Recursion Theory: Computational Aspects of Definability 9783110275643, 9783110275551

Table of contents :
Preface
Contents
Part I: Fundamental theory
1 An introduction to higher recursion theory
1.1 Projective predicates
1.2 Ordinal notations
1.3 Effective transfinite induction
1.4 Recursive ordinals
1.5 ?1/1-completeness and S1/1 boundedness
2 Hyperarithmetic theory
2.1 H-sets and ?
2-singletons
2.2 ?1/1-ness and hyperarithmeticity
2.3 Spector’s Uniqueness Theorem
2.4 Hyperarithmetic reducibility
2.5 Some basis theorems and their applications
2.6 More on O
2.7 Codes for sets
3 Admissibility and constructibility
3.1 Kripke–Platek set theory
3.2 Admissible sets
3.3 Constructibility
3.4 Projecta and master codes
3.5 ?-models
3.6 Coding structures
3.7 The Spector–Gandy Theorem
4 The theory of ?1/1-sets
4.1 A ? 1/1-basis theorem
4.2 ?1/1-uniformization
4.3 Characterizing thin ?1/1-sets
4.4 S1/2-sets
5 Recursion-theoretic forcing
5.1 Ramified analytical hierarchy
5.2 Cohen forcing
5.3 Sacks forcing
5.4 Characterizing countable admissible ordinals
6 Set theory
6.1 Set-theoretic forcing
6.2 Some examples of forcing
6.3 A cardinality characterization of ?1/1-sets
6.4 Large cardinals
6.5 Axiom of determinacy
6.6 Recursion-theoretic aspects of determinacy
Part II: The story of Turing degrees
7 Classification of jump operators
7.1 Uniformly degree invariant functions
7.2 Martin’s conjecture for uniformly degree invariant functions
7.3 The Posner–Robinson Theorem
7.4 Classifying order-preserving functions on 2?
7.5 Pressdown functions
8 The construction of ?1/1-sets
8.1 An introduction to inductive definition
8.2 Inductively defining ?1/1-sets of reals
8.3 ? 1/1-maximal chains and antichains of Turing degrees
8.4 Martin’s conjecture for ?1/1-functions
9 Independence results in recursion theory
9.1 Independent sets of Turing degrees
9.2 Embedding locally finite upper semilattices into ?D, =?
9.3 Cofinal chains in D
9.4 ?-homogeneity of the Turing degrees
9.5 Some general independence results
Part III: Hyperarithmetic degrees and perfect set property
10 Rigidity and biinterpretability of hyperdegrees
10.1 Embedding lattices into hyperdegrees
10.2 The rigidity of hyperdegrees
10.3 Biinterpretability
11 Basis theorems
11.1 A basis theorem for ?1/1-sets of reals
11.2 An antibasis theorem for ?0/1-sets
11.3 Perfect sets in L
Part IV: Higher randomness theory
12 Review of classical algorithmic randomness
12.1 Randomness via measure theory
12.2 Randomness via complexity theory
12.3 Lowness for randomness
13 More on hyperarithmetic theory
13.1 Hyperarithmetic measure theory
13.2 Coding sets above Kleene’s O
13.3 Hyperarithmetic computation
14 The theory of higher randomness
14.1 Higher Kurtz randomness
14.2 ?1/1 and ?1/1-Martin-Löf randomness
14.3 ?1/1-randomness
14.4 ?1/2 and S1/2-randomness
14.5 Kolmogorov complexity and randomness
14.6 Lowness for randomness
A Open problems
A.1 Hyperarithmetic theory
A.2 Set-theoretic problems in recursion theory
A.3 Higher randomness theory
B An interview with Gerald E. Sacks
C Notations and symbols
Bibliography
Index

Citation preview

Chi Tat Chong, Liang Yu Recursion Theory

De Gruyter Series in Logic and Its Applications

| Editors Wilfried A. Hodges, University of London, United Kingdom Menachem Magidor, The Hebrew University of Jerusalem, Israel Anand Pillay, University of Notre Dame, USA

Volume 8

Chi Tat Chong, Liang Yu

Recursion Theory

| Computational Aspects of Definability

AMS subject classification (2010) 03-01, 03-02, 03D, 03E Authors Prof. Dr. Chi Tat Chong National University of Singapore Department of Mathematics 10 Lower Kent Ridge Road 119076 Singapore [email protected] Prof. Dr. Liang Yu Nanjing University Institute of Mathematical Science 210093 Nanjing People’s Republic of China [email protected]

ISBN 978-3-11-027555-1 e-ISBN (PDF) 978-3-11-027564-3 e-ISBN (EPUB) 978-3-11-038129-0 Set-ISBN 978-3-11-027565-0 ISSN 1438-1893 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2015 Walter de Gruyter GmbH, Berlin/Boston Typesetting: PTP-Berlin, Protago TEX-Produktion GmbH Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

| Chi Tat Chong dedicates this book to the memory of his parents; Liang Yu dedicates it to Li Zhang.

