Higher Recursion Theory 9781316717301

Table of contents :
Contents......Page 13
Part A. Hyperarithmetic Sets......Page 18
1. Analytical Predicates......Page 20
2. Notations for Ordinals......Page 25
3. Effective Transfinite Recursion......Page 27
4. Recursive Ordinals......Page 32
5. Ordinal Analysis of ∏ 1 1 Sets......Page 35
1. Hyperarithmetic Implies ∆ 1 1......Page 39
2. ∆ 1 1 Implies Hyperarithmetic......Page 45
3. Selection and Reduction......Page 49
4. ∏ 0 2 Singletons......Page 54
5. Hyperarithmetic Reducibility......Page 59
6. Incomparable Hyperdegrees Via Measure......Page 63
7. The Hyperjump......Page 65
1. Basis Theorems......Page 69
2. Unique Notations for Ordinals......Page 72
3. Hyperarithmetic Quantifiers......Page 76
4. The Ramified Analytic Hierarchy......Page 79
5. Kreisel Compactness......Page 87
6. Perfect Subsets of ∑ 1 1 Sets......Page 88
7. Kreisel's Basis Theorem......Page 91
8. Inductive Definitions......Page 93
9. ∏ 1 1 Singletons......Page 98
1. Measure-Theoretic Uniformity......Page 105
2. Measure-Theoretic Basis Theorems......Page 109
3. Cohen Forcing......Page 111
4. Perfect Forcing......Page 115
5. Minimal Hyperdegrees......Page 120
6. Louveau Separation......Page 124
Part B. Metarecursion......Page 130
1. Fundamentals of Metarecursion......Page 132
2. Metafinite Computations......Page 138
3. Relative Metarecursiveness......Page 141
4. Regularity......Page 146
1. Hyperregular Sets......Page 152
2. Two Priority Arguments......Page 155
3. Simpson's Dichotomy......Page 163
Part C. α-Recursion......Page 166
1. ∑1 Admissibility......Page 168
2. The ∑1 Projectum......Page 174
3. Relative α-Recursiveness......Page 178
4. Existence of Regular Sets......Page 182
5. Hyperregularity......Page 184
1. α-Finite Injury via α*......Page 192
2. α-Finite Injury and Tameness......Page 195
3. Dynamic Versus Fine-Structure......Page 201
4. ∑1 Doing the Work of ∑2......Page 211
1. Shore's Splitting Theorem......Page 221
2. Further Fine Structure......Page 224
3. Density for ω......Page 229
4. Preliminaries to α-Density......Page 233
5. Shore's Density Theorem......Page 235
6. β-Recursion Theory......Page 244
PartD. E-Recursion......Page 248
1. Partial E-Recursive Functions......Page 250
2. Computations......Page 254
3. Reflection......Page 259
4. Gandy Selection......Page 261
5. Moschovakis Witnesses......Page 266
1. Set Forcing over L (k)......Page 276
2. Countably Closed Forcing......Page 282
3. Enumerable Forcing Relations......Page 287
4. Countable-Chain-Condition Forcing......Page 290
5. Normann Selection and Singular Cardinals......Page 296
6. Further Forcing......Page 298
Chapter XII. Selection and k-Sections......Page 300
1. Grilliot Selection......Page 301
2. Moschovakis Selection......Page 304
3. Plus-One Theorems......Page 307
4. Harrington's Plus-Two Theorem......Page 316
5. Selection with Additional Predicates......Page 321
1. Regular Sets......Page 326
2. Projecta and Cofinalities......Page 330
3. van de Wiele's Theorem......Page 342
4. Post's Problem for E-Recursion......Page 345
5. Slaman's Splitting and Density Theorems......Page 350
Bibliography......Page 356
Subject Index......Page 360

Citation preview

Higher Recursion Theory

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the 2nd publication in the Perspectives in Logic series, is an almost self-contained introduction to higher recursion theory, in which the reader is only assumed to know the basics of classical recursion theory. The book is divided into four parts: hyperarithmetic sets, metarecursion, α-recursion, and E-recursion. This text is essential reading for all researchers in the field. GERALD E. SACKS works in the Department of Mathematics at Harvard University and at Massachusetts Institute of Technology.

PERSPECTIVES IN LOGIC

The Perspectives in Logic series publishes substantial, high-quality books whose central theme lies in any area or aspect of logic. Books that present new material not now available in book form are particularly welcome. The series ranges from introductory texts suitable for beginning graduate courses to specialized monographs at the frontiers of research. Each book offers an illuminating perspective for its intended audience. The series has its origins in the old Perspectives in Mathematical Logic series edited by the ^2-Group for "Mathematische Logik" of the Heidelberger Akademie der Wissenschaften, whose beginnings date back to the 1960s. The Association for Symbolic Logic has assumed editorial responsibility for the series and changed its name to reflect its interest in books that span the full range of disciplines in which logic plays an important role. Arnold Beckmann, Managing Editor Department of Computer Science, Swansea University Editorial Board: Michael Benedikt Department of Computing Science, University of Oxford Elisabeth Bouscaren CNRS, Département de Mathématiques, Université Paris-Sud Steven A. Cook Computer Science Department, University of Toronto Michael Glanzberg Department of Philosophy, University of California Davis Antonio Montalban Department of Mathematics, University of Chicago Simon Thomas Department of Mathematics, Rutgers University For more information, see www.aslonline.org/books_perspectives.html

PERSPECTIVES IN LOGIC

Higher Recursion Theory

GERALD E. SACKS Harvard University

ASSOCIATION f o r s y m b o l i c l o g i c

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 4843/24, 2nd Floor, Ansari Road, Daryaganj, Delhi - 110002, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University's mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107168435 10.1017/9781316717301 First edition © 1990 Springer-Verlag Berlin Heidelberg This edition © 2016 Association for Symbolic Logic under license to Cambridge University Press. Association for Symbolic Logic Richard A. Shore, Publisher Department of Mathematics, Cornell University, Ithaca, NY 14853 http://www.aslonline.org This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. A catalogue record for this publication is available from the British Library. ISBN 978-1-107-16843-5 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

In Memory Of My Father Irwin "Pete" Sacks

Preface to the Series Perspectives in Mathematic Logic (Edited by the "Q-group for Mathematical Logic" of the Heidelberger Akademie der Wissenschaften)

On Perspectives. Mathematical logic arose from a concern with the nature and the limits of rational or mathematical thought, and from a desire to systematise the modes of its expression. The pioneering investigations were diverse and largely autonomous. As time passed, and more particularly in the last two decades, interconnections between different lines of research and links with other branches of mathematics proliferated. The subject is now both rich and varied. It is the aim of the series to provide, as it were, maps of guides to this complex terrain. We shall not aim at encyclopaedic coverage; nor do we wish to prescribe, like Euclid, a definitive version of the elements of the subject. We are not committed to any particular philosophical programme. Nevertheless we have tried by critical discussion to ensure that each book represents a coherent line of thought; and that, by developing certain themes, it will be of greater interest than a mere assemblage of results and techniques. The books in the series differ in level: some are introductory, some highly specialised. They also differ in scope: some offer a wide view of an area, others present a single line of thought. Each book is, at its own level, reasonably selfcontained. Although no book depends on another as prerequisite, we have encouraged authors to fit their book with other planned volumes, sometimes deliberately seeking coverage of the same material from different points of view. We have tried to attain a reasonable degree of uniformity of notation and arrangement. However, the books in the series are written by individual authors, not by the group. Plans for books are discussed and argued about at length. Later, encouragement is given and revisions suggested. But it is the authors who do the work; if, as we hope, the series proves of values, the credit will be theirs. History of the Q-Group. During 1968 the idea of an integrated series of monographs on mathematical logic was first mooted. Various discussions led to a meeting at Oberwolfach in the spring of 1969. Here the founding members of the group (R.O. Gandy, A. Levy, G.H. Muller, G. Sacks, D.S. Scott) discussed the project in earnest and decided to go ahead with it. Professor F.K. Schmidt and Professor Hans Hermes gave us encouragement and support. Later Hans Hermes joined the group. To begin with all was fluid. How ambitious should we be? Should we write the books ourselves? How long would it take? Plans for authorless books were promoted, savaged and scrapped. Gradually there emerged a form and a method. At the end of an infinite discussion we found our name, and

VIII

Preface to the Series

that of the series. We established our centre in Heidelberg. We agreed to meet twice a year together with authors, consultants and assistants, generally in Oberwolfach. We soon found the value of collaboration: on the one hand the permanence of the founding group gave coherence to the overall plans; on the other hand the stimulus of new contributors kept the project alive and flexible. Above all, we found how intensive discussion could modify the authors' ideas and. our own. Often the battle ended with a detailed plan for a better book which the autor was keen to write and which would indeed contribute a perspective. Oberwolfach, September 1975 Acknowledgements. In starting our enterprise we essentially were relying on the personal confidence and understanding of Professor Martin Earner of the Mathematisches Forschungsinstitut Oberwolfach, Dr. Klaus Peters of Springer- Verlag und Dipl.-Ing. Penschuck of the Stiftung Volkswagenwerk. Through the Stiftung Volkswagenwerk we received a generous grant (1970-1973) as an initial help which made our existence as a working group possible. Since 1974 the Heidelberger Akademie der Wissenschqften (MathematischNaturwissenschaftliche Klasse) has incorporated our enterprise into its general scientific program. The initiative for this step was taken by the late Professor F.K. Schmidt, and the former President of the Academy, Professor W. Doerr. Through all the years, the Academy has supported our research project, especially our meetings and the continuous work on the Logic Bibliography, in an outstandingly generous way. We could always rely on their readiness to provide help wherever it was needed. Assistance in many various respects was provided by Drs. U. Feigner and K. Gloede (till 1975) and Drs. D. Schmidt and H. Zeitler (till 1979). Last but not least, our indefatigable secretary Elfriede Ihrig was and is essential in running our enterprise. We thank all those concerned. Heidelberg, September 1982

R.O. Gandy A. Levy G. Sacks

H. Hermes G.H.Muller D.S. Scott

Author's Preface

Higher recursion theory (HRT) has been one of my two major obsessions for the last twenty years. Nonetheless my interest has not waned. Perhaps because, as Browning claimed: "The best is yet to be." I was talked into the subject, skittish all the way, by G. Kreisel. The old devil insisted, in several conversations beginning in 1961, on the existence of golden generalizations of recursion theory in which infinitely long computations converged. I listened for hours, without understanding a word, to his tales of the mother lode of recursion theory hidden far below the peaks of effective descriptive set theory. My initial reaction to his yarns was naive. One could readily generalize the static (or syntactic) aspects of classical recursion theory (CRT) such as the enumeration theorem, but one could not hope to lift dynamic results such as the Friedberg-Muchnik solution of Post's problem to higher domains. Clearly the dynamic facts of CRT were inseparable from certain combinatoric properties of finite sets, and these properties, being "truly finite" in nature, could not be generalized fruitfully. In 1961 I was trying to prove the density of the recursively enumerable degrees, and that investigation had a no-nonsense flavor that rendered all exotic pursuits unpalatable. In 1963 I began to understand what Kreisel was talking about. He had unearthed a compactness theorem for co-logic in which hyperarithmetic played the part of finite. His result was: if A is a n } set of axioms of co-logic and every hyperarithmetic subset of A has a model, then A has a model. (This eye-opener was the forerunner of Barwise's compactness theorem for Zx admissible sets.) Kreisel's proposal of 1961, as understood by me in 1963, was: replace the natural numbers by a n } set J of indices for the hyperarithmetic sets, and recursively enumerable by IT}; most important of all, let "finite" mean hyperarithmetic. The more one scrutinized his initially obscure proposal, the more lucid it became. Right away it was clear that a "finite" union of "finite" sets was "finite". A set A was "recursive" if A and I-A were n } ; in Kreisel's terminology A was hyperarithmetic on /. It was immediate that a "recursive" function restricted to a "finite" set was "finite". Now it was plausible that the dynamic results of CRT would lift up. The new notions were dubbed metarecursively enumerable, metafinite, and metare-

X

Author's Preface

cursive. For smoothness the set / of indices of hyperarithmetic sets was replaced by a IT} set of unique notations for recursive ordinals, and finally by the set of recursive ordinals. Thus a metarecursively enumerable set was simply a set of recursive ordinals whose unique notations constituted a 11} set. The first test of metarecursion theory, Kreisel unerringly pronounced, was to prove the Friedberg-Muchnik theorem. A major technical obstacle stood in the way. It was possible for a metarecursively enumerable subset of a metafinite set not to be metafinite. (For me this obstacle was a source of meta-delight.) It was overcome in 1963 by a partial metarecursive map of co onto (DCXK, a foretaste of Jensen's projectum techniques in L. In 1966 I finished my work on metarecursion theory and conceived a plan for writing a book called Higher Recursion Theory. Part A would develop hyperarithmetic theory from scratch and would include connections with forcing and compactness. Part B would expound metarecursion, and Part C would deal with Zx admissible ordinals. The principal flaw in the plan was my ignorance of Kleene's theory of recursion in normal objects of finite type. Platek's lecture on the superjump in Manchester in 1969 gave me my first toehold on the Kleene theory, and long talks with Gandy and Grilliot in 1970 brought me to the top of the subject. Thus Part D was born. In the 1970's and early 1980's considerable progress took place in admissible ordinals and in finite types, and so Parts C and D had to be started over and over again. Even hyperarithmetic theory, fairly stable for some years, saw new developments. By 1980 I had formed a definite view of the contents of higher recursion theory (HRT). There is only one fork in the road upward. The natural numbers turn into ordinals or into sets. If ordinals, then recursively enumerable becomes Z x . If sets, then belonging to a recursively enumerable set means there exists a convergent computation presented as a wellfounded, possibly infinite tree. The ordinal approach was blazed by G. Takeuti, the setcomputational by S. C. Kleene, both in the 1950's. Happily each leads in a different way to a proof of the Friedberg-Muchnik theorem. One of the early hopes for HRT was the discovery of new theorems in CRT via ideas from above. It is now obvious (as usual, after the case) that CRT is not "low" enough for applications of HRT. A theorem of HRT is proved by overcoming the lack of some combinatoric facts taken for granted in CRT. Moving upward means leaving behind some of the power of CRT. The same loss occurs when moving downward. It is not surprising, then, that HRT has been applied successfully "below" CRT. Consider the splitting theorem of CRT: each non-recursive, recursively enumerable degree is the join of two incomparable such degrees. T. Slaman and H. Woodin have shown that splitting is not provable from a fragment of Peano arithmetic known as Zx bounding. Their argument draws on tricks from HRT. M. Mytilinaios has shown that splitting does follow from Zx induction. His result is inspired by R. Shore's proof of splitting for every Zx admissible ordinal; it uses Z 2 blocking. J. Shinoda and T. Slaman have shown that the theory of polynomial-time degrees of recursive sets interprets first order arithmetic. They apply an idea from Moschovakis's work (on Kleene's theory of recursion in normal objects of finite type) that views divergence as Zj in character.

Author's Preface

XI

The book, on an informal level, is almost self contained. Some of the arguments present, in an easygoing fashion, material that a typical reader has encountered elsewhere in a more formal setting. For example a course on Godel's L is not assumed. The essential facts about L are given ab initio, but some readers may want more details. No previous acquaintance with forcing is necessary, but it would help clarify the various effective notions of forcing studied here. The more one knows of CRT, the better, but little more than the enumeration theorem is assumed, and at least one well known logician managed to learn priority arguments in the setting of Hl admissible ordinals before applying them to CRT. The book has four parts: A. Hyperarithmetic Sets B. Metarecursion C. a-Recursion D. ^-Recursion Part A is perhaps longer than some would wish. It lingers, as if life were not short, on effective transfinite recursion (ETR), the method invented by Church and Kleene in their study of notations for ordinals. My treatment of ETR follows that of H. Rogers, the first to present it intuitively. The classic theorems are proved, most of them in the original spirit, theorems such as recursive ordinals equal constructive ordinals, Kleene's 0 is 11} complete, Z} bounding, and hyperarithmetic equals A}. In addition measure and forcing are developed and applied in a hyperarithmetic context. The set of all real X such that the ordinals recursive in X are the recursive ordinals has measure 1. Forcing with hyperarithmetically encoded perfect sets yields a minimal hyperdegree. Forcing with Z} sets (originated by Gandy) leads to Louveau's separation theorem. Hyperarithmetic theory (HT) is often regarded, and rightly so, as the source of effective descriptive set theory. In this book it is the prologue to higher recursion theory. Many of the major developmets of HRT are foreshadowed in HT. Part B carries out of the program of metarecursion sketched above. It is easy to follow once one becomes accustomed to thinking of hyperarithmetic as "finite". Part B speedily verifies that the priority method of Friedberg and Muchnik can be executed in a higher domain. Simpson's dichotomy applies metarecursion to create new categories of 11} sets. Part C tackles a, an arbitrary Zx admissible ordinal. Post's problem is solved by combining fine structure of L with priority. The catch phrase here is "Zj doing the work of Z 2 ". The priority method, when applied in CRT, needs Z 2 replacement. In a-recursion Zj suffices with the assistance of effective approximations, downward projecta and Godel-Jensen condensation. Shore's density theorem is an example of a Z 3 (more precisely Z2.5) argument of CRT lifted to every Zx admissible a. An early result that points to the flexibility of such a's is the regular sets theorem. A subset of a is said to be regular if its intersection with each ordinal less than a is a-finite (that is, belongs to L(a)). An arecursively enumerable set may fail to be regular, but it always has the same adegree as some regular, a-recursively enumerable set. Part D assigns a meaning to {e}(x) for every set x via a notion of computation following schemes devised by Normann, and (subsequently) by

