Q,
UT
33f S(L,(T, S ) k
9).
Fix S and consider the mechanics of adding a T t o La(* in the style of Section 2. The afinite forcing conditions of Section 2 are retained so that will equal a,but all sentences of &(9,3) are to be forced with 3 held fixed at S. The forcing relation P 9 , restricted to the afinite P’s of Section 2 and to the ranked sentences of &(9, J) with J interpreted as S, is C l over L a ( S ) . This last is true uniformly in S ; that is, there is a single Z1 formula of & ( J ) that defines the forcing relation in question for all S. The definition of Q has three cases. 91. Choose such an S and such Case I . (ES)(EQ)p>Q [S 4 33 & Q a Q. Devote infinitelymany of the steps remaining in the construction of T to making T generic in the sense of p. Thus when the construction T is finished, it will be the case that S 4 33 and La(T, S) k 9. 9 ] . Proceed as in Case 1. Cose 2. (ES) (EQ)p>Q [ S E 73 & Q Cose 3. Otherwise. T t follows that for all S,
UT

S 4 q * ( E Q ) ~ > Q [ Q P91 But then by 4.1, 33 is acomputable; so Case 3 never applies. The overall construction of T has the following form: T is generic in the sense of I so that aT = a;T is generic in the sense of p for countably many S’s chosen in the course of the overall construction of T so that 33 not be Borel in T . Let F be normal and regular, and let a be a countable, Fadmissible ordinal. We say CM is FBore1 in a if 33 is Borel in every T such that U: = a.
FRECURSIVENESS
301
We say 93 is uniformly FBore1 in a if there exist 2 ; formulas P ( X , Y ) and Q ( X , Y ) such that
93 = SP(S, T )
and
2w

