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Descriptive Set Theory Second Edition Mathematical Surveys and Monographs Volume 155 Descriptive Set Theory Second Edition Yiannis N. Moschovakis American Mathematical Society http://dx.doi.org/10.1090/surv/155 Descriptive Set Theory Second Edition Mathematical Surveys and Monographs Volume 155 Descriptive Set Theory Second Edition Yiannis N. Moschovakis American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen, Chair J. T. Stafford Benjamin Sudakov 2000 Mathematics Subject Classification. Primary 03–02; Secondary 03D55, 03E15, 28A05, 54H05. For additional information and updates on this book, visit www.ams.org/bookpages/surv-155 Library of Congress Cataloging-in-Publication Data Moschovakis, Yiannis N. Descriptive set theory / Yiannis N. Moschovakis. – 2nd ed. p. cm. — (Mathematical surveys and monographs ; v. 155) Includes bibliographical references and index. ISBN 978-0-8218-4813-5 (alk. paper) 1. Descriptive set theory. I. Title. QA248.M66 2009 511.322—dc22 2009011239 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 This book is dedicated to the memory of my father Nicholas a good and gentle man CONTENTS Preface to the second edition ............................................... ix Preface to the first edition ................................................. xi About this book ............................................................ xiii Introduction ............................................................... 1 Chapter 1. The basic classical notions .................................... 9 1A. PerfectPolishspaces.............................................. 9 1B. TheBorelpointclassesoffiniteorder............................... 13 1C. Computing with relations; closure properties . ...................... 18 1D. Parametrizationandhierarchytheorems............................ 26 1E. Theprojectivesets................................................ 29 1F. Countable operations ... ......................................... 33 1G. Borel functions and isomorphisms . ................................ 37 1H. Historicalandotherremarks...................................... 46 Chapter 2. κ-Suslin and -Borel .......................................... 49 2A. TheCantor-BendixsonTheorem................................... 50 2B. κ-Suslinsets...................................................... 51 2C. TreesandthePerfectSetTheorem................................. 57 2D. Wellfounded trees ................................................. 62 2E. TheSuslinTheorem............................................... 65 2F. Inductive analysis of projections of trees. .......................... 70 2G. TheKunen-MartinTheorem...................................... 74 2H. Categoryandmeasure............................................. 79 2I. Historicalremarks................................................ 85 Chapter 3. Basic notions of the effective theory .......................... 87 3A. Recursivefunctionsontheintegers................................. 89 3B. Recursivepresentations............................................ 96 3C. Semirecursivepointsets............................................101 3D. RecursiveandΓ-recursivefunctions................................110 3E. TheKleenepointclasses........................................... 118 3F. UniversalsetsfortheKleenepointclasses...........................125 3G. Partialfunctionsandthesubstitutionproperty..................... 130 3H. Codings,uniformityandgoodparametrizations.................... 135 3I. Effectivetheoryonarbitrary(perfect)Polishspaces.................141 vii viii CONTENTS 3J. Historicalremarks................................................ 142 Chapter 4. Structure theory for pointclasses ............................ 145 1 4A. The basic representation theorem for Π1 sets....................... 145 4B. Theprewellorderingproperty......................................152 4C. Spectorpointclasses...............................................158 4D. The parametrization theorem for ∆ ∩X............................165 1 1 4E. The uniformization theorem for Π1,Σ2 .............................173 1 4F. Additional results about Π1 ....................................... 184 4G. Historicalremarks................................................ 202 Chapter 5. The constructible universe....................................207 5A. Descriptive set theory in L.........................................208 5B. Independenceresultsobtainedbythemethodofforcing.............214 5C. Historicalremarks................................................ 215 Chapter 6. The playful universe .......................................... 217 6A. Infinite games of perfect information ...............................218 6B. TheFirstPeriodicityTheorem.....................................229 6C. TheSecondPeriodicityTheorem;uniformization...................235 6D. The game quantifier G .............................................244 6E. TheThirdPeriodicityTheorem.................................... 254 6F. ThedeterminacyofBorelsets..................................... 272 6G. Measurablecardinals..............................................280 6H. Historicalremarks................................................ 290 Chapter 7. The recursion theorem ........................................ 293 7A. Recursion in a Σ∗-pointclass.......................................293 7B. TheSuslin-KleeneTheorem....................................... 298 7C. Inductive definability . .............................................309 7D. The completely playful universe. ...................................323 7E. Historicalremarks................................................ 339 7F. ResultswhichdependontheAxiomofChoice......................341 Chapter 8. Metamathematics ..............................................353 8A. Structuresandlanguages.......................................... 355 8B. Elementary definability. ...........................................365 8C. Definability in the universe of sets..................................371 8D. Godel’suniverseofconstructiblesets...............................381¨ 8E. Absoluteness..................................................... 390 8F. The basic facts about L............................................401 8G. Regularity results and inner models . ............................... 416 8H. Onthetheoryofindiscernibles.....................................446 8I. Some remarks about strong hypotheses. ............................468 8J. Historicalremarks................................................ 473 The axiomatics of pointclasses .............................................. 475 References ..................................................................477 Index ....................................................................... 491 PREFACE TO THE SECOND EDITION There was no question of “updating” this book nearly thirty years after it was first published—in 1980, volume 100 in the Studies in Logic series of North Holland. The only completely rewritten sections are 6F, which gives a proof of the determinacy of Borel sets (a version of Martin’s second proof not available in 1980) and 7F, where the question of how much choice is needed (especially) to prove Borel determinacy is examined. There is also a new, brief section 3I on the relativization method of proof, which has baffled some of the not-so-logically minded readers. Beyond that, the main improvements over the first edition are that - this one has many fewer errors (I hope); - the bibliography has been completed and expanded with a small selection of relevant, more recent publications; - and many passages have been rewritten. (It has been said that the most basic instinct in man is not for food or sex but to edit someone else’s writing—and the urge to edit one’s own writing is, apparently, even stronger.) There have been two major developments in Descriptive Set Theory since 1980 which have fundamentally changed the subject. One is the establishment of a robust connection between determinacy hypotheses, large cardinal axioms and inner model theory, starting with Martin and Steel [1988] and Woodin [1988], to such an extent that one cannot now understand any of these parts of set theory without also understanding the others. I have added some “forward references” to these developments when they touch on questions that were formulated in the book. The other is the explosion in applications of Descriptive Set
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