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RECURSION THEORY PROCEEDINGS of SYMPOSIA in PURE MATHEMATICS Volume 42 http://dx.doi.org/10.1090/pspum/042 RECURSION THEORY PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume 42 RECURSION THEORY AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 42 PROCEEDINGS OF THE AMS-ASL SUMMER INSTITUTE ON RECURSION THEORY HELD AT CORNELL UNIVERSITY ITHACA, NEW YORK JUNE 28-JULY 16, 1982 EDITED BY ANIL NERODE AND RICHARD A. SHORE Prepared by the American Mathematical Society with partial support from National Science Foundation grant MCS 8120074 1980 Mathematics Subject Classification. Primary 03D25, 03D30, 03D45, 03D55, 03D60, 03D70, 03D80, 03E05, 03E10, 03E35, 03E45, 03E47, 03F30, 03G05, 13E05, 14M05, 28A12, 90D05, 90D13. Library of Congress Cataloging in Publication Data Main entry under title: Recursion theory. (Proceedings of symposia in pure mathematics; v. 42) Papers presented at the 1982 AMS Summer Research Institute held at Cornell University from June 28 to July 16, 1982, and co-sponsored by the ASL. Bibliography: p. 1. Recursion theory—Addresses, essays, lectures. I. Nerode, Anil. 1932— II. Shore, Richard A., 1946- . III. American Mathematical Society. IV. AMS Summer Research Institute (1982: Cornell University) V. Series. QA9.6.R4 1984 511.3 8448525 ISBN 0-8218-1447-8 COPYING AND REPRINTING. Individual readers of this publication, and nonprofit librar• ies acting for them are permitted to make fair use of the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews provided the customary acknowledgement of the sources is given. Republication, systematic copying, or multiple reproduction of any material in this pub• lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Executive Director, American Mathematical Society, Box 6248, Providence, Rhode Island 02940. The appearance of the code on the first page of an article in this volume indicates the copy• right owner's consent for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law, provided that the fee of $1.00 plus $.25 per page for each copy be paid directly to Copyright Clearance Center, Inc., 21 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copyright © 1985 by the American Mathematical Society. Printed in the United States of America. All rights reserved except those granted to the United States Government. The paper used in this book is acid-free and falls within the guidelines established to in• sure permanence and durability. Table of Contents Preface vii I. Classical Recursion Theory REA operators, R.E. degrees and minimal covers CARL J. JOCKUSH, JR. AND RICHARD A. SHORE 3 The embedding problem for the recursively enumerable degrees MANUEL LERMAN 13 Major subsets and automorphisms of recursively enumerable sets WOLFGANG MAASS 21 The structure of the degrees of unsolvability RICHARD A. SHORE 33 Tree arguments in recursion theory and the 0"'-priority method ROBERT I. SOARE 53 Major subsets and the lattice of recursively enumerable sets MICHAEL STOB 107 II. Generalized Recursion Theory Unimonotone functions of finite types (recursive functionals and quantifiers of finite type revisited IV) STEPHEN C. KLEENE 119 Canonical forms and hierarchies in generalized recursion theory PHOKION G. KOLAITIS 139 Aspects of the continuous functionals DAG NORMANN 171 Post's problem in E'-recursion GERALD E. SACKS 177 The E-recursively enumerable degrees are dense THEODORE A. SLAMAN 195 HI. Fine Structure and Descriptive Uncountable ZF-ordinals RENE DAVID AND SY D. FRIEDMAN 217 Another look at gap-1 morasses H. D. DONDER 223 v VI CONTENTS Condensation-coherent global square systems H. D. DONDER, R. B. JENSEN AND L. J. STANLEY 237 Fine structure theory and its applications SYD. FRIEDMAN 259 Determinancy and the structure of L(R) ALEXANDER S. KECHRIS 271 Recursivity and capacity theory ALAIN LOUVEAU 285 A purely inductive proof of Borel determinancy DONALD A. MARTIN 303 IV. Effective Mathematics Decidable Ehrenfeucht theories T. MILLAR 311 A survey of lattices of r.e. substructures A. NERODE AND J. REMMEL 323 Survey of constructions in Noetherian rings A. SEIDENBERG 377 V. Foundations and Complexity Theory Elements de logique 11^ JEAN-YVES GIRARD AND JEAN PIERRE RESSAYRE 389 Paris-Harrington incompleteness and progressions of theories KENNETH MCALOON 447 Reverse mathematics STEPHEN G. SIMPSON 461 Infinite fixed-point algebras ROBERT M. SOLOVAY 473 The "slow-growing" IJ2 approach to hierarchies S. S. WAINER 487 Godel theorems, exponential difficulty and undecidability of arithmetic theories: an exposition PAUL YOUNG 503 Appendix I. List of participants 523 Appendix II. List of short courses and hour lectures 527 Preface The 1982 AMS Summer Research Institute was cosponsored by the ASL and was devoted to Recursion Theory. It met at Cornell University from June 28 to July 16. Our intention was to consider recursion theory in the broadest sense. This is reflected in the lists of participants and lectures (Appendices I and II) as well as in the contents of this Proceedings volume. The hour talks were roughly grouped around seven short courses—two in Classical Recursion Theory and one each in Generalized Recursion Theory, Fine Structure of L, Descriptive Set Theory, Effective Mathematics and Complexity Theory (Computer Sciences). These series correspond to the sections of this volume except that we have put the two set theoretic subjects into one section and have combined the papers on the founda• tional topics with those on computer science. Both of these are natural alignments since the talks in Descriptive Set Theory dealt mainly with the structure of L(R) and the papers in complexity theory are strongly related to classical undecidabil- ity and incompleteness results. This volume contains contributions representing most of the short courses and hour lectures given at the Institute. The papers representing the talks by Martin, Steel and Woodin in Descriptive Set Theory have however appeared in Cabal Seminar, 1979-81 Springer Lecture Notes in Mathematics no. 1019. In addition to the courses and hour lectures there were many shorter talks given by the participants and organized in sections. Extended abstracts for thirty-two of these talks appeared as a special bound issue of the Recursive Function Theory Newsletter edited by Iraj Kalantari under the auspices of Western Illinois and Cornell Universities. As Cochairmen of the organizing committee we would like to thank our fellow committee members, S. Feferman, Y. Moschovakis, H. Putnam, G. Sacks, J. Shoenfield and R. Soare for their advice and encouragement. The excellent logistical support which helped make the Institute a success was provided by Peter Fejer who was the local coordinator and especially by Dottie Smith of the AMS who helped run just about everything. Both deserve our thanks. Anil Nerode Richard A. Shore vn Appendix I LIST OF NAME AFFILIATION Klaus Ambos- Spies University of Dortmund, West Germany Arthur Apter Rutgers University Ferdinando Arzarello University of Torino, Italy Jos Baeten University of Minnesota, Minneapolis Joshua D. Bernoff Massachusetts Institute of Techonology Mark Bickford University of Wisconsin Robert Birmingham University of Connecticut Jaime Bohorquoz Cornell University Ronald Book University of California, Santa Barbara Jerald Bope Bard College Egon Borger University of Dortmund, West Germany Sheryl Brady Cornell University Tim Carlson University of California, Berkeley Douglas Cenzer University of Florida John C. Cherniavsky National Science Foundation, Washington, D.C. Chi Chong National University, Singapore Eugeniusz Cichon Pennsylvania State University, University Park Peter Clote University of Paris VII, France Daniel E. Cohen London University, Queen Mary College, England Barry Cooper Leeds University, England John Cowles University of Wyoming James Cremer University of Wisconsin John Crossley Monash University, Australia Rene David Toulouse University, France Martin Davis NYU, Courant Institute of Mathematical Sciences Jacob Dekker Rutgers University Robert A. DiPaola CUNY, Graduate Center Carlos A. DiPrisco Inst. Venezolano de Investigaciones Cientificas, Venezuela 523 524 APPENDIX I Hans-Dieter Donder University of Boon, West Germany Solomon Feferman Stanford University Peter Fejer Cornell University Matthew Foreman University of California, Los Angeles Harvey Friedman Ohio State University Sy D. Friedman Massachusetts Institute of Technology Jean-Yves Girard University of Paris VII, France Warren Goldfarb Harvard University Edward Griffor Uppsala University, Sweden Marcia J. Groszek Massachusetts Institute of Technology Leon Harkleroad University of Notre Dame Leo Harrington University of California, Berkeley Juris Hartmanis Cornell University Louise Hay University of Illinois at Chicago James Henle Smith College Christer Hennix SUNY, College at New Paltz Harold Hodes Cornell University Bernard R. Hodgson University Laval, Canada Steven Homer DePaul University Wen Qi Huang Huashong Institute of Technology, Peoples' Republic of China Michael Ingrassia Western Illinois University Ljubomir, Ivanov Sofia University, Bulgaria John Jayne University College, England Thomas
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