93 = SQ(S, T )
for all T such that a; = a .
Theorem 4.3. Suppose F is C,, (n 2 l), Em,or RI. Let 33 C 2w and let a be a countable Fadmissible ordinal. Then (i), (ii), and (iii) are equivalent. (i) '93 is FBore1 in a. (ii) 93 is uniformly FBore1 in a. (iii) 93 is acomputable. The proof of 4.3 is similar in form to that of 4.2. 4.3 is one mode of generalbation of 4.2. Another mode is given by 4.4 for Z, (n 2 1). Let 93 2 w , n 2 1, and a be a countable, admissible ordinal. We say is C,, over La if
for some
9 a C,,
sentence of la(&). We say % is C,, over L a ( T ) if
93 = S(L&
S ) I=
9)
for some 9 a C,, sentence of & ( a , 6). We say 93 is A,, over La if 93 is A,, over L a ( T ) if 33 and and 2w  93 are C, over La. We say 2w  CM are C n over L , ( T ) .
lheorem 4.4. Let 93 C 2W and let a be a countable, C,,admissible ordinal (n 2 1). Then (i) and (ii) are equivalent. (i) CM is A,, over La. (ii) 93 is A,, over La(T) for every T such that a : , = a.
9 is introThe proof of 4.4 is like that of 4.2. The forcing relation P duced as before, and then the following observation is made. The relation P 9,restricted to the C,, sentences of L a ( 9 , 3 ) with 3 interpreted as S, is C, over La(S). The obvious extension of 4.4 to the case of Zm is true.
G.E. SACKS
302
5. Further extensions and questions
(Ql) Under what conditions: is a normal Fjump regular?; is a normal, regular Fjump firstorder absolute? (Q2) Is there a natural concept, necessarily broader than firstorder absoluteness, which is definable for all normal, regular Fjumps, and which covers the partial improvement of 3.1 expressed by 5.1?
Theorem 5.1. Let X C_ a and let a be a countable, Z:,admissible ordinal for some n 2 1. Then (i) and (ii) are equivalent. (i) X is Z, over La. = a. (ii) X is Zn over L a ( T ) for every T such that (Q3) Is there a concept called second order absoluteness which is definable for all normal, regular Fjumps, and which covers the results of Section 4? (44) Is there a result about RI which corresponds to 4.4? to 5.1? (QS) Is there a proof by the Barwise compactness approach of Friedman and Jensen [12] of 2.1? 2.2?, 4.2? (Q6) Let F be Z, (n 2 l), Zoo,or RI. Does every countable set of Fdegrees have a minimal upper bound? We conjecture it does, but at the moment we know little more than 2.3. For each F define Fl as follows. An ordinal a is F1admissible if a is Fadmissible and the limit of Fadmissible ordinals. Thus RI = (Z1)1. If is defined for all T and F is normal, then Fl is normal. (Q7) Suppose F is regular and Fl is normal. What further conditions on F guarantee that Fl is regular?
References [ 11 G. Sacks, Normal Fjumps, in preparation. [ 21 G. Sacks, Countable admissible ordinals and hyperdegreeq to appear. [ 31 S. Kripke, Transfinite recursions on admissible ordinals I and I1 (abstracts). J. Symbolic Logic 29 (1964) 161162. [4] R. Platek, Foundations of Recursion Theory, Ph. D. Thesis, Stanford University
(1965). [S] G. Sacks, Forcing with perfect closed sets, in: Proceedings of the 1967 UCLA Set Theory Institute, to appear. [6] H. Friedman, Killing admissible ordinals (mimeomgraphed notes), Stanford University (1969).
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[ 71 J. Barwise, Implicit definability and compactness in infiiitary languages, Lecture Notes in Mathematics 7 2 (1968) (SpringerVerlag) 135. [ 81 K. Godel, The consistency of the axiom of choice and of the generalized continuum hypothesis (Princeton University Press, 1966). [ 91 S.C. Kleene, Hierarchies of numbertheoretic predicates, Bull. Am. Math. SOC. 61 (1955) 193213. [ l o ] R. Candy and G. Sacks, A minimal hyperdegree, Fund. Math. 61 (1967) 215223. [ 111 S.C. Kleene and E.L. Post, The upper semilattice of degrees of recursive unsolvability, Ann. Math. 5 9 (1954) 379407. [ 12) H. Friedman and R. Jensen, A note on admissible ordinals, Lecture Notes in Mathematics 7 2 (SpringerVerlag, 1968).
A SIMPLIFIED PROOF FOR THE UNSOLVABILITY OF THE DECISION PROBLEM IN THE CASE AVA
H.HERMES Freiburg LBr., Germany
1 . Introduction Using earlier ideas of Wang [ 11 and Buchi [ 2 ] t..e first pro0 for the unsolvability of the decision problem in the case AVA has been given by Kahr et al. [3] . (The knowledge of this paper is pressuopposed.) Wang, who invented the domino games, proved that the decision problem for the corner game is unsolvable. Buchi inferred that the decision problem for the case V A VAV is insolvable. Kahr et al. treated the diagonal game. The unsolvability of the decision problem for the diagonal game leads to the result that the decision problem for AVA is unsolvable. The essential idea of Kahr et al. was to represent each configuration of a Turing machine M periodically on a diagonal of the first quadrant. The dominoes of the game associated with M have two different pruposes: (a) to present the machine M , (b) to generate the periodic patterns on the diagonals. In order to achieve (a) and (b) the authors used 26 dlfferent kinds of dominoes. In t h s paper a simplified proof for the unsolvability of the decision problem in the case AVA will be indicated, where only 10 different kinds of dominoes occur. The modification of the proof of Kahr et al. may be described as follows : (1) the dominoes are not used to describe the pattern, (2) it is not required that each pair of adjoining dominoes match in color, (3) the pattern is given by a function, (4) the pattern is used to indicate the edges where the adjoining dominoes should match in color. A minor change consists in replacing the diagonals by the rows of the first quadrant. 307
H. HERMES
308
2. Fundamental concepts Let be Q the set of unit squares of the first quadrant. Let be cp a mapping of Q into (0, 1 }. cp is called a mosaic iff (a) on the lowest row we have alternating values of cp, (b) if for each quadruple of squares A , , A 2 , A ?, A,, where A 2 is the right neighbor of A and A , is the right neighbor of A and A 3 is immeaiately above A we have
,
cp(A3) = cp(A4) iff
cp(A 1) = 1 and
cp(A2) = 0
.
If we associate to the squares of the lowest row the values 0, 1 , 0, 1, ..., to the squares of the next now the values 0, 0, 1 , 1, 0, 0, 1 , 1, ..., to the squares of the next row the values 0, 0, 0, 0, 1 , 1, 1, 1 , 0, 0, 0, 0, ..., etc., we get an example for a mosaic. A domino game is given by three finite sets D, Do, D * of dominoes, where Do C D and D* C D. Let be cp a mosaic and D, Do, D* a domino game. A mapping F of Q into D is called a D, Do, D*,cpcovering of Q , iff (1) F ( A ) € D o for eachA of the lowest row, where cp(A) = 0, (2) F ( A ) ED* for each square A , where cp(A) = 1, (3) F ( A 1) and F ( A 2) are matching in color for each pair A 1, A of squares where A is the right neighbor of A and cp(A ,)= cp(A 2) = 0, (4) F ( A and F ( A 2) are matching in color for each pair A 1, A 2 of squares where A is immediately above A and cp(A 2) = 0. A domino game D, Do, D* is called a good game iff there is a mosaic cp and a mapping F such that F is a D, Do, D*, cpcovering. In order to show the unsolvability of the decision problem for the case A V A i t is sufficient to associate in an effective way to every Turing machine M a domino game DM,D;, Dh and to every domino game D, Do, D* a formula OlDDoD* € AVA such that (*)
(**)
M does not halt iff DM,D;, Dh is good, D, Do, D* is good iff CYDDOD*is satisfiable.
The definition of D M ,DL,D; is given in the next section. The definition o the p proofs for (*) and (**) can be performed without diffiof a ~ ~ and culty, knowing [3] . Cf. also [4] .
UNSOLVABILITY OF A DECISION PROBLEM
3 09
3 . The definition of the domino game D,, D&, Dk associated with a Turing machine M (which is operatmg on a oneway tape).
M is identified with its table, which may be considered as the set of its rows. We use a as a variable for the letters of the alphabet and 4 as a variable for the states. 0 is the blank letter and also the initial state. For each pair 4,a we have a row qabq of M , where 4 is a state and b (the behaviour) is a letter a or one of the symbols r (right), 1 (left), s (stop). The colors of the dominoes are a, 4 4 gr,ql, qar, qal, B (beginning), C,, C, (corner), H(horizontal), W (west), S (stop). We introduce a color function @ which is defined for each pair 4, a, and where @qa is a color, as follows: __
i
qa
@qa =
Gar qal S,
if if if if
qaZj E M
qarj E M qulq E M qasij E M
In DM we have four single dominoes (C), (C'), (B), (W)and six lunds of dominoes (a), (qar),( 4 4 , [qa] , [qar], [qal] . (C) is the only element of D i and (0) the only element of D&. The colors of the dominoes are given by the following diagrams :
310
H. HERMES
References [ 1J H. Wang, Proving theorems by pattern recognition. 11, Bell Systems Technical J. 4 0
(1961) 141. [ 2) J.R. Buchi, Turing machines and the Entscheidungsproblem, Math. Ann. 148 (1962) 201213. (31 A.S. Kahr, E.F. Moore and H. Wang, Entscheidungsproblem reduced to the 'd3vcase, Proc. Natl. Acad. Sci. U.S. 48 (1962) 365377. [4] H. Hermes, Entscheidungsproblem und Dominospiele, in: Selecta Mathematica 11, hrsg. von K. Jacobs (Springer, 1970) pp. 114140.
AN INTRINSIC CHARACTERIZATION OF THE HIERARCHY OF CONSTRUCTIBLE SETS OF INTEGERS') Stephen LEEDS and Hilary PUTNAM Department of Philosophy, Harvard University, Cambridge,Mass
Introduction 1. Notation
Some of our results will be in secondader number theory, others will be in set theory; we will use the standard (incompatible) notations for each. In secondorder number theory, small Roman letters will represent (i.e., be used as constants to designate, as variables to range over) integers; capital Roman letters will represent sets of integers. In set theory, small Roman letters will represent sets. Small Greek letters will always represent ordinals. We will use Roger's notations [9] in recursive function theory: the eth Apartial recursive function will be designated by '@ 1; the domain of will be designated by 'W$'.J will be a fixed primitive recursive 11 onto map from N X N to N ; the inverse maps will be denoted by 'K' and 'L', so that
KJ(x, y ) = x
W(X,
y ) =y
.
The definition of jumps is as follows: Let A be a set of integers. We define
A ( " + l ) = {x : ( 3 y ) 0,= & x ) } ( = ( A @ ) ) ' )
A(W) = { J ( x ,y ) : x E A Q ) } .
(We will denote N ( l ) by 'K'.) 1)This paper is essentially the first author's Ph.D. thesis (of the same title) written under the direction of Professor Putnam and submitted to M.I.T.,June, 1969.
311
3 12
S. LEEDS and H. PUTNAM
Notation in set theory varies little from author to author; ours will be familiar. It should be pointed out that, hke Godel (but unlike, e.g., Cohen), we take M A = U MP,for limit A. ‘L’ will denote U MP,the class of construcP. The last clause may be translated as ‘z has x ‘members’ ’, which has simple translations into expressions arithmetic in Ea.) It will be noticed that if E is a complete set of order a,so are E(l),E(2),..., nor is there any obvious way to choose a complete set, or a complete Turing degree, as being somehow unique (e.g., minimal) among the complete sets of a given order. We shall deal with this problem by speaking, not of Turing degrees, but of arithmetical degress, defined as follows: If A is arithmetical in B , we write ‘A Ga B’. If A Ga B and B Ga A , we write ‘A a B’. We define degaA = { B : B E a A } . is an equivalence relation; also if A E Ma, deg, a E M a + l z ) . Clearly Also, the complete sets of order a constitute a single arithmetical degree. We designate this degree by ‘deg,(a)’. The dega(a) have properties quite analogous to those of the d ( K ( ” ) )in the arithmetical hierarchy: (1) ifA E L n P ( w ) , (301)(degA Ga deg(a))
(2) if a and 0 are indices, a < 0 
deg(a)
< deg(0) .
If we restrict our attention to index a,(1) remains true, and (2) becomes true unconditionally. We find that the sequence of deg(a) has the comprehensiveness and monotone feature characteristic of the familiar hierarchies. We call the sequence of deg(cu) for index a > w the hierarchy of arithmetical degrees. This hierarchy is, to this point, apparently less interesting than the
5

GaB * A B(“) for some n , 3 a sequence of n + l sets bo. E l ... Bn such that BO = B, Bj+l = B: and A B,. This last expression is clearly firstorder definable over MOr
’) Because A
5
S. LEEDS and H. PUTNAM
3 14
familiar hierarchies. What is lacking in its characterization is a feature which we may call ‘minimality’: we need something analogous to the sense in which K(4), and not td5),or any set r.e. in is chosen as the successor of K ( 3 ) in the arithmetic hierarchy. We would like to define ‘next least’ in such a way that if a is an index, and /3 is the least index > a,then deg(p) is the next least degree after deg(a). We are encouraged in thinking that something of this kind is possible, by the following result of Boolos: The hierarchy of arithmetical degrees is identical with the ramified analytic hierarchy, where the latter is defined; further, it is identical with the hyperarithmetic hierarchy where that hierarchy is defined  taking into account in the latter case the difference between Turing degrees and arithmetical degrees. More precisely, Boolos proved
(1)letting/30=pa(Aa=Aa+1)wehave: i f a < b 0 t l , A a = M a n P ( w ) (2) if a < constructive w 1 (Kleene wl) then order A = a * A GaHb for all b such that Ib 1 = w * a ,and if I c I < w .a,then A $, H , . Also, if la 1 = w * a, then order H, =a. The sequence of complete degrees is identical with the sequence of deg(Hb) for I b I a limit ordinal; to the extent to which deg(Hb) may be seen as a natural ‘next’ degree after all deg(H,), la I < 1 b 1, our question has been answered. There are two ways in which the answer falls short. First, 0 may appear somewhat artificial, and the hyperarithmetic hierarchy may seem t o inherit this artificiality. Secondly, and more significantly, constructive w 1 < ‘ 3 4 , indeed O0 <  the hierarchy we are considering far outstrips the hyperarithmetic hierarchy and even the ramified analytic hierarchy. One might deal with the first difficulty by citing results like those of Enderton [S] on the minimality of 0 as a notational system. To the extent to which this answer is persuasive, one will also be able to deal with large stretches of our hierarchy, even beyond constructive wl. For, as Boolos shows, if G is a complete set of order p, then result (2) above holds, relativized to G . That is, between /3 and /3 + a?,the hierarchy of arithmetical degrees is identical with the limit members of the Ghyperarithmetical hierarchy. We will not give details here, as we will make virtually no use of this fact. The cases that make even the relativized hyperarithmetic hierarchies useless for our purposes are those in which an index /3 is a limit of nonindices or a successor of a nonindex. In such cases, we have the following situation: Let G be a complete set of order a < /3. Then for limit 6 < a?,there is a
04
3 15
THE HIERARCHY OF CONSTRUCTIBKE SETS
&‘
member of U G ,say d , such that w * Id1 = 6 . Then if Ib I = w . Idl, is of order a + 6. It follows that order (Y + w? $w1, by the restriction on forcing conditions that no ordinal may appear twice. Then Q IF {x, : Ord x } = { x n , : Ord x }, & {x6 : Ord x } = {xg r : Ord x } for either n #n’, 6 #6’. Extending Q to a complete sequence, we arrive at a model N ‘ in which either ‘n=n” is true, and n f n l , or ‘6 = 6 ” is true, and 6 f 6 ’ . By the induction we used to show the same result in N, this is not possible.
‘ l ) This
THE HIERARCHY OF CONSTRUCTIBLE SETS
339
appear in w 1 for any q. Let w ; = w1 U ( ( r , 6)). Let P’be like P,except in having w ; for w l . Then P‘ 2 P and P’ I @.


(b) If @ is ‘n E K ’, where n = J(r, s) If (P I n EK 1 ) and (P k n EK 1 ) then k l has fewer than s members. Let K l l ... K l , be the members of k , . Let K I J + ,... K l , be distinct nonempty members of Ma nP(w),distinct from the members of k l , and such that r 4 K l s . Let k; = ( K l l ... K1,, K l , t + l ... K l s ) . Let P’ be like P except in having ki for k l . Then P’ 2 P and P’ I @.


(c) If @ is V,x $x If h,(P I @), then for some c ES, P’ i $ c . Then P’ V,x $x.

. (P i $ c ) . Let P’2 P and


(d) If @ is $ If (P b $),then for some Q, Q 2 P, Q I JI. Then Q I $ (a standard result), i.e., Q I @.
2. P
[email protected] 
I @
Proof 4 trivial +if P
[email protected] but (P I @), then some P’2 P forces + P’
[email protected]; but P’2 P a p ’ k 4
  
 @.
But then
We now introduce transformations on forcing conditions: We define a transformation r to be a set of four functions, and four associated sets in Ma nP(w),r l , r 2 ,p l , p 2 , f l , f> r l , r 2 , satisfying the following conditions:
p i : w + w 11 onto, with pi a product of disjoint 2cycles
q ( x ) = n J(x,
n) E fi
= n J(x,
n) E ri
&(X)
.
If P = (k1, k2, wl,w 2 )is a forcing condition, wi = ( m i l , t j i l )
...(mili,6ili> ,
S. LEEDS and H. PUTNAM
340
we define
T ( K ~=~{ J) ( r , j ) : Tj(J(r,j ) ) = 1 & r E Kij .v. ri (J(r, j ) ) = 0 & r 4 Kij}
Then
( ~ ( m l 6l 11) ,
...r ( m l r l ,6 l r l ) ) ,
( ~ ( m621) ~ ... ~ T,( m 2 z , 622)>) . Notice that n ( P ) = P. Corresponding to each transformation 7,we define, by simultaneous induction on rank, ~ ( c )T,( @ ) for each constant and formula in L. For constants and formulas of rank < a ,we have ~ ( c=) c,
r ( @= ) @ [we may regard occurrences of W i
in such formulas as illformed] For constants of rank a,that is, K , and K 2 , we define
r ( K j )= { x u :x EKi & J ( x , 1) E ti .v. x
4 K j & J ( x , 0 )E fi}
where ‘fi’ is a name of t i , of rank < a. For formulas of rank a:T ( P )= the result of replacing in @ all occurrences of Ki by r(Ki) . For constants of rank 6 + 1 : T { X & : @x}= { x 6 : 7(@)} . For formulas of rank 6 + 1, or A: T ( @ ) = the result of replacing in 9,first, all constants c by r(c);next, all occurrences of Wjc1c2by ( 3 , x ) ( J ( c l , x) E ri & Wixc2),and all occurrences of Wyc2 where ‘y’ is a variable, by ‘(3,x) (JCy, x ) E rj & Wixc2). Unlimited formulas are treated in exactly the same way. We have a simple
34 1
THE HIERARCHY OF CONSTRUCTIBLE SETS
Lemma. For any constant c, we have

Proof. * Let Q 2 P, Q l m(c). Extend Q to a complete sequence, and consider the corresponding model N ' . In N ' , we have @(c) and @(m(c)), hence c # n ( c ) is true in N ' . A simple induction shows this to be impossible. * Similarly.

Lemma. Let T(P)be a forcing condition. For any @, we have
Proof. By induction on rank. Here are the interesting cases: (1) Let @ be V,x$x. Then P t V,x$x * (Vc E S , ) P I $c. By inductive assumption, (Vc E S,)T(P) l~($x):('). Let c ES, . Then ~ ( c E ) S, ,hence 7 ( P ) l ~($x):('); hence, by the lemma above, T ( P ) lr($x);. Hence, 7 ( P ) I V6x$x. The converse is similar. (2)
[email protected]$.IfPFN$,assume7(P) ~  ~ ( $ ) ) . L ~ ~ Q I > T ( P ) , T ( $ ) . By inductive assumption, T ( Q ) I J/.But 7 ( Q ) 2 P * e.The Q converse is similar. (3) @ is c E K , . Let c be {x, : $x, }. Let P c E Ki.Then for some n, P k c = n. If 6 < a , then c is absolute, hence all P force c = n , and all P force ~ ( c = n )i.e., , c = n. Hence, T ( P ) t~(c=n). If 6 > a , then N
{x, : Ord x }) [(inductive step)]
* T ( P ) k.T ( c ) = n ,
i.e., T(P)k ~ ( c = n.)
Now we show that T ( P ) k n E 7 ( K i ) .We have n = J ( r , s) and r E K i s in P. Now eitherP b n E K , & J ( n , 1) E tior P k n E K , &J(n,O) €ti. In the first case, we have r EK , in 7 ( P ) ,and J ( n , 1) E t j is absolute, so T ( P ) k n E K , & j ( n , 1) E ti,i.e., T ( P )1~(nE K i ) . In the second case, we have r 4 Kis in T(P),andJ(n, 0) E ti is absolute, so 7 ( P ) ln @ Ki & J ( n , 0) E ti, i.e., ~ ( E n Ki). In either case, we have 7(P) I ~ ( c = n&) ~ ( EnKi), i.e., T(P)
T ( P )IT ( C E Kj).
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(4) The case where qi is Wic1c2is quite similar; once again, the transformations on formulas have been set up expressly to allow the desired result. The remaining cases proceed by a straightforward induction.
We can now apply the results on transformations to prove the independence lemma.
Lemma. If P k qi and qi does not contain ‘ W 2 ’ , ‘K2’,then if Q differs from P only in k2, 9 , Q I 4.