XII

Author's Preface

Moschovakis. A structure is E-closed if it is closed under application of the partial ^-recursive function [e] for all e. The biggest twist in E-closed structures is the existence of reflection phenomena and their application to priority and forcing thanks to a crucial connection between reflecting ordinals and Moschovakis divergence witnesses uncovered by Harrington and Kechris. On a more mundane level, the key to progress in ^-recursion is usually a new selection theorem, such as those proved by Gandy, Grilliot, Moschovakis and Normann. This is true in priority arguments, and even more so in forcing arguments. Intuitively, a selection theorem provides an effective method of selecting a member of a nonempty, "recursively enumerable" set. van de Wiele's theorem explains why some Z t functions are not ^-recursive. The book ends, fittingly I think, with Slaman's density theorem for E-closed L(K:)'S. I owe a great deal to those of my students who wrote theses on HRT: C. Bailey, G. Driscoll, S. Friedman, E. Griffor, L. Harrington, S. Homer, F. Lowenthal, M. Machtey, J. Macintyre, D. MacQueen, J. Owings, R. Shore, S. Simpson, T. Slaman, J. Sukonick and S. Thomason. Also to colleagues who contributed to HRT: C.T. Chong, J. E. Fenstad, R. Gandy, T. Grilliot, B. Jacobs, S. C. Kleene, G. Kreisel, A. Leggett, M. Lerman, W. Maass, J. Moldstad, Y. N. Moschovakis, D. Normann, R. Platek, J. Shinoda, C. Spector, G. Takeuti and T. Tugue. I am grateful for the long support provided by the National Science Foundation (Division of Mathematical Sciences). Several inspiring trips to the Mathematics Institute at Oberwolfach came about through the generosity of the Heidelberg Academy and the dedication to logic of G.H. Muller. The book was typed by M. Beucler and L. Schlesinger, whose patience was admirable. The manuscript was read by Brian O'Neill, who made numerous suggestions, mathematical and grammatical. Lastly there is the debt to J.B. Rosser (1907-1989), A.Nerode, H.Rogers and B. Dreben. The final four gave me my start and kept me going. If they are pleased with HRT, then so am I. Gerald E. Sacks June 1990 Cambridge, Chicago, Pasadena and Princeton

Contents

Part A. Hyperarithmetic Sets

1

Chapter I. Constructive Ordinals and U { Sets

3

1. 2. 3. 4. 5.

Analytical Predicates Notations for Ordinals Effective Transfinite Recursion Recursive Ordinals Ordinal Analysis of n } Sets

Chapter II. The Hyperarithmetic Hierarchy 1. 2. 3. 4. 5. 6. 7.

Hyperarithmetic Implies A} A} Implies Hyperarithmetic Selection and Reduction I\°2 Singletons Hyperarithmetic Reducibility Incomparable Hyperdegrees Via Measure The Hyperjump

Chapter III. E{ Predicates of Reals 1. 2. 3. 4. 5. 6. 7. 8. 9.

Basis Theorems Unique Notations for Ordinals Hyperarithmetic Quantifiers The Ramified Analytic Hierarchy Kreisel Compactness Perfect Subsets of I { Sets Kreisel's Basis Theorem Inductive Definitions U\ Singletons

3 8 10 15 18

22 22 28 32 37 42 46 48

52 52 55 59 62 70 71 74 76 81

XIV

Contents

Chapter IV Measure and Forcing 1. 2. 3. 4. 5. 6.

88

Measure-Theoretic Uniformity Measure-Theoretic Basis Theorems Cohen Forcing Perfect Forcing Minimal Hyperdegrees Louveau Separation

88 92 94 98 103 107

Part B. Metarecursion

113

Chapter V Metarecursive Enumerability

115

1. 2. 3. 4.

Fundamentals of Metarecursion Metafinite Computations Relative Metarecursiveness Regularity

115 121 124 129

Chapter VI. Hyperregularity and Priority

135

1. Hyperregular Sets 2. Two Priority Arguments. 3. Simpson's Dichotomy

135 138 146

Part C. a-Recursion

149

Chapter VII. Admissibility and Regularity

151

1. 2. 3. 4. 5.

l,l Admissibility TheZi Projectum Relative a-Recursiveness Existence of Regular Sets Hyperregularity

Chapter VIII. Priority Arguments 1. 2. 3. 4.

a-Finite Injury via a* a-Finite Injury and Tameness Dynamic Versus Fine-Structure 2X Doing the Work of E 2

*

151 157 161 165 167 175 175 178 184 194

Contents

XV

Chapter IX. Splitting, Density and Beyond

204

1. 2. 3. 4. 5. 6.

Shore's Splitting Theorem Further Fine Structure Density for co Preliminaries to a-Density Shore's Density Theorem /^-Recursion Theory

204 207 212 216 . 218 227

PartD. ^-Recursion

231

Chapter X. £-Closed Structures

233

1. 2. 3. 4. 5.

Partial ^-Recursive Functions Computations Reflection Gandy Selection Moschovakis Witnesses

Chapter XL Forcing Computations to Converge 1. 2. 3. 4. 5. 6.

Set Forcing over L (/c) Countably Closed Forcing Enumerable Forcing Relations Countable-Chain-Condition Forcing Normann Selection and Singular Cardinals Further Forcing

Chapter XII. Selection and /c-Sections 1. 2. 3. 4. 5.

Grilliot Selection Moschovakis Selection Plus-One Theorems Harrington's Plus-Two Theorem Selection with Additional Predicates

Chapter XIII. ^-Recursively Enumerable Degrees 1. 2. 3. 4. 5.