Preface . . . logic is about definability. – Gerald E. Sacks [124]

The above quote underlines the authors’ view that recursion theory is anchored on the notion of definability. Although classically the theory began with Turing’s conception of effective computability (of various types of objects), it soon took on a much broader perspective. A fundamental theorem states that a set of natural numbers is computable if and only if it is Δ01 -definable in first order arithmetic. This sets the stage for a general study of sets of natural numbers and their syntactic classes. The view on definability does not conflict with the fact that computation is the heart of recursion theory. Indeed a definability problem naturally translates into one about (possibly relativized) computation and vice versa. Kleene’s characterization of the Δ11 -sets of numbers as precisely those that are hyperarithmetic, and the corresponding theorem of Spector and Gandy that these are all the reals in L ωCK , is a case 1 in point. Once the syntax (the language of first and second order arithmetic) and the structure in which to interpret members of ω and its subsets are specified, definability naturally takes its place. Tools and construction techniques have been introduced to study problems concerning computability for syntactic classes over ω, 2ω and even the power set of 2ω or ω ω . Post’s problem for the recursively enumerable sets, the Gandy basis theorem for Σ11 -sets of reals, characterization of Δ11 -sets in terms of hyperarithmeticity, the definability of the jump operator, are immediate examples. Indeed it is hard to find a problem in the area that has little to do with definability. The title of this book is chosen to reflect this point. The idea of writing a book in recursion theory, focusing on the computational aspects of the set of reals and its subsets, evolved over a period of time. The subject was established by Kleene who introduced the notion of hyperarithmetic hierarchy relative to a real and its associated ordinals. It grew in the hands of Spector, Gandy, Kreisel and Sacks. These were major developments in what one now calls higher recursion theory. The theory was a subject of intensive study for almost thirty years beginning in the 1950s, but seemed to lose momentum in the 1980s. In an interview reproduced in the Appendix, Sacks remarked: “I tended over the years to keep predicting the immediate end of recursion theory, but I have been wrong every time.” Despite the rather pessimistic pronouncement by one of the founders of the field, recursion theory is alive and well. However, higher recursion theory in the grand tradition of the pioneers received only occasional attention since, even though a number of highly significant contributions to the subject, such as Slaman and Woodin’s theorem on the rigidity of the hyperdegrees, were made towards the end of the last century, and many interesting as well as challenging problems remain. Indeed Sacks himself bade farewell to higher recursion theory in 1998. Today the subject is not pursued with as much vigor

VIII | Preface

or zeal as it deserves. We view this as rather unfortunate since in it one sees a beautiful illustration of the unity of mathematical logic, where ideas and tools from set theory and model theory are fruitfully applied to study problems about computability. It was a tempting proposition to write another book (following Sacks’s Higher Recursion Theory [123]) to exhibit this interplay. Recent advances in the study of the global theory of Turing and hyperdegrees, and the emergence of higher randomness as a subject of investigation, provided the impetus for us to turn the idea into reality. It is hoped that the reader would agree, at the end of it, that the enterprise of generalizing recursion theory that began more than half a century ago has been mathematically rewarding, and that the subject is both attractive and highly challenging, with many deep and elegant theorems. This book consists of four main parts. The first six chapters constitute Part I and develop the fundamental theory of Π11 -sets and fine structure theory of the constructible universe, the ramified analytical hierarchy, and set theory. They lay the groundwork for the rest of the book. Part II focuses on the structure of Turing degrees. Here we discuss classification of the jump operator in relation to Martin’s conjecture and the construction of Π11 -sets, as well as degree-theoretic independence results. We turn our attention to hyperdegrees in Part III. In particular, we prove the rigidity of the structure of the hyperdegrees and its biinterpretability with second order arithmetic. We also include a chapter on basis/antibasis theorems as well as perfect subsets in L. Part IV is devoted to the study of higher randomness, a subject which is naturally linked to the topics covered in the first three parts. Exercises are sprinkled throughout the chapters. Many of them are intended to supplement the topics covered. The most challenging ones are labelled with a ⋆ . The book ends with an interview with Gerald Sacks (in Appendix B) conducted in 2009. We think that it is fitting to conclude the book with a tribute to one whose work has arguably substantially shaped the development of recursion theory, both classical and higher, in the last century. A word about prerequisites: While this book attempts to cover as much background material as possible, it is not intended to be encyclopedic. The reader is assumed to be familiar with basic logic and set theory, including classical recursion theory, basic set theory (including forcing) and algorithmic randomness. A number of the results are used in proofs, sometimes without explicit mention. In line with our preference and taste, some of the proofs of theorems in set theory presented are discernibly recursion-theoretic in flavor. These proofs are not necessarily the simplest. Our aim is to offer the reader a different perspective to these theorems by highlighting the effective content that is not immediately apparent. Hopefully these provide additional insights to the ideas that went behind the proofs. The materials in the first two chapters closely follow Sacks’s Higher Recursion Theory [123]. Mansfield and Weitkamp’s monograph Recursive Aspects of Descriptive Set Theory [88] was also one of our key references. Of course one should not fail to mention Hartley Rogers’s classic Theory of Recursive Functions and Effective Computabil-

