Table Of Content

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. ROBINSON, New Haven P. SUPPES, Stanford Advisory Editorial Board Y. BAR-HILLEL, Jerusalem K. L. DE BOUV~RE,S anta Clara H. HERMES, Freiburg i. Br. J. HIN TI K K A, Helsinki J. C. SHEPHERDSON, Bristol E. P. SPECKER, Ziirich NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM LONDON LOGIC COLLOQUIUM '69 PROCEEDINGS OF THE SUMMER SCHOOL AND COLLOQUIUM IN MATHEMATICAL LOGIC, MANCHESTER, AUGUST 1969 Edited by R. 0. GANDY C. M. E. YATES 1971 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM LONDON 0 North-Holland 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, record- ing or otherwise without the prior permission of the copyright owner. Library of Congress Catalog Card Number 71-146188 International Standard Book Number 0 7204 2261 2 PUBLISHERS : NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NORTH-HOLLAND PUBLISHING COMPANY, LTD. - LONDON 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 to 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 Associa- tion 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 men- tioned organisations, the Vice-Chancellor and University of Manchester, Mrs E.C. Connery (Administrative Assistant to 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 to 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 to 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. Dymacek, M. Hart, J. Ash, J.F. Eklof, P.C. (Ja) Hay, L. Aczel, P. (Ca) Elliot, J.C. Hayes, A. Aloisio, P. Engelfriet, J. Henrard, P. Bacsich, P.D. (Ca) Feiner, L. (Ca) Herman, G.T. (Ca) Barendregt, H.P. Felgner, U. (Ca) Hermes, H. (Ib) Barwise, J. (Ib, Ja) Fenstad, J.E. (Jab) Hinman, P.G. (Jb) Bezboruah, A. Fischer, G. Hirscheimann, A. Bud, M.R. Fitting, M. Hodges, W. Boffa, M. (Ca) Flum, J. Hopkin, D.R. Boos, W. Fredriksson, E. (Ca) Howson, C. Booth, D.V. (C) Friedrichsdorf, U. de Iongh, D.R. Bridge, J.E. Fris, M.A. Isaacson, D.R. Biichi, J.R. (I) Gabbay, D. (Cab, Ta) Jackson, A.D. Bukovsky, L. (C) Candy, R.O. Jahr, E. Carpenter, A.J. Gavalu, M. Jech, T. Chudacek, J. Gielen, J. Jensen, R.B. (I) Clapham, C.R.J. Gielen , W. Johnson, D.M. Cleave, J.P. Gilmore, P.C. (C) de Jongh, D.H.J. (Ca) Cooper, S.B. (Ta, Ta) Gjone, G. Jugasz, I. Cramp, G.A. Gloede, K. Jung, J. Cusin, R. Glorian, S. Karp,C. (L) Cutland, N.J. (Ca) Goodman, J.G.B. Keune, F.J. Daguenet, M. Gordon, C. Koppelberg, B.J. (C) Daub, W. Gornemann, S. Krcalova, M. Davies, E. Gostanian, R. Kripke, S. (I) Derrick, J. Grant, P.W. Kunen, K. (Ib) Devlin, K.J. Greif-Muhlrad, C. Lablanquie, J.C. Dickmann, M.A. Grigorieff, S. Lacombe, D. (L) Drabbe, J. Grivet, C. Lassaigne, R. Drake, F.R. Hajek, P. (Ib) Lolli, G. Dubinsky, A. Hamilton, A.G. (Ca) Lucas, T. xiii xiv LIST OF PARTICIPANTS Lucian, M.L. Pincus, D. (Ca) Smith, J.P. Lynge, 0. Platek, R. (Ib) Sochor, A. McBeth, C.B.R. Potthoff, K. (Ca) Solovay, R.M. (1) Macchi, P. Pour - El, M. Staples, J. Machover, M. Prestel, A. Stepanek, P. (Ja) Machtey, M. (Ca) Putnam, H. (Ib) Stomps, H.J. Macintyre, A. (Ca, Ta) Rabin, M. (L, I) Strauss, A. Magidor, M. (C) Renc, 2. Suzuki, Y. Marek, W. (Ja) Richter, M. De Swart, J. Martin, D.A. (1) Richter, W. (Ib) Tall, F.D. Meyer, E.R. (Ja) Rodino, G. Telgarsky, R. Minsky, M. (1) de Roever, W.P. Thomas, M. Monro, G.P. Rogers, H. Treherne, A.A. van der Moore, W.A. Rogers, P. Truss, J. Morley, M. Rose, H.E. Ungu, A.M. Moschovakis, Y.N. (Ib, Jb) Rousseau, G. Villemin, F.Y. Moss, B.J. Rowse, W.G. Volger, H. Moss, M. (Cab) Sabbagh, G. Wainer, S.S. Myhill, J. Sacks, G. (Ib) Wdsilewska, A. Nyberg, A.M. (Jab) Sakarovitch, J. Waszkiewicz, J. (C) Ore, T. Jr. Scholten, F.P. Wcglorz, B. (I) Pacholski, L. (Ja) Scott, A.B. Weiner, M. Paris, J.B. (Ta) Shepherdson, J.C. van Westrhenen, S.C. Paterson, M.S. (la) Shoenfield, J.R. (Lb) Whitehouse, R.M. Pelletier, D.H. Siefkes, D. (Ca) Wilmers, G.M. Perlis, D. (C) Silver, J. (L) Worboys, M.F. Philp, B.J. Simmons, H. Worrall, J. Pilling, D.L. Slomson, A. 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 intro- duce 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 them- - selves. 