Regular Sets Projecta and Cofinalities van de Wiele's Theorem Post's Problem for ^-Recursion Slaman's Splitting and Density Theorems

Bibliography Subject Index

233 237 242 244 249 259 259 265 270 273 279 281 283 284 287 290 299 304 309 309 313 325 328 333 339 343

Part A Hyper arithmetic Sets Hyperarithmetic theory is the first step beyond classical recursion theory. It is the primary source of ideas and examples in higher recursion theory. It is also a crossroads for several areas of mathematical logic. In set theory it is an initial segment of Godel's L. In model theory, the least admissible set after o. In descriptive set theory, the setting for effective arguments, many of which are developed below. It gives rise directly to metarecursion theory (Part B), and yields the simplest example of both a-recursion theory (Part C) and £-recursion theory (Part D).

Chapter I Constructive Ordinals and IT} Sets

It is shown that a universal quantifier ranging over the real numbers is equivalent in certain circumstances to an existential quantifier ranging over the recursive ordinals, a countable set. Along the way notations for ordinals and the method of defining partial recursive functions by effective transfinite recursion are developed.

1. Analytical Predicates The analytical predicates are obtained by applying function quantifiers to recursive predicates. Chapter I focuses on analytical predicates in which at most one function quantifier occurs, since in that case an analysis based on ordinals goes smoothly. 1.1 Partial Recursive Functions. Some conventions, occasionally violated, in this book are: co is {0, 1 , 2 , . . . } , the set of natural numbers. h, c, e, m, n are constants that denote natural numbers. x, y, z9 . . . are variables that range over co. f9 g9h9 . . . are total functions from co into co. X9 Y9 Z, . . . are subsets of co. 9 \j/9 0 are partial functions from co into co9 that is functions whose graphs are subsets of co2. (j)(b)~c is true iff (if and only if) (b) is defined and equal to c. cj)(b) ~ \j/(c) iff both (j){b) and ij/(c) are defined and equal, or neither is defined. {e}f is the £-th item in the standard enumeration of functions partial recursive in / There exist a recursive predicate T and a recursive function U, both devised by Kleene, such that (1)

{e}'(b)*c iff (Ey)lT{f(y)9e9b9y)

f(y) encodes {|i}:

f(y)=I\pl+m' i

&

U{y) = c].

4 I. Constructive Ordinals and 11} Sets pt is the i-th smallest prime; p0 = 2. The right side of (1) says there is a computation y derived from the e-th set of equations, and the values of/restricted to i 0. For simplicity R(f x) will be used somewhat ambiguously to denote a recursive predicate with an arbitrary number of function and number variables. A predicate is analytical if it is built up from recursive predicates by application of propositional connectives, number quantifiers and function quantifiers. Thus (1)

(Ex)(f)(Eg)R(x9y9f9g9h)

and (Ef)(h)S(f ft, z)

are analytical ifR and S are recursive. A predicate is arithmetic if it is analytical but includes no function quantifiers. There is an aspect of the classification of predicates which will seem picayune now but which will matter a great deal later. A predicate may be classified by virtue of its explicit form, as were the predicates of (1), or by being proved equivalent to another predicate already classified. For example, the predicate, "/is constructible in the sense of Godel", is seen to be analytical only after it is shown that every constructible number-theoretic function is constructible via a countable ordinal. 1.3 Theorem (Kleene 1955). IfP(f following forms:

x) is analytical, then it can be put in one of the

(Eg)(y)R(f x, g9 y),

(Eg)(h)(Ey)R(f x,

g,h,y)...

(g)(Ey)R(f x, g9 y)9

(g)(Eh)(y)R(f x, g9ft,y). . .

A(fx) where A is arithmetic and R is recursive. Proof First P(f x) is put in prenex normal form with a recursive matrix by the usual quantifier manipulations associated with first order logic. Then the resulting prefix is put in one of the desired forms by applying the following rules. K is arbitrary. (1)

(x)(Ef )K(f x)~(Ef)(x)K((/) x , x).

( / ) , is defined by (f)x(y) =/(2*-3>).

1. Analytical Predicates 5 / i s thus interpretable as a code for {fn\n 1). The most important of all II} predicates is: X encodes a countable wellordering. It turns out to be universal n j , hence not Z}. It gives rise to a bounding principle with numerous applications. For example, it is used in Chapter IV to compute the Lebesgue measure of a II} set of reals. If X ^ 2 W , then K is said to be nxn (Ej respectively) if XeK is n^ (I* respectively). Similar conventions are in force when K ^ co or K ^ co2 etc.

1. Analytical Predicates 7 1.6 Theorem (Spector 1955). Suppose A(X) is X}. (i) n{X\A(X)}isU\. (ii) If(E1X)A(X\ then the unique X that satisfies A(X) is A}. Proof. (i) Let B be n {X\A(X)}. Then

(ii) Let C be the unique solution of A(X). Then xeC~(EX)|>l(X)

& xel] xeX].

•

In Chapter III, Section 6, it will be shown that every Z} set with a non-A} member has a continuum of members. The proof will require more than trivial quantifier manipulations, namely an analysis of E} predicates by means of recursive trees with infinite branching. Part (i) of Theorem 1.6 is often alluded to as follows: a set (of numbers) is II} if it is the closure of a II} set under a £} closure condition. A predicate A(X) is a closure condition if the intersection of any non-empty collection of solutions of A(X) is a solution of A(X)9 and if every set (of numbers) is contained in some solution of A(X). Let A(X) be a closure condition. It follows that for each Y there is a least X, call it Y09 such that Y c X and A{X): Y0=n{X\Y^X

& A(X)}.

Yo is called the closure of Y under A. By Theorem 1.6(i), YoeU\ if YeU\ and A(X) G £}, because then Y /(*) # z].

If the left side of (1) is Zj (11^ respectively), then the right side is 11^ (Z" respectively). • 1.8-1.12 Exercises 1.8. Show there exists a universal 11^ predicate, that is a 11^ predicate P(e,f x) such that for each 11^ predicate Q(f x) there is a c for which P(c,/, x) and Q(f x) are equivalent for all/and x.

8 I. Constructive Ordinals and n} Sets

1.9. show ni s si+1, si £ ni+1, ni $ si and si $ ni. 1.10. Let co have the discrete topology and co" the product topology. Basic closed subsets of cow can be coded by finite sequences of natural numbers, hence by natural numbers. A closed subset of cow, regarded as an intersection of basic closed sets, can be coded by a subset of co. Show "X codes a closed subset of co^" is arithmetic. Show "X codes a countable, closed subset of co60" is n j . 1.11. Let L be a first order language whose set of primitive symbols is recursive. Let S(X) be "X codes a countable set of sentences of L". Show "S(X) and X is consistent (that is yields no contradiction via first order logic)" is arithmetic. Show "S(X) and X has a model" is Sj. 1.12. Suppose O. By Theorem 3.4(i), 2M*)}(»)60, and so {r(e)}(n)eO. Now assume We c 0. Then for each rc, {r(e)}(n)eO, and by Theorem 3.4(iii) {s(e)}(n) 0 so that {r(e)}(n) = a. Then

Hence 2° |x| < 0]. The notion of height is useful for proving theorems about wellfounded relations by transfinite induction. Let Re be the e-th recursively enumerable binary relation, that is R x e( >

y)4^ M(x, y) is defined.

Thus {Re\e < a>} is a simultaneous recursive enumeration of all recursively enumerable binary relations. 4.3 Lemma. There exists a recursive f such that for all e: (i) Re is wellfounded /(e) e O; and (ii) Re is wellfounded -• \Re\ < \f(e)\.

4. Recursive Ordinals 17 Proof. The idea is to define a one-one, order-preserving map from the field of Re into 0 by an effective transfinite recursion on Re. One difficulty is the uncertain nature of the field of Re. It is recursively enumerable, but may be empty or finite. Let h be a total recursive function such that Rk(e,*)(x9y)"Re(x,y)

& Re(x9 ri) & Re(y, n)

for all e, n, x and y. Rhie,n) is the initial segment of Re below n. Rh{e,n) is empty if n is not in the field of Re. There exists a total recursive t such that 1 ;

tib e)

' ~ {{6}(fc(e, n))\n < co} otherwise.

Recall g from Lemma 4.1. Let k be a recursive function such that

{k(b)}(e)^g(t(b,e)); let c0 be a fixed point offc,that is {fc(c0)} ^ {c0}. Define J [€) — {^o}(^)» and t(e) = f(c0, e). Then = t(e)

0 if H, = ^, {f(h(e, n))\n < co} otherwise,

Suppose Re is wellfounded to show f(e)eO and \Re\ < \f(e)\. If the field of Re is empty, then Wt{e) = (\) and f(e)eO by Lemma 4.1. Assume #e#. Then l^*(c«)l < \Re\ f° r aU n) to those sequence numbers /(x) such that (i)i/(l) > / ( 2 ) . . . is an infinite descending sequence in SR(y) iff SR(y) is not wellfounded. D Proposition 5.3 equates the problem of checking membership in II} sets with the problem of checking wellfoundedness of recursive relations. Thus recursive ordinals suffice to analyze II} sets. 5.4 Theorem (Kleene). Each II} set is many-one reducible to O. Proof. Suppose B e H}. According to subsection 5.2 there is a recursive R such that yeB~(f)(Ex)R(f(x),y) for all y. By proposition 5.3, yeB0) until an n is found such that ae Wp{{z)m. Such an n must exist if a0) until an n is uncovered such that {z} (n) is defined and ae Wp({z)(n)). Let no be the first such n uncovered. (If no does not exist, then {0*(e, a, z)}x(v) is undefined for all v.) Then {6m(e9 a, z)}x^{6({e}(a,{z}(no)),

h(no))}x.

The recursive iterater / needed for the definition of k by effective transfinite recursive is given by: eo {I(e)}(a,b)= 9({e}(aJ),eo) 0*(e,a9z) 0

iffe = 2fl, ifb = 2d*2a, ifh = 3-5 z ' otherwise.

Let {c}~{I(c)}. k is {c}. k is total because 0 and 6^ are.

D

1.3 Theorem. Each of the following predicates is U{. (0 xeO & yeHx. (ii) xeO & y$Hx. Proof (i) Let A(X) be the conjunction of:

(e) [3-5 e e0^(X) 3 . 5 e = { < x , n >

{ } ( )

Recall that (X)m = {n|?K, T) in which Tdoes not appear. Hb is represented by a formula developed by recursion on \ choice downward closed under many-one reducibility. Proof. Suppose (x)(EY)yeHYP A(x, Y) for some arithmetic A(x, Y) with hyperarithmetic parameters. Then (1)

(x)(Ea)(Ee)[aeO

&

A(x,{e}H°)l

Let H(a,X) be the n ^ predicate of Theorem 4.2.II. Then (1) is equivalent to (2)

(x)(Ea)(Ee)[aeO

&

(Z)(H(a,Z)^A(x,{e}z))l

Since the matrix of (2) is IIJ, Lemma 2.6.II provides hyperarithmetic functions a(x) and e(x) such that (x)[a(x)eO

& (Z)(if(a(x),Z)^A(x,{e(x)} z ))].

The range of a(x) is bounded by some beO by Corollary 5.6.1. To be precise, (x)(a{x)eOb). It follows from Lemma 2.1.II and Theorem 1.3.II that the predicate (3)

me{e{x)}H«»

is A} hence hyperarithmetic. Thus there is a FeHYP such that me(Y)x equivalent to (3). Clearly (x)A(x,(Y)x). •

is

4.14 S j Dependent Choice. A typical instance of Zj dependent choice is (Y)(EZ)A{Y,Z)^(EY)(x)A((Y)x9(Y)x+1). A(Y,Z) is an arithmetic predicate that may contain set parameters. According to Exercise 4.22 HYP is an co-model of I j dependent choice. £} choice is an

4. The Ramified Analytic Hierarchy 69 immediate consequence of Z} dependent choice; H. Friedman found an co-model of the former in which the latter fails. In Chapter IV it will be shown that J^(co^K9 T) is a model of 1} dependent choice for almost all T, that is a set of T's of measure 1. At that time the equivalences of Lemma 4.16 will prove useful. The proof of 4.16 requires the following Corollary of Lemma 4.6. Then there exists a beO such that X is 4.15 Lemma. Suppose XGJ^(CO^K9T). recursive in HJ. (b depends only on a ramified analytic definition of X from T) Proof Let G(x) be a ranked formula such that (1)

neX^Jf{ (ii). Lemma 4.15 implies Jf{co^9 T) £ HYP(T). Suppose Be HYP(r). Let B be HJ. \b\T < co^K. There is a formula jfb of rank \b\ + 1 such that (1)

beHl'~J((a>?>9T)tjrh{n)

for all n. (1) is proved as was Lemma 4.7 with the aid of Corollary 4.4.II, now relativized to T. (ii) -• (iii). HYP(T) is an co-model of A} comprehension by a relativization to 7* of the proof of Theorem 4.10. (iii)->(i). By Theorem 4.10 relativized to T, HYP(T) £ M(co?K, T). It follows from Lemma 4.15 that each set hyperarithmetic in Tis recursive in some Hi, where \b\T < coJK. But then there cannot be an aeOT such that \a\T = co^K. •

4.17-4.21 Exercises 4.17. Prove 4.2(1) and 4.2(2). 4.18. Show < , defined in subsection 4.4, is wellfounded. 4.19. Prove 4.13(3) is A}. 4.20. Show 2} dependent choice implies 11} choice.

70 III. 1} Predicates of Reals 4.21. Prove there is an Jf b of rank \b\ + 1 that satisfies (1) of 4.16. 4.22. Show HYP is an co-model of E} dependent choice.

5. Kreisel Compactness Some of the boundedness properties of the hyperarithmetic hierarchy lead to a compactness theorem for co-logic discovered by G. Kreisel. His result is: if / is a II} set of sentences of co-logic such that every hyperarithmetic subset of / has a model, then / has a model. Note that "hyperarithmetic" takes the place of "finite" in the compactness theorem for first order logic. Kreisel's theorem paved the way for Barwise's compactness theorem for countable £ x admissible sets. Since this is not a book on model theory, the Kreisel result is presented in set theoretic terms. Recall what is meant by the compactness of 2™. 2 is the two-element space with the discrete topology. 2C0 is the product of countably many copies of 2 and has the product topology. If / is a family of closed subsets of 2m such that every finite subfamily of / has a nonempty intersection, then I has a nonempty intersection. Kreisel compactness is concerned with families of Aj subsets of 2C0. At first this idea seems hopeless, because it is easy to find a family of A} subsets with the finite intersection property whose intersection is empty. Kreisel's insight provides a way out. First ask that every hyperarithmetic subfamily have a nonempty intersection. Second require that the family be ITJ. As has been observed earlier, finite bears to recursively enumerable a relation similar to that borne by hyperarithmetic to II}. If n is a A i-index for a subset of 2®, then Dn denotes the subset indexed by n. 5.1 Theorem (Kreisel). Let I be a II} set of indices ofA{ sets. Suppose n{Dn\neH}*0 for every hyperarithmetic H^I. Then

Proof. Suppose not. Then (1)

(X)(Ey)lyeI &

The matrix of (1), call it Q{X,y), is 11}. The proof of Theorem 2.3.II shows Q(X,y) can be uniformized by some 11} P(X,y). The virtue of P(X,y) is its A.}-ness, since

Hence the set J of all n such (EX)P(X,n) is a £} subset of /, and n{Dn\neJ} = 0.

6. Perfect Subsets of 1} Sets 71 By the remark following Theorem 3.7.II, there is a hyperarithmetic H separating J and co-1, that is J c H c /. But then

0.

•

5.2 Corollary. Let J be a III set of ranked sentences of t2; (iv) G(si,ti) and Q(s29t2); (V)

| i < co & j < X } is defined by recursion on i. s°0 and t° are null. Fix i and ;'. Choose < s2j"x, t2+j1 > and < 4 ; +1 > *2/ +1 > to be incomparable extensions of . The choices can be made effectively from 0 since 0 is a complete 11} set (Theorem 5.4.1). Let P be the perfect set encoded by {slj\i < co & j < 21}. D Countable 11} subsets of 2W are much more complicated than their Z} counterparts. Their elements tend to be scattered throughout the constructible hierarchy. They will be discussed in Section 9. The next result is a uniform version of part of Theorem 6.2. 6.3 Theorem. There exists a recursive function h such that: if c is an index for a 2} predicate C(X) with no perfect set of solutions, then h(c)eO and

Proof Let P(X9b\ clearly IIi, be beO &

[C{X)^X, where < s, # > is a node of T, w is a sequence number that encodes an initial segment of a function, length of w = length of s, and R(s, w) holds. An infinite branch through U delivers an XeK, a. witness to the fact that XEK, and a witness to the fact that X£L(cof). As in the proof of 9.3, XeK iff{e}* is wellordered. Choose X o , a solution of (1), to minimize \{e}Xo\ = 5. Define Ud by restricting U to l = c o ^ .

1. Measure-Theoretic Uniformity 91 Proof. Lemma 4.16.III.

•

The next proposition says that J2?(a^K, T) is closed under formation of hyperarithmetic conjunctions. 1.7 Proposition. Let 3P\(i^K, &~) is to consider how to approximate a hyperarithmetic set of reals by an open set. At the ground level a subbasic open set is approximated by itself. At higher levels the approximation is built up from approximations available at lower levels. It turns out that Kj{p\p\\- ^} is a good approximation of { T\Jt(afi^, T)¥&}9 and that u {p|p 11 ~ J^ } is slightly better. Some calculation is needed to show that the latter open set is hyperarithmetic and regular, and that the error of approximation is meager (cf. Exercise 3.10). 3.2 Lemma. The predicate p \\- &, restricted isl,\ &'s9 is U\. Proof. Same as that of Lemma 4.5.III. The clauses of the definition of IK when ^ e E { , correspond to closure conditions whose conjunction is some Z} formula A(X, T). An induction on full ordinal rank and logical complexity shows

iff

~

3.3 Genericity. Tis generic (in the sense of Cohen) with respect to a sentence;) v

The matrix of (2) is equivalent to (3)

(Ea)fleO [r Ih (E7'fl') [A(n, Y^) v B(n,

(3) is n j by Lemma 3.2. Kreisel's selection Lemma (2.6.II) yields r and a as hyperarithmetic functions of q and n. By Spector's boundedness theorem (5.6.1)

3. Cohen Forcing 97 there is a recursive upper bound y on \a(q, n)\. Thus (n)(q)(Er)qzr [ H h ( E ^ ) \_A(n, Y^) v ~B(n9 r>)]], and so p forces (4) Since Tis generic and a member of p, J/(co^K, T) satisfies (4). But then J/(a)^K, T) satisfies (EX)(x)txeX~(EY)A(x,Y)l (The UX" is x(EYy)A(n9 Yy)).

D

3.7 Theorem. Assume T is generic and (EX) 3* (X) has no unranked variables save X.If

Proof. Suppose p \\-(EX)^(X). Then p \\-(^(x9(x)) for some 9(x) of rank a < co^K. The relation p I h JT, restricted to sentences JT of rank at most a, is A} by Exercise 3.14. Recall the construction of a generic T following Proposition 3.4. Repeat that construction with the J^'s replaced by a hyperarithmetic enumeration of all formulas of rank at most a. Then the constructed set, call it H, can be taken to be hyperarithmetic, since it can be defined by recursion on co relative to a hyperarithmetic predicate. H obeys Lemma 3.5 with respect to all sentences of rank at most a. Hep, since the construction of H can start with p. Hence Jt(aff, H) 1= ^(x^(x)\ Jt(off, H) is Jf{o^) by Lemma 4.16(ii).III. • 3.8 Corollary. / / T is generic, then 0 is Zj definable over Jt(aff, T). Proof. By Lemma 3.5.III,

xeO~(EY)YeliYPA(x,Y) for some arithmetic A. Theorem 3.7 implies ff, T) N(EY)A{n9 Y).

•

3.9 Category Versus Measure. There is an analogy between the results of Sections 3 and 2 based on a standard analogy between category and measure. Let A be a subset of 2W. (Recall the topology assigned to 2^ in subsection 6.1.II.) A is said to be nowhere dense if it has an empty interior. A is said to be meager if it is contained in a countable union of nowhere dense closed sets. Baire's category theorem states: the

98 IV. Measure and Forcing complement of a meager set is dense. Consequently meager sets are thought to be small and analogous to sets of measure 0. It follows from Proposition 3.4(ii) that the set of J ' s such that 7 is generic with respect to a given sentence 3* is dense open, and the set of all generic T is comeager. Thus Theorem 3.6 is analogous to Theorem 1.5, and Exercise 3.11 to Theorem 2.2.

3.10-3.18 Exercises 3.10. An open subset of 2W is said to be regular if it equals the interior of its closure. Let A be a A} subset of 2°\ Find a A | 5 such that B is regular open and (A — B) v (B — A) is meager. 3.11. (Hinman, Thomason). Let A be a non-meager, 11} subset of 2*° . Show A has a hyperarithmetic element. 3.12. If Tis generic, then T^HYP. 3.13. If Tis generic, then OT ^K. 3.15. Assume Tis generic. Let Tt(i is generic (cf. Exercise 2.6). 3.18. Show there exists a generic TQf, X) ¥ P\

then P I h # \

(2) If J^( Y*) is unranked and P I h ^(x Q take each recursive ordinal to its unique notation. Thus \n(p)\ = p.

1. Fundamentals of Metarecursion 117 Assume A £ co^K. A is called metarecursively enumerable (Kreisel and Sacks 1963) if n\_A~\ is II}. {n\_A~\ = {n(a)\aeA}.) A is called metarecursive if A and co^K — A are metarecursively enumerable. A is said to be metafinite if n\_A~\ is hyperarithmetic. The choice of Q makes no difference. Suppose Qi and Q2 are II} sets of unique notations. Let 0: Qt -* Q2 be the unique map such that \0(n)\ = \n\. Then 9(x) = y iff (1)

xeQx

& yeQ2 & (Ef) [/is an order-preserving map of Qfx onto Q^*].

(Qf^K x co^K be a partial function. is said to be partial metarecursive if

1.2 Proposition. L^t K9 A c (y^K (0 K is metafinite +-+K is metarecursive and bounded above by some p < (ii) A is meta r.e. 1 there is a partial metarecursive function (l)n(z,x1, . . . ,xn) such that: for each partial metarecursive i//(x1, . . . , xn) there is an e < co such that

Proof. Let n = 1. As in subsection 5.2.1, each II} predicate Pe(x, y) can be put in the form

(f)(Eu)T(f(u),e,x9y) for some e determined by P. The proof of Kreisel's uniformization Theorem (2.3.II) yields a recursive function g such that g(e) is an index for the II { predicate that uniformizes

Pe(x,y) & xeQ & yeQ. Define cj)1(e,x)~y by (f)(Eu)T(f(u),g(e\n(xln(y)). (n is the notation function of subsection 1.1.)

•

1.5 Lemma. There exist metarecursive functions j and k such that: for each metafinite set K, there exists a unique S < co^ such that K={x\x