Preface

| IX

ity [116], the book that first made recursion theory accessible to several generations of students in logic and which largely set the style for books on this subject. In this book we give preference to calling the subject recursion theory rather than computability theory (as it is otherwise known) in deference to its historical origin, although both terms are used interchangeably throughout. During the preparation of the manuscript, the authors benefited from comments and suggestions of colleagues and students, some of whom read through various portions of the book. Certain chapters were used by the second author in a graduate course given at the National University of Singapore. We thank Laurent Bienvenu, Peter Cholak, Decheng Ding, Qi Feng, Su Gao, Junle Goh, Noam Greenberg, Wei Li, Yiqun Liu, Yong Liu, Benoit Monin, André Nies, Dilip Raghavan, Gerald Sacks, Richard Shore, Stephen Simpson, Theodore A. Slaman, Frank Stephan, Wei Wang, Yue Yang and Jing Zhang for their interest, ideas, suggestions and/or support of this project. Finally, for this work Chong was partially supported by NUS grant WBS C140000-025-00, while Yu was supported in part by the National University of Singapore, the Humboldt Foundation and the National Science Foundation of China grant no. 11071114 and 11322112. Their generosity is gratefully acknowledged. Chitat Chong and Liang Yu 15 June 2015

Contents Preface | VII

Part I: Fundamental theory 1 1.1 1.2 1.3 1.4 1.5

An introduction to higher recursion theory | 3 Projective predicates | 3 Ordinal notations | 10 Effective transfinite induction | 13 Recursive ordinals | 16 Π11 -completeness and Σ11 -boundedness | 18

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Hyperarithmetic theory | 23 H-sets and Π20 -singletons | 23 Δ11 -ness and hyperarithmeticity | 25 Spector’s Uniqueness Theorem | 29 Hyperarithmetic reducibility | 31 Some basis theorems and their applications | 33 More on O | 36 Codes for sets | 40

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Admissibility and constructibility | 45 Kripke–Platek set theory | 45 Admissible sets | 51 Constructibility | 53 Projecta and master codes | 58 ω-models | 62 Coding structures | 64 The Spector–Gandy Theorem | 69

4 4.1 4.2 4.3 4.4

The theory of Π11 -sets | 75 A Π11 -basis theorem | 75 Π11 -uniformization | 77 Characterizing thin Π11 -sets | 79 Σ21 -sets | 81

5 5.1 5.2

Recursion-theoretic forcing | 85 Ramified analytical hierarchy | 85 Cohen forcing | 88

XII | Contents

5.3 5.4 6 6.1 6.2 6.3 6.4 6.5 6.6

Sacks forcing | 93 Characterizing countable admissible ordinals | 99 Set theory | 105 Set-theoretic forcing | 105 Some examples of forcing | 109 A cardinality characterization of Π11 -sets | 115 Large cardinals | 118 Axiom of determinacy | 121 Recursion-theoretic aspects of determinacy | 123

Part II: The story of Turing degrees 7 7.1 7.2 7.3 7.4 7.5

Classification of jump operators | 131 Uniformly degree invariant functions | 131 Martin’s conjecture for uniformly degree invariant functions | 134 The Posner–Robinson Theorem | 137 Classifying order-preserving functions on 2ω | 141 Pressdown functions | 142

8 8.1 8.2 8.3 8.4

The construction of Π11 -sets | 147 An introduction to inductive definition | 147 Inductively defining Π11 -sets of reals | 150 Π11 -maximal chains and antichains of Turing degrees | 154 Martin’s conjecture for Π11 -functions | 159

9 9.1 9.2 9.3 9.4 9.5

Independence results in recursion theory | 165 Independent sets of Turing degrees | 165 Embedding locally finite upper semilattices into ⟨D , ≤⟩ | 170 Cofinal chains in D | 172 ω-homogeneity of the Turing degrees | 177 Some general independence results | 179

Part III: Hyperarithmetic degrees and perfect set property 10 10.1 10.2 10.3

Rigidity and biinterpretability of hyperdegrees | 183 Embedