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

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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 and automorphisms of recursively enumerable sets WOLFGANG MAASS 21 The structure of the degrees of unsolvability RICHARD A. SHORE 33 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 . KLEENE 119 Canonical forms and 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


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 , 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 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 . Both of these are natural alignments since the talks in Descriptive 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 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 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, 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, Daniel E. Cohen London University, Queen Mary College, 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 Stanford University Peter Fejer Cornell University Matthew Foreman University of California, Los Angeles Harvey Friedman 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 Jech Pennsylvania State University, University Park Carl Jockusch University of Illinois Iraj Kalantari Western Illinois University Akihiro Kanamori CUNY, Baruch College Miodrag Kapetanovi Mathematics Institute, Yugoslavia Alexander Kechris California Institute of Technology, Pasadena Clement Kent Lakehead University, Thunder Bay, Canada Dennis Kfoury Boston University Stephen C. Kleene University of Wisconsin Phokion Kolaitis Occidental College, California Spiros Kourouklis University of California, Los Angeles Evangelos Kranakis University of Heidelberg, West Germany Ralph M. Krause National Science Foundation, Washington, D. C. Antonin Kucera Charles University, Czechoslovakia Stuart Kurtz Robert LaGrange University of Wyoming Anne Leggett Western Illinois University Stephen Legrand Pennsylvania State University, University Park Manuel Lerman University of Connecticut, Storrs APPENDIX I 525 Alain Louveau CNRS, France Robert Lubarsky Massachusetts Institute of Technology Wolfgang Maass University of California, Berkeley Hebrew University, Israel Mark Manasse Bell Telephone Laboratories, New Jersey Alfred Manaster University of California, San Diego, LaJolla Richard Mansfield Pennsylvania State University, University Park David Marker Yale University Donald Martin University of California, Los Angeles Pierre Matet Pensylvania State University, University Park Adrian R. D. Mathias Peterhouse, England Kenneth McAloon CUNY, Brooklyn College Charles McCarthy Wolfson College, Oxford University, England George Metakides University of Patras, Greece Robert Mignone College of Charleston Terrence Millar University of Wisconsin Arnold Miller University of Texas Douglas Miller Schlumberger-Doll Research, Connecticut William Mitchell Pennsylvania State University, University Park Thomas Moran Columbia University Michael Morley Cornell University Yiannis N. Moschovakis University of California, Los Angeles Kanji Namba University of Tokyo, Japan Anil Nerode Cornell University Yosaku Nishiwaki Keio University, Japan Dag Normann University of Oslo, Norway Iain Phillips Oxford University, England Marian Pour-El University of Minnesota, Minneapolis William Powell Swampscott, Massachusetts Hilary Putnam Harvard University Nicholas Reingold Massachusetts Institute of Technology Jeffrey Remmel University of California, San Diego, LaJolla Ian Richards University of Minnesota, Minneapolis David Rosenthal Ithaca College John Rosenthal Ithaca College Gerald Sacks Harvard University Ramez Sami Cairo University, Egypt Andrej Scedrov University of Pennsylvania, Philadelphia Steven Schwarz Massachusetts Institute of Technology Philip Scowcroft Cornell University Abraham Seidenberg University of California, Berkeley 526 APPENDIX I Juichi Shinoda Nagoya University, Japan Joseph Shoenfield Duke University Richard A. Shore Cornell University Stephen Simpson Pennsylvania State University University Park Theodore Slaman University of Chicago Kay Smith St. Olaf College Rick L. Smith University of Florida Robert I. Soare University of Chicago Robert Solovay University of California, Berkeley Dieter Spreen Technical High School, West Germany Lee Stanley Dartmouth College John Steel University of California, Los Angeles Michael Stob Calvin College Viggo Stoltenberg-Hanson Uppsala University, Sweden It Tan Massachusetts Institute of Technology Gregory Taylor Columbia University Simon Thompson Oxford University, England Tosiyuki Tugue Nagoya University, Japan Daniel Velleman University of Texas, Austin Catherine Wagnei Cornell University Stanley S. Wainer Leeds University, England Shiqiang, Wang Beijing Normal University, People's Republic of China Richard Watnick University of Connecticut, Stamford Klaus Weihrauch University of Hagen, West Germany Scott Weinstein University of Pennsylvania, Philadelphia Galen Weitkamp Western Illinois University Lawrence Welch Western Illinois University Philip Welch Oxford University, England W. Hugh Woodin California Institute of Technology, Pasadena Li Xiang Kweichow University, People's Republic of China Dong-Ping Yang Institute of Computing Technology, People's Republic of China Alexander C. Yessenin-Volpin Boston, Massachusetts Paul Young Purdue University Appendix II Short Courses The fine structure of L, SY. D. FRIEDMAN, Massachusetts Institute of Technology (3 lectures) Complexity theory, JURIS HARTMANIS, Cornell University (4 lectures) Descriptive set theory, A. KECHRIS, California Institute of Technology and YlANNIS MOSCHOVAKIS, University of California, Los Angeles (2 lectures each) Effective mathematics, ANIL NERODE, Cornell University (4 lectures) E-recursion theory, GERALD E. SACKS, Massachusetts Institute of Technology and Harvard University (4 lectures) The degrees of unsolvability, RICHARD A. SHORE, Cornell University (4 lectures) R. E. sets and degrees, ROBERT I. SOARE, University of Chicago (4 lectures) Invited Hour Addresses Absolute II2 real singletons, RENE DAVID, Toulouse University, France Strong D-principles, HANS-DlETER DONDER, University of Bonn, West Germany Constructive presentations, SOLOMON FEFERMAN, Stanford University IT2 and generalized recursion theory, JEAN-YVES GlRARD, University of Paris VII, France A gentle approach ot priority arguments, LEO HARRINGTON, University of California, Berkeley Pseudo-jump operations, CARL G. JOCKUSCH, JR., University of Illinois, Urbana Recursive functional and quantifiers of finite types, STEPHEN C. KLEENE, University of Wisconsin, Madison Inductive and recursion in E, PHOKION G. KOLAITIS, Occidental College The embedding problem for the recursively enumerable degrees, MANUEL LERMAN, University of Connecticut Some effective results in measure theory, ALAIN LOUVEAU, CNRS, France Isomorphisms in the lattice of R. E. sets, WOLFGANG MAASS, University of California, Berkeley Countable decomposable admissible sets: Kleene structures, MENACHIM MAGIDOR, Hebrew University, Israel

527 528 APPENDIX II Homogeneous trees, DONALD A. MARTIN, University of California, Los Angeles Semantic methods in complexity theory, KENNETH MCALOON, Brooklyn College, CUNY Decidable Ehrenfeucht theories, TERRENCE MILLAR, University of Wisconsin, Madison Aspects of the continuous functional, DAG NORM ANN, University of Oslo, Norway Constructions in Noetherian rings, ABRAHAM SEIDENBERG, University of California, Berkeley The fundamental problem of infinite injury, JOSEPH R. SHOENFIELD, Duke University Mathematical practice and subsystems of second order arithmetic, STEPHEN G. SIMPSON, Pennsylvania State University The E-recursively enumerable degrees are dense, THEODORE S LAM AN, University of Chicago Lindebaum algebras with additional structures, ROBERT M. SOLOVAY, University of California, Berkeley An introduction to gap-one morasses, LEE STANLEY, Dartmouth College Scales in L(R), JOHN R. STEEL, University of California, Los Angeles Major subsets and the lattice of recursively enumerable sets, MICHAEL STOB, Calvin College Reducibility of substitutions, LESLIE G. VALIANT, Harvard University The slow growing approach ot hierarchies, STANELY S. WAINER, Leeds University, England Determinacy of real games, scales, and inner models, W. HUGH WOODIN, California Institute of Technology Subrecursive computational complexity, PAUL YOUNG, Purdue University