Proof. We show that no extension of Q forces 4. We call 6 a w2ordinal (of a forcing condition, say P1)if ( 3 n ) ( ( n , 6 ) E W 2 in P1).We call n a w2integer of P l if (36) ((n,6 ) E W 2 in P l ) . Let Q 2 Q. Choose P* 2 P, P* = (kl, k 2 , wl, w2) with the following properties: kl in P* = k , in Q’ k2 in P* has as many members as does k2 in Q’ w1 in P* = w1 in Q’ the w2 ordinals of P* are precisely those which are w2adinals in either P or Q’. Choose Q* 2 Q’ 2 Q with the following properties: k l in Q* = k l in Q’ k2 in Q * = k 2 in Q’ w1 i n Q * = w , i n Q ’ the w2 ordinals of Q * are precisely those which are w2ordinals in P*. We now have P* 2 P, Q* 2 Q, P* and Q* have identical k l and w1 sets, k2 sets of equal length, and w2 sets with exactly the same ordinals. We can now find a transformation that takes P* to Q*: with respect to the k2 sets, closure properties of Ma!r l P(w) clearly allow us to find such a transformation; however, because p2 in any transformation must be a product of disjoint 2cycles, we proceed as follows: Let T be a transformation that leaves k l and w1 fmed, and takes each Kzi of P* to the K2i of Q*, and takes each w2 integer of P* to an integer that is not a w2 integer of either P* or Q*. Let 7’ be a transformation that leaves k,, w l , and k 2 fixed, and takes each w2 integer of T ( P * )to a w2 integer of Q* in such a way that T’T(P*)= Q*. qi; so r ( P * ) ~ ( q i ) ,hence T‘T(P*)k ~ ’ ~ ( q i ) . We have P kqi,hence P* iBut 4 contains neither ‘K2’ nor ‘W2’, so that TIT($) is just @. Also T‘T(P*)= Q*. We have, then, Q * t qi. But Q* 2 Q’, so (Q’ k 4). But Q ‘ was any extension of Q, so Q k qi, hence Q k qi.



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By parity of reasoning, the theorem is true with ‘K,’, ‘W,’for ‘K,’, ‘W,’.
Proof of the theorem We return now to the model N . We had found two sets of integers KW 1R 1 and KW both definable over N as classes, and in such a way that the defi2R 2 nition of KW ‘y did not contain the symbols ‘K2’ or ‘W2’ and the definition 1R 1 of KW‘7 did not contain the symbols ‘K,’ or ‘W,’. Both KWS T and KW ,R2 1R 1 2R 2 were w yjumps of uubs on Ma nP(w). Or,
Let A be a set of integers, A GaKW
A GaKW Then A is also a class ,R2 in N , and is given by two definition, of which one is K 2 , W2free, and the other K , , W,free. It follows that there are two formulas in Oe, $1 and $2 such that Or,
1R 1
where G1 is K , , W,free, and $2 is K1, Wlfree. We have

Let Pi € {P, }force (Vx)($,(x, K , , W,) $J,(x, K,, W,)).Let Pi be ( k l , k,, w l , w 2 ) . We introduce the symbol ‘Q2 P‘ to mean ‘Q 2 P and ki i
and wi are identical in Q and P’. Using this notation, we have the following lemma: Lemma. (Vn) (n E A

(3Q)(Q2 Pi & Q Ii 2
K , , W1 J
Proof: * Let n € A . Let P* 2 Pi, and P* k Gl(n, K , , W,).Let Q be like P* in k , , w1 and like Pi in k,, w 2 . Then, by the independence lemma, Q k 4l(n, K , , W l ) , also Q > P i 2
=Let Q >Piand Q l$l(n, K , , W1). 2 Assume n !$ A . Let Pj 2 Pi, Pi I $,(n, K , , W,),Pi E {P, }. Let Q *be like Pi in k,, w 2 , and like Q in k,, w , . Then, again by the independence $2(n, K , , W 2 ) ; also Q* k $ , ( n , K 1 , W1).But Q* 2 Pi, so lemma, Q * Ithat Q* f $,(n, K , , W,)CJ $,(n, K , , W,). This is a contradiction, hence n €A.


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We assume the following
Lemma Under a suitable Godelnumbering of forcing conditions and sentences, { (P, @) : @ is limited, or, unlimited, of complexity < n , and P k @ }is a class in Ma+ (i.e., € Ma+,,). From the preceding two lemmas, we immediately deduce the
Theorem. If A
5 K;"R':
and A ,where t l , ..., tbi are aterms. Definition 1.9. Let a be a definitional type. Then the aconditions are strings of the form E l & E2 & ...E,, 0 < t , where each Ei is an aatomic formula, and no Ei is Ep i # j . Two aconditions are called incompatible just in case one of the conjuncts of one is the negation of one of the conjuncts of the other.

Definition 1.10. Let a = (n, Fcn, a, b, c) be a definitional type. Then the aclauses are strings ( C  t ) , where Cis an acondition and t is an aterm.
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(C+ c) is interpreted as “if Cholds then it is cat arguments ( u l , ...,un)”. Definition 1.11. Let a = (n, Rel, a, b, c ) be a definitional type. Then the aclauses are strings (C+ true) or (Ct false), where Cis an acondition.
(C+true), ( C + false) are interpreted as “if C holds then it is true at arguments (ul, ...,un)”, and “if Cholds then it is false at arguments (ul , ...,un)”. Definition 1.12. Let a b e a definitional type. Then an adefinition is a set (possibly infinite) of aclauses such that any two distinct elements have incompatible antecedents. Definition 1.13. Let a be a definitional type. Then an effective definitional scheme of type a is an adefinition which is, as a set of strings, recursively enumerable in the sense of ordinary recursion theory. Definition 1.14. Let p be an effectibe definitional scheme of type (n, Rel, a, b, c ) and let M = (D, co, ..., ck, fo, ...,f,.,Ro, ..., R , ) be a relational structure of relational type (a, b, c). Then we write DREL(0, M) for that nary partial relation on D given by DREL(P, M) (xl,..., x n ) is true if and only if there is some element ( C + true) of p such that C holds when each ui is replaced by x i and each diby ci and each
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H. FRIEDMAN
appropriately either P or T or D , and n is mentioned whenever X ( Y ) is T . One defines nRelequivalence similarly. We now wish to prove that any gTa of type (a, b, c) is nFcnequivalent (nRelequivalent) to some eds to type ( n , Fcn, a, b, c) ((n, Rel, a, b, c)), and vice versa. We first consider the first half. We must first define the auxiliary notion of pseudo eds of type a. Definition 1.1 7. Let a be a definitional type. Then a pseudo adefinition is a set of aclauses such that any two elements whose antecedents are not incompatible have the same consequent. Definition 1.18. Let a be a definitional type. Then a pseudo eds of type a is a pseudo adefinition which is, as a set 0 s strings, recursively enumerable in the sense of ordinary recursion theory. In addition we extend the notions DREL and DFCN and nFcnequivalent and nRelequivalent to pseudo eds’s. We also need the auxiliary notion of aformula, for definitional types a. Definition 1.19. Let (n,x, a, b, c) be a definitional type. The aformulae are given by ( I ) each aatomic formula is an aformula (2) F , & ... & F,, 0 < r , is an aformula if each Fi is (3) F is an aformula if F is.

It is clear what it means for an acondition to imply an aformula, in the sense of propositional logic. Lemma I . I. 1. If 0 is a pseudo eds of type a = (n, x, a, b, c) then there is an eds y of type a such that 0 is xequivalent to y.
Proof. Let f be a total function from w onto 0 which is recursive in the sense of ordinary recursion theory. Define a total function g from w into aformulae by g(0) = that acondition Co such that for some x we have f ( 0 ) = (Co + x), g(k + 1 ) = C , & g(i), where there is some y with
M
OdiGk

f ( k + 1) = ( C , + y ) . Then g is recursive in the sense of ordinary recursion theory. Let h be a total function w into sets of aconditions given by h(k) = { C : C is an acondition and C implies g(k) and C mentions every aatomic
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formula or its negation that is mentioned in g ( k ) } . Then h will be recursive in the sense of ordinary recursion theory. Finally set y = {(C+x) : for some k we have C E h(k) and ( f ( k )+x) €0). Then y is the desired eds of type a. This Lemma 1.1.1 is essentially an effective version of the theorem of topology that every open set in the Cantor space can be written as the union of pairwise disjoint basic open sets.
Theorem Z.I. Let (a, b, c ) be a relational type and let 0 < n and let B1 be a fap of some type (n, e, p , 4 , a, b, c ) and let 0 2 be a gTa of type (a. b, c). Then there are eds yl, 7 2 of type (n, x, a, b, c ) such that p 1 is nxequivalent to rl and 82 is nxequivalent to y2.
Proof. Let y be any (n, x , a, b, c)condition. We consider a modified action, called by yaction, of S 1 and p2. For p l , in the imput registers one places each variable ui in register ri. One proceeds with the instructions “if E holds then d o B ; z” with the following conventions. Each noncounting register will either be empty or else will contain an (n, x, a, b, c)term. Unless E is an (n, e, p , q, a, b, c)atom E is selfexplanatory. If E is an (n, e, p , 4, a, b, c)atom and not all of the registers mentioned in E are nonempty, then the standard action is used; namely, E does not apply, and one passes to the next instruction. If they all are nonempty, then E is interpreted as an (n. x, a, b, c)atomic formula. If when so interpreted, it is a conjunct o f y , then it is considered to apply. If its negation is a conjunct of y then it is considered not to apply, and one passes to the next instruction. If neither it nor its negation is a conjunct of y then terminate but do not yield any output whatsoever. If E is considered to apply, then d o B . Doing B is selfexplanatory unless B is “replace ri by the result of app1yingf;jto the contents of ( r t l ,..., rrr)if all I registers are nonempty; otherwise d o nothmg”. In case they all are nonempty, they all contain (n, x, a, b, c)terms, and so replace ri by the (n, x, a, b, c)term Ff(sl, ..., sl), where si is the contents of rti. For the yaction of B 2 , choose n adjacent squares and place the variables u l , ...,u, on them in order and place the reader at the leftmost of the n squares chosen, and start in state q o . Now for the subsequent yaction. We now describe the subsequent action of p2 at any stage: the device discontinues computation and does not yield an output if there is a 1 G j G m such that the bi squares to the right of the square being read all have (n, x, a, b, c)terms printed on them, t l , ..., t ,and both A i i ( t l , ..., tb> and bi  A i i ( t l , ..., t . are not a conjunct o f y . If there is so suchj, then let bl) (~7~. x, eo, ..., e m )be such that the device is in state 4i, x is the set of auxiliary
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symbols printed on the square being read, and ei = 0 if R p t l , ..., t , are as bI above, and A k j ( t l , ..., tb> is a conjunct o f y ; 1 otherwise. If there is no element ((qi,x, eo, ..., e m ) + s) of p2 then: the device discontinues computation and yields as output the contents of the square being read. If there is an element ((qi,x, c l , ..., c m ) + s) of B2, then follow the (u, b, c)command s if it of kind (1)(4). If s is of kind (5) then follow s after replacing “element of D, .o, ..., y b .)’’ by “(n, x, a, b, c)term, ~ i . ( t...~, tb, .)” and replacing I I ‘)j” 6y “ti”,and “elements of D” by “(n, x, u, b, c)terms”. If this modified s cannot be obeyed then terminate computation and yield as output the contents of the square being read. Now take y; to be the pseudo eds of type (n. x, u, b, c) given by 7; { (y + t) : the yaction of yields the (n, x, u, b, c)term t } if x = Fcn; { ( y + false : the yaction of p1 yields  } U (0,+ true : the yaction of p1 yields something other than  } i f x = Rel. Take y; to be the pseudo eds of type (n, x, a, b, c) given by 7;= { 0,+ t ) : the yaction of 132 yields the (n. x, a, b, c)term t } if x = Fcn; (0,+ false) : the yaction of 132 yields a blank} U (0,+ true : the yaction of 132 yields something other than a blank}. Finally choose yl,72 to be eds’s of type (n, x, a, b, c) which are, respectively, nxequivalent to y; ,y;, by Lemma 1.1 .l.
FA
Theorem 1.2. Let (n, x, a, b, c) be a definitional type and let /3 be an eds of type (n, x , a, b, c). Then there is a gTa y of type (a, b, c) such that 13 is nxequivalent to y, Proof. We only give an intuitive argument for the case x = Fcn, of the action when applied t o a structure M of type (a. b, c) at imput (xl, ..., x n ) . Generate all the elements of 0.Doing this much does not require use of the constant, function, and relation symbols, although the elements of 13 mention them, and are used as auxiliary symbols. Whenever an element (C+ t ) is generated, replace all occurrences of variables ui by xi.Next all (n, x, a, b, c)terms occurring in this (C + t) are replaced by their values (elements of D)by first evaluating subterms of lowest complexity, and then of higher complexity. Then the (n, x, a, b, c)atomic formuae are evaluated as true of false, and then finally C i s evaluated as true or false. If C is evaluated as true, then the value o f t previously computed is given as output. If it is evaluated as false, then the tape is cleared of (C+ t ) and the generation of 0is continued. We continue with our comparisons between f a p , gTa, and eds. Let M ( 2 ) be the structure (0, 0, 1, =) of type (( l), ( 2 ) , (2)), where D o
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is given by (a) 0 , l E D o (b) (s, t ) EDo if s, t ED. Thus every element of M ( 2 ) is given by a term; take f ( x , y ) = ( x , y ) . Let D+ be given by (a)O, 1,mED' (b) (s, t ) ED+ if s, t E D + . Let g : D+ + D+ be given by g(x, y ) = (x, y ) .
Definition Z.20. Let 0 < p . Then x EDo is called pregular if and only if there is a finite sequence ( y l ,..., y,) such that (1) each y i is a sequence of elements of D+ U {  } of length p ( 2 ) some entry of y, is x ( 3 ) each y S l differs from y i in at most one entry, where some z ED' was changed to some (zl, z2), or to 0, or to 1, or to , or to z l , or to 00, where z 1 , z 2 are both entries o f y , (4) each entry ofyl is either 0,1 ,m, or . Such ( y l , ..., y,) are called pgood.
Lemma 1.3.1. For each 0 < p there is a x p E D o which is not pregular; furthermore, no y € Do is pregular if x p occurs as a subexpression of y .
Proof. Define the desired x p inductively by x 1 = (0,I ) , X ~ =+ ((y. ~ z), (z, y)), where x k = 0.1,z). Then one proves by induction on p that x p is not pregular. The basis case is trivial. If (jl, ...,y,) is k t 1good and x ~ is+an ~entry in y,, then note that x ~ =+( y ,~z), and by induction hypothesis both y and z are not kgood. But there has to be an entry in (jl, ...,y,) where y occurs and z does not (or vice versa), unless k = 1, in which case the lemma is true by trial and error. As long as y stays, one is handicapped in one's efforts to obtain z together withy;in fact, one cannot make use o f y for this purpose, and can only use k entries. Hence z cannot be obtained together w i t h y in a k + 1good sequence since z is not kregular. (Of course, once y disappears, one has to obtain it all over again.)
Lemma 1.3.2. Let a = ( 1 , e, p , q, ( l ) , (2), (2)) be a procedural type, and let f.3 be any fap of type a.Then PREL(f.3,M ( ~ ) ) ( X , + ~EZ+PREL(f.3, ~) M(2))(xn+2+,).
Proof. Note that there are n t l t e registers in which to place elements of Do in following f.3 applied to M ( 2 ) . Using lemma 1.2.1 it is obvious that the action of f.3 applied to M ( 2 ) at x , + ~ + is ~ isomorphic to the action of f.3 applied to M(2) at x,+2+e.
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Theorem 1.3. There is an eds of type (1, Rel, (l), (2), (2)) which is not 1Relequivalent to any fap of type (1, e, p , 4, (I), (2), (2)). Proof: Consider the eds P given by P = {(A!(ul, f k ) + true : k is even, 0 < k } U { ( A ! (ul, f k ) + false : k is odd, 0 < k } , where f l
[email protected](do, dl), fk+l = F0 ~ 0( F ~ s2),@(s2, ( ~ ~ , sl)>,where f k = F : ( S ~s2). , Then use lemma 1.2.2 to show that DREL(0, M(2)) # PREL(y, M(2)) for all fap y of type (1, e, p , 4 , ( l ) , (2), (2)). Note that we have shown much more than the theorem. Namely, we have shown that there is a futed structure with equality and such that every element is given by a term, on which there is a simple eds which cannot be simulated by any fap. From the proof of Theorem 1.2 we easily obtain the following. Corollary 1.3.1. There is a structure M with equality and with every element of the domain given by a term and eds’s S1, 02 of type (1, Rel, ( l ) , (2), (2)), (l,Fcn,(1),(2),(2))such that for all fap’syl,y2 o f t y p e ( l , e 1 , p l , 41,(1), (21, (211, (1,e2, P 2 , 4 2 , (1), (21, (2)) we have DREL(P,? M ) f PREL(y1, M ) and DFCN(P2, M ) f PFCN(y2, M ) .
~ r o o f TakeM=M(2),P1 i asPabove,P2={(A!(ul, f k ) + o : k i s even, O < k } U {(A!(vl, t k ) + 1 : kisodd,Of2)(n2) < r ( f i 9 f 2 ) ( n 1 ) holds. We shall show in the next section: if (1) is satisfied, then S2 is recursive in
Sl,f,>f,,T . Applying this result to the function r(f2,f1) we obtain: if (2) is satisfied, then Slis recursive in S 2 , f l , f2,T.This and the following remark will complete the proof of the theorem: Let S,,S 2 be two recursively enumerable sets and let fi(i= 1,2) be recursive onetoone functions from N onto Si.Let T'
RAMSEY’S THEOREM
44 1
be an infinite recursively enumerable Rhomogeneous subset of N ; T ’ contains an infinite recursive subset T which is also Rhomogeneous. f l ,f2 and T being recursive, a set recursive in S1 , f l ,f 2 , T is recursive in S,. The existence of an infinite recursively enumerable Rhomogeneous set T’ implies therefore that either Slis recursive in S2 or S 2 is recursive in Sl. Neither of this need be, as there exist recursively enumerable sets of incomparable degrees.
5. Let fi, i = 1 , 2 , be onetoone functions from N onto Si,let r be the function defined by
and let T be an infinite set such that for all n 1, n 2 in T such that n < n 2 the inequality r ( n l ) d r ( n 2 ) holds. Let g be a function as established in the lemma of Section 3: g is recursive in f i ,Sl, T ;g(rn) is an element of T for all rn in N a n d for all k, rn in N such that g(rn) d k we have rn + 1 dfl(k). Define the set S; as follows:
rn E S; if and only if (3x) (x d g(rn)
A
m =f2(x))
S2 is clearly recursive in f2,g and therefore recursive in S,, f l , f2, T . We show Si = S 2 . (1) If rn ES; then (3x)rn = f 2 ( x ) and rn ES2. ( 2 ) Assume rn E S 2 . In order to prove (3x) (x d g(rn) A rn = f 2 ( x ) ) it
suffices to show the following: For all j such that g(m) + 1 d j we have f 2 ( j ) # m. Assume therefore g(rn) + 1 d j and choose n in T such that j d n.
g(m)and n being elements of T such that g(rn) < n we have
therefore n
If g(rn) + 1 d k , then rn have
n
+ 1 dfl(k); the functionfl being onetoone, we
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442
(q being a sufficiently big number).
From n
we conclude m f f 2 ( k ) for all k such that g ( m ) + 1 < k < n ; as n can be chosen arbitrarily large, we finally have nz2 f J i ( k ) for all k such that g(m)+ 1 G k . 6 . The known proofs of Ramsey’s theorem show that there exist arithmetical Rhomogeneous sets for recursive relations R . A somewhat careful analysis of one of these proofs establishes the fact that there exists such a set in the class V 3 V n 3 V 3 . Problem. What is the first class C in the arithmetical hierarchy such that for every recursive (binary symmetric) relation R there is a Rhomogeneous set in C ? (Editors’ note: this problem has been completely answered by Jockusch  see [4] and [ S ] .)
References [ 11 F.P. Ramsey, On a problem in formal logic, Proc. London Math. Soc. 30 (1930) 264 286. [ 21 R.M. Friedberg, Two recursively enumerable sets of incomparable degrees of unsolvability, Proc. Natl. Acad. Sci. U.S. 43 (1957) 236238. [ 31 A.A. MuCnik, On the unsolvability of the problem of reducibility in the theory of algorithms, Dokl. Akad. Nauk. SSSR (N.S.) 108 (1956) 194 197. [4j C.J. Jockusch, Ramsey’s theorem o n recursion theory (to appear). (51 C.E.M. Yates, A note on sets of indiscernibles, (these proceedings).
A NOTE ON ARITHMETICAL SETS OF INDISCERNIBLES C.E.M. YATES Manchester University, England
Specker has proved that there is a recursive partition (of the set N of natural numbers) which possesses no infinite recursively enumerable (Z1) 0 set of indiscernibles; the existence of some infinite set of indiscernibles is simply the familiar theorem of Ramsey [ 3 ] .Specker’s proof depends on the solution of Post’s Problem by making use of two 27 sets of incomparable degrees of unsolvability. Our main purpose in this note is to present (in Section 1) an entirely different proof based on the theory of retraceable sets.’). A shorter proof has been obtained more recently by Jockusch [ 11 ,who in fact obtains a stronger theorem (replacing Z7 by Z!) discussed at the end of Section 1. Specker observes at the end of his paper that the usual proof of Ramsey’s theorem guarantees the existence of an infinite A: set of indiscernibles (for a recursive partition) and asks to what extent this can be improved. In terms of arithmetical classification this question has been completely answered by Jockusch by means of the negative result just mentioned combined with the positive result that A! can be improved to II;. We present our own proof of this second result in Section 2 to serve as a focal point for the discussion there. Once again it depends on an elementary ‘priority argument’. As usual, a set is called Z ,: n:, A: if it is definable in prenex normal form with n alternating quantifiers where the first quantifier can be chosen to be respectively existential, universal or both. We use P,(S) for the set of all nelement subsets of a set S g N . A kpartifion of Pn(S) is simply a sequence of k disjoint sets whose union is Pn(S). I f A = ( A o , ..., A k ) is a kpartition of Pn(S) and I C S then Z is a set of indiscernibles for A if Pn(Z) E A j for some j < k ’). For other background concepts the most useful ’)I first became aquainted with Specker’s theorem at his Manchester talk in March 1966
and soon afterwards discovered the different proof expounded in Section 1. I am grateful to Georg Kreisel for informing me of Jockusch’s work and relivening my interest in this topic, and also to Jockusch for various informative and critical comments. ’1 Specker uses the term ‘Rhomogeneous set’ in his contribution to this volume [ 51 .
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reference is Rogers’ book [4] ;in particular, there is some lengthy discussion of retraceability in the exercises to Chapter 9 of [4] ,and in combination with [2], [ 6 ] and [7] this should be more than adequate. For convenience, however, we recall the basic definitions. A set X (with elements x o , x l , ..., in their natural order) is retraceable if there is a partial recursive function F such that F ( X , + ~=)x n for all n and F ( x o ) = x o ; F is said to retrace X . A partial recursive function is a retracing function if it retraces at least one infinite set. There is no loss of generality for most purposes in assuming a retracing function F to be normalised (or special as it has previously been called): i.e. im(F) dom(F) and F ( x ) < x for all x . Our notation will differ somewhat from [4] . We use Re to denote the eth Zy set in some standard recursive enumeration of these sets, and R i to denote the finite subset of Re enumerated at stage s in some standard recursive approximation to R . Small Greek letters such as u and T will be used to denote strings, i.e. finite sequences of integers. For our present purposes all strings may be supposed to be strictly monotonic. The length lh(u) of a string u is the number of elements in it, and for any two strings u and 7 we use u * 7 to represent the concatenation of u and 7,i.e. the string obtained by listing the elements of T after those of u. We use (4 to denote the null string (this should cause no confusion with the null set). Finally, for any function F we use F [ n ] to denote the string (F(O),..., F(n));this notation is also used for the corresponding substrings of strings. 1. The proof we give of Specker’s theorem depends on the existence of a retracing function F which retraces exactly one infinite set X and which is such that both X and f a r e immune. The existence of such a function F was first proved by McLaughlin [2] , and its construction involves an interesting elementary ‘priority argument’ which is roughly of the same level as that used to solve Post’s Problem. In order to make this note selfcontained and because this theorem is probably not wellknown, we begin this section with a short proof of it which uses the estate method of presentation. Recall that a set is immune if it is infinite but has no infinite Zy subset. It is easy to prove that any retraceable set is either recursive or immune and so it will be sufficient to arrange that x i s immune. Also, it is easy to see that if F retraces an infinite set X such that x i s immune then F can not retrace any other infinite set; for, if ro belongs to such another infinite set, then : ( 3 z ) ( F ( z ) ( y )= ro) } is an infinite Zy subset of Hence it is sufficient to prove:
x.
Theorem 1.I (McLaughlin). There is a (finiteone) retracing function F which retraces a set X such that x i s immune.
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hoof. F will be defined as the limit of a recursive sequence of finite functions { P } the , construction of FSproceeding stagebystage alongside the approximation {R:}to {Re }. Before describing the construction of FSwe shall need some preliminary definitions. First, an estate p is a string (eo, ..., e k ) with eo < ...< ek Q e . An estate p is higher than an estate p' if the least number in one but not the other belongs to p. Notice of course that there are only finitely many estates for any particular e . Secondly, a string u satisfies e at stage s if u(x) ER,"for some x E dom(u). More generally, u satisfies an estate p = (eo, ..., e k ) at stage s if for each j < k there is an xi such that u(xi) ERii. Lastly, a string u is eligible if it is strictly monotonic and u(x) 2 2r for
all X. Now, we can turn to the definition of F . The essential part of this definition is in fact the definition of an eligible string us. Let uo and fl both be empty. For s > 0 there are two cases. Case I : there is a number e and an eligible string u which satisfies a higher estate than uSl at stage s, and either (a) u = d for some r < s or (b) u = uSl * 'k' for some k 4 dom(FSI). In this case we let es be the least such e and us the first string (in some standard enumeration of the strings) corresponding to 8. Case 2: otherwise. We leave us = us1. Having defined us it is easy to define F S :it is simply the smallest function extending FS' which retraces the elements of us.The existence of F S is made possible by the condition k 4 dom(FSl) in Case l(b) above (which is the only case in which FS# Fsp1).Hence it is easy to see by induction on s that FSl FSfor all s > 0. The required function F is the union of the sequence {FS}. The set X which we claim has the properties stated in the theorem is in a sense the limit of the sequence { u s } .Before this can be made clear, we need the simple but important lemma that follows. Notice that among the estates which are satisfied at various stages, there must be a highest one p e . Let s(e) be the first stage s at which us satisfies pe. Lemma A . U S 2 ude) for all s 2 s(e).
Proof. Suppose there is, for reductio ad absurdurn, a least stage so 2 s(e) such that us 3
[email protected]).Then there exists an fsuch that us0 satisfies a higher fstate than 0 ~ 0  l at stage so. Moreover, f > e because ~ ~ 0 2  lus(e).Now, there is no k such that us0 = uS0l * 'k', and so us0 = d for some r < so. But r can not
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be 2 s(e) because then uS0 2
C.E.M. YATES
and also r can not be < s(e) because each
ur for r < s(e) satisfies a lower estate and hence a lower fstate (since f > e) than us(e) and 0 ~ 0  l . This is the required contradiction.
Let ue = It follows from Lemma A that uo E u1 ...; let X be the range of lime ue. Then X is infinite because u, is eligible for each e. It is not, however, immediately clear that X is infinite. This follows from the immunity of F which in turn follows from the next lemma. Lemma B . For each e , if R e is infinite then ue satisfies e.
Proof. Suppose, for reductio ad absurdum, that e is not an element of pe. Let so > s(e) be such that there exist a number k o in R$’ which exceeds 21h(ue) and all the elements of us for s < s(e). We claim that uSO satisfies a higher estate than ve, which is the required contradiction. For, either ur satisfies a higher estate than ue, for some r < s, or uSO is inevitably defined t o be ue * ‘ko’.To see this, we have only to verify that k o 4 dom(FSol). But if ko E dom(FS0’) then k o belongs 10 ur for some r such that s(e) < r < so; in which case ur satisfies a higher estate than p,. It follows immediately from Lemma B that x i s immune, and in particular that X is infinite. It is clear that X i s retraced by F since u, is retraced by Fs(e)for each e. This concludes the proof of Theorem 1 . l .
As we observed before Theorem 1.1, we can immediately deduce the stronger result we need: Corollary 1.2. There is a (finiteone) retracing function which retraces exactly one infinite set X and is such that both X and F a r e immune. Now, we turn to deducing Specker’s theorem. Definition 1.3. Let F be a retracing function. The partition of sociated with F is { Ao, A } where
P, (im(F))as
Notice that any subset of a set retraced by F is a set of indiscernibles for the associated partition.
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Theorem 1.4. Let F be a retracing function which retraces exactly one infinite set X and is such that both X and 2 are immune. Then the partition of P~(irn(F))associated with F has no infinite Zy set of indiscernibles. Proof. Let A = { A , , A ,} be the partition of P2(im(F)) associated with F . If I is an infinite set of indiscernibles for A then either I C X or I differs from a subset of X by one element; for, if P 2 ( I ) C A o then J = {F(Z)(x): x € I } is an infinite set retraced by F , and if P2(I) C A , then I cannot contain more than one element of X . Since X and 1are both immune, the theorem follows.
Coro1Zu.y 1.5 (Specker). There exists a recursive partition { B o ,B , }of P,(N) which possesses no infinite 2 ; set of indiscernibles. Proof. Let F be a retracing function having the properties required by Theorem 1.4; its existence was proved in Corollary 1.2. Let G be a oneone recursive function whose range is im (F). Define {x,y}EBi++ { C ( x ) , G ( y ) } E A ,
i = 0, 1
where A = { A , , A } is the partition associated with F. Then B = {B,, B , } is a recursive partition of P2(N) because G is oneone. Let I be an infinite set of indiscernibles for B. It follows that if
I’ = {G(x) : x € I } the I’ is an infinite set of indiscernibles for A . Clearly, if I is Zy, then I‘ is Zy; we deduce that I is not Zy. Jockusch has announced [ 11 a stronger theorem, namely that there is a recursive partition of P2(N) which has no infinite Z! set of indiscernibles. Unfortunately, the present method can not be used to deduce this stronger theorem, because if F retraces X and F is nor finiteone then for some x, y such that F ( y ) = x the set { u : F(u) = x & u # y } is an infinite Zy subset of f.Also, by Theorem 5.2 of our paper [7] ,if a finiteone retracing function retraces a unique infinite set then that set must be of degree < O(l). These two observations make it unlikely that any significant extension of this particular method will be found.
2. As Specker observes in [S] , an analysis of the usual proof of Ramsey’s Theorem reveals that any partition A of P2(N) possesses a set of indiscer
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nibles w h c h is A! in A . In the present section we give a proof of Jockusch’s recent observation that A! can be replaced by ns. The proof is a nice example of a special procedure for reducing the quantifierform of a set having certain prescribed properties. This general procedure is usually referred t o as ‘the priority method’, although this now familiar nomenclature is rather inappropriate and could be more accurately replaced by ‘the approximation method’ or ‘the trialanderror method’. It will probably be helpful to first sketch the proof for AS. So let A = { A O , A ’ } be a partition of P2(N). We make an approximation { I , } to the required set I of indiscernibles as follows. I , is defined by induction on e, simulLettingIO= N w e define taneously with apartition A, = {A:,A:}ofI,.
A { = { x :xEI,&{I,(O),x}€Aj}
j=O,l,
where I,(O), Ie(l), ..., is a listing of the elements of I, in their natural order. Then we define:
o
i f A 2 is infinite ,
1
otherwise,
J(e)= and let = Ad(e).It is clear that I , is uniformly of degree Q a(2) where a is the degree of A , but we can say no more than this (equivalently, I , is uniformly A! in A). Let E o ( 0 ) , E o ( l ) ,..., and E l ( 0 ) ,E1(l), ..., be listings of those e such that J ( e ) = 0 and J ( e ) = 1 respectively. If for each j E (0,1 } we set
f o r a l l x , a n d l e t I j = {Ij(O),Ij(l), ...}, thenitiseasy tosee t h a t I O , I 1 are sets of indiscernibles for A which are of degree < a(2)(equivalently, belong to A,(a)). At least one of 1 0 , I must be infinite. Using the extension of the KreiselShoenfield basis theorem which we mentioned in [7] ,it is possible to obtain a set of indiscernibles of degree <
[email protected]).
fieorem 2 (Jockusch). Let A be any 2partition of P2(N). Then A possesses an infinite set of indiscernibles which is IlS in A . hoof. Let A = { A o , A
1 be a 2partition of P2(N). We make a double ap
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proximation { I e f } to the required set I of indiscernibles; the sequence { I e f } will be uniformly of degree < a(') where a is the degree of A . It is this extra approximation which enables us to reduce the quantifierform of I . For each e , limfIef will exist, and lettingle = limfIefwe then f o r m I o , I 1 from the sequence {I,} in the usual way as outlined before this theorem. I o will E @(a) and if I o is finite then I' will be infinite and E @(a), so one of these two sets satisfies our demands. We define I,, for each f by induction on e , simultaneously defining a 2partitionAef= {A:f,A$}ofIef. Let Zof= N f o r allfand define
A&= {x : x € I e f & { I e f ( 0 ) , X } € A j } for each j E (0, 1 1, where Ief(0),Ief(l), ..., is a listing of the elements of Ief in their natural order. We next define
Note that if e < el then Ierf C_ Ief for all f. Clearly, for each e if limfIef exists then J(e, f ) and I e + ' i can change at most once for f > F(e), where F(e) is the least fsuch that I e f = limfIef. It is important to note that F is monotonic, as can be verified by careful examination of the construction. Since limfIof exists by definition, we conclude that J ( e ) (= limf J(e, f)) and Ie (= limfIef) are both defined. Similarly,AL (= limfAif) exists for all e and j E (0, 1 }. Another important observation is that if Ief #Ie,,, then Ief(0)>fi this ensures that Ie(0)2 F(e). Next, note that for each e , if I is infinite then one or other of A$,,Aif must be infinite, so that is infinite. Hence we conclude that Ie is infinite for all e . Finally, it is not difficult to verify that J , { I e f }and { A d f }are uniformly of degree < a('). We now define I o , I' from { I e } as before. Let Eo(0),EO(l),..., and E1(0),E1(l), ..., be listings of those e such that J ( e ) = 0 andJ(e) = 1 respectively (both taken in their natural order). For each j E (0, 1 }we set
ef
for all x , and let I j = { I j ( O ) , I j ( l ) , ...}. This concludes the construction. Our first claim is that both I o and I1 are sets of indiscernibles for A . We
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claim in fact that
for e a c h j E ( 0 , 1 }. To prove this, let { I i ( x 1 ) , I i ( x 2 )be ) a typical element of P 2 ( I j ) w i t h x , < x 2 . Then
IJ(x2 ) = IEj(X2),F(Ej(X2))(0)
k j ( X 1 )+l,F(Ej(X*))
C Ie,f for allfand e, e' such that e' > e. But then it follows that because I J ( x 2 )EIE.(x )+' because F(Ei(x2)) 2 F ( E i ( x 1 ) + 1 )and so IE.(xl)+l= 1
. I
1
I E ~ ( ~ ~ ) + ~ , F ( E Hence ~ ( ~ ~ )I)j (.x 2 ) € A &I. ( , 1) and so { I i ( x l ) , I 1 ( x 2 ) } = {IE.cx )(O), I i ( x 2 ) )E A i , which is the desired result. Our second claim is that 1 1
I o E n!(a) and that if I o is finite then I' E n!(a). The latter part of the claim follows from the membership in @(a) of I , where / = Zo U I' = {Ie(0) : e 2 0 ) . To see first that I E @(a), observe that
&L
(Vh)h,fW, h ) = 0)) .
(The final clause holds because, for h > F(e), J(e, h ) can only change from 0 to 1, not from 1 to 0.) Either I o is infinite in which case it is the required set, or I' differs from I by a finite set and so E @(a). This is in a sense the best possible result because of Jockusch's other theorem mentioned in Section 1 : there is a recursive partition with no infinite Cs set of indiscernibles. There are, however, other ways in which one might seek to strengthen it. For example, one might hope to prove that if b is r.e. in and > O(l) then every recursive partition has an infinite set of indiscernibles which is of degree < b. We can not see, however, how to adapt the
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the proof above to the usual techniques that are appropriate for this sort of problem. A much fuller investigation of the various possible extensions of Theorem 2 can be found in [ I ] .
References [ 11 C.G. Jockusch, Ramsey’s Theorem and Recursion Theory (to appear). [ 21 T.G. McLaughlin, Splitting and decomposition by regressive sets, 11, Can. J. Math.
19 (1967)291311. [ 31 F.P. Ramsey, On a problem in formal logic, Proc. Lond. Math. SOC. 30 (1930)
264286. [ 41 H. Rogers, Theory of recursive functions and effective computability (McGraw Hill, 1967). [S] E. Specker, Ramsey’s theorem does not hold in recursive set theory, this proceedings. [6]C.E.M. Yates, Recursively enumerable sets and retracing functions, Z.Math. Logik und Grund. Math. 8 (1962) 331345. [7]C.E.M. Yates, Arithmetical sets and retracing functions, Z. Math. Logik und Grundlagen der Math. 13 (1967) 193204.