Southeast Asian Conference on Logic: Proceedings of the Logic Conference Singapore, 1981 PDF
Preview Southeast Asian Conference on Logic: Proceedings of the Logic Conference Singapore, 1981
STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 111 Editors J. BARWISE, Stanford D. KAPLAN, Los Angela H. J. KEISLER, Madison P. SUPPES, Stanford A. S. TROELSTRA, Amsterdam NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD SOUTHEAST ASIAN CONFERENCE ON LOGIC Proceedings of the Logic Conference Singapore, 1981 Edited by C.-T. CHONGandM. J. WICKS National University of Singapore Kent Ridge Singapore 1983 NORTH-HOLLAND AMSTERDAM 0 NEW YORK 0 OXFORD ELSEVIER SCIENCE PUBLISHERS B.V. (NORTH-HOLLAND), 1983 @ All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the priorpermission of the copyright owner. ISBN: 0 444 86706 6 Published by: Elsevier Science Publishers B.V. (North-Holland) P.O. Box 1991 1000B Z Amsterdam The Netherlands Sole distributors for the U.S .A . and Canada: Elsevier Science Publishing Company, Inc. 52 Vanderbilt Avenue New York, N.Y. 10017 U.S.A. Library of Congress Cataloging in Publication Data Southesst ~8i.aC onference on Logic (1981 : singagore) Proceedings of the Logic Conference, Singapore, 1981. (Studies in logic and the foundations of mathe- maticn ; v. xu) BibUOcaPhY: P. 1. Logic, Gymbolic and msthcmatical-Congresses. , I. Chow, C.-T. (Chi-Tat), 1949- 11. wick6 M. J. (Hilcolm John), 1928- 111. Title. I&V. fis&t8r 6. 1981 5ll. 3 83-11458 mm 0-444-86706-6 (u .s .I PRINTED IN THE NETHERLANDS V PREFACE The visit of Gerald Sacks to the National University of Singapore in 1981 provided an opportunity to organize a short conference in Mathematical Logic. We were fortunate to receive encouragement and material support for this venture from several sources. Specific acknowledgements are made below. Sponsorship of the conference by the Association for Symbolic Logic was received and gave added inspiration. A final word in this connexion concerns the debt we owe to invited speakers who were able to provide for travel expenses from their own resources. Their presence a t the conference would not have been possible otherwise. The publication of these Proceedings came about through an initiative of North Holland. The progress in producing the volume has been somewhat fitful, and we appreciate their forbearance and understanding. The items herein, with some exceptions, are written versions of invited talks given at the conference. Abstracts of contributed papers have appeared in the Journal of Symbolic Logic. The two workshops: in Recursion Theory by Mark Tamthai and Model Theory by Chris Ash, which were held in conjunction with the conference, are not recorded. We were fortunate to receive editorial assistance from John Bell during his visit to NUS. A great debt is owed to Mimi Bell and Madam Lam for producing the splendid typescript for the vol ume. Malcolm Wicks vi ORGANIZATION AND SPONSORS LOGIC CONFERENCE: November 9 - 13, 1981 ORGANIZERS Department of Ma thema tics, National University of Singapore and Singapore Mathematical Society SPONSOR Association for Symbolic Logic ORGANIZING M. J. Wicks, Chairman COMMITTEE C. T. Chong, Secretary J. N. Crossley H. H. Teh VU ACKNOWLEDGEMENTS It i s a pleasure to express our gratitude for financial assistance and other forms o f material support to the follo wing International Union o f History and Philosophy of Science, Division for Logic, Methodology and Philosophy of Science Singapore Mathematical Society National University o f Singapore Department o f Mathematics, NUS ix SUMMARY OF PROGRAMME Opening speech by H.H.Huang, Deputy Vice Chancel 1 or, National University of Singapore One-hour Lectures: J.N.Crossley, Monash University, Austral a: Some undecidability results M.Lerman, University o f Connecticut, U.S.A.: Recursion-theoretic structures A.McIntyre, Yale University, U.S.A.: Decision problems for real and p-adic exponentiation A.Nerode, Cornell University, U.S.A.: Recent developments in recursive content of mathematics, with special attention to analysis G.E.Sacks, Harvard University and Massachusetts Institute o f Technology, U.S.A.: On the limits of recursive enumerability Half-hour lectures: M.W.Bunder, University o f Wol ongong, Australia: Combinatory logic as a basis for logic and set theory R.Goldblatt, Victoria Univers ty of Wellington, New Zealand: Logic and the Baire Category Theorem T.Kawai, Kagoshima University, Japan: An axiom system of non- standard set theory Li Xiang, Kweichow University, People's Republic of China: The t-computable and t-complex numbers N.Motohashi, University of Tsukuba, Japan: Some remarks on Barwise Approximation Theorem on Henkin Quantifiers K.Namba, University of Tokyo, Japan: Formal systems and Boolean valued combinatorics X Summary of programme I.B.Tan, Singapore and Massachusetts Institute of Technology, U.S.A.: Sequentially large cardinals M.J.Wicks, National University o f Singapore, Singapore: Representations of a commutator in a free group P.C.Wong, University o f Malaya, Malaysia: Some decision problems in group theory - A survey of recent results Workshops: M.Tamthai, Chulalongkorn University, Thailand: Recursion Theory C.Ash, Monash University, Australia: Mode2 Theory xi LIST OF PARTICIPANTS C.Ash M. H. Liew A. J. Berrick A.McIntyre M.W.Bunder N.Motohashi K. N . Cheng K.Nakatagawa C .T. Chong K. Namba S. K.Chua B.F.Nebres J . N. Cross1e y A. Nerode R. Go1 dbl a tt W.J.O'Connor S. Hayas hi T .A. Peng J. G. Hee G. E. Sacks T. Kawai Y .T. Soon P.H.Koh M.Takahashi P.Y.Lee M. Tamthai M.Lerman 1.B.Tan Y. K. Leong H. H. Teh B. C. Leuar M. J. Wicks A. Lewis P. C. Wong Li Xiang SOUTHEASTASIAN CONFERENCE ON LOGIC C:T. Chong and M.J. Wicks (editors) 0 Elsevier Science Pu bliahers B. V. (North-Holland), 1983 1 YODEL -THEORETIC FORMS OF THE AXIOM OF CHOICE C. J. Ash Department of Mathematics Monash University Austral ia We consider those consequences of the Axiom of Choice (AC) which can be expressed i n a certain model-theoretic form. Very many of the standard weak forms of AC can be expressed i n this way. We describe a single uniform method for showing that such statements are not provable i n Zermelo-Fraenkel set theory (ZF) without AC. Among those weak forms of AC which can be expressed i n a more special way, we show that there i s a weakest, namely : Min : Every infinite set can be partitioned either into two infinite components or into infinitely many components each having more than one ele ment, 51. Model -theoretic forms Let L be a countable language and l e t J I ,b~e s entences of LO W. 1 Let L1 be the set of symbols occurr ng i n $ and L2 be those occur- ring i n either IJJ or $. We l e t Ax($ +) denote the statement that every L1-structure satisfying IJJ has an expansion to an L2-structure +. satisfying For example, the Axiom o f Cho ce itself i s equivalent to the statement that every structure (A,R for which R i s an equivalence relat on on A has an expansion o f the form (A,R,S) where S i s a unary relation on A intersecting each equivalence class i n exactly one e ement. Thus AC can be expressed in the form Ax($,$). Another examp e i s the Boolean Prime Ideal Theorem (BPI) which asserts that
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