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.Nw ill denote the set of nonnegative integers and E will be any set. By means of relations {Ri}a nd functions {pj}w e 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 E A 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 4 D. LACOMBE - - (ii) SupposeAs’PE, E=N,, A=No. If we put on A the condition (VXEA)( VY)(X=Y (modu1o the finite subsets ofE) Y € A ) , + o1-n1t i, then there exist continuously many maps T : E E for which TA = A.T his holds because we only consider monadic relations. We obtain a counter- example if we take E = Nand let A = the set of all two-place 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 EE)({x)EA), (d)(VXEA)(Xinfinite +(3Y,ZEA)(Y,Z infinite A YUZ = x A Y ~ z = ~ ) ) . Let E‘,A‘ satisfy the same conditions. Then we can prove that they are isomorphic: (37 : E ozE’)(TA= 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 iso- morphism may fail. There are interesting problems involving denumerable lattices of subsets of N. Consider the following lattices: (a) the set of recursively enumerable sub- n: sets of N, (b) the 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 one-one. + RECURSION THEORETIC STRUCTURE 5 Given a structure S= {Rj}U {pj} on E, a numbering a : N + E will enable us to compare this structure with various well-known 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 a-lS has this property? Definition 2.2. (i) s is 3-recursive (of recursive type, decidable) iff 1-1 (3a : N o-E)(a-'i~s r ecursive). 1-1 (ii) S is V-recursive iff (Va : N .=E) (CIS is recursive). Property (i) looks like a Zi relation. If we limit S then it may become arith- metical. 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 two-place 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 Zi . 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,f l E Ore, where the sub-group generated by a,f l is not only not 3- recursive 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 counter- example 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 well-ordering on a denumerable E to be of recursive type is equivalent to it being of primitive recursive type, or of arith- metic type, or even of Zt type: it is to be a recursive ordinal. FraissC [2] makes two interesting conjectures about necessary and sufficient conditions for 3-recursiveness. 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 V-recursiveness in (Definition 2.3). Since we may replace functions by graphs we will only consider relations (el, p = (PI, ..., P, 1, Q = ..., Qn) on E. 6 Li. LACOMBE 5 Definition 2.3. (i) P is 3 -recursive in Q iff (3a : N E)( a-l P is recursive os in a-lQ). (ii) P is V-recursive in Q iff ( ~: Na E)( a-1 P is recursive in a-1~). We cannot immediately define a notion of 3!-recursive (in) since if a is a numbering for which a-lP is recursive (in a--IQ)t hen 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 al , ..., 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 yl, ..., pm) and suppose the structure is 3-recursive. Then it is 3 !-recursive. And a necessary and sufficient condition for the structure to be 3-recursive 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 1-1 map of N onto E such that =G is recursive but successor is not. (b) (Q( the rationals), +) is 3 !-recursive (and the only such CU also makes X recursive). (c) (RA (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 F@)E = EEp (i.e., the set of functions EP + E) , FE= U F(p)E, PEN R@)E = PEP (i.e., the set of p-place relations on E), We will see later that V-recursiveness is equivalent to the simple property of being reckonable (Definition 2.4). Corresponding to V-recursive in there will be two equivalent notions: (1) reckonable in (Definition 2.1 l), and (2) the notions due to FraissC of Fo-recursiveness and F-recursiveness (Definition 2.12). Definition 2.4. Let P E R(p)Ew here E, may be denumerable or not. Then: (i) P is radical1,v reckonable if P(x , ..., xp) '3 (xi = xi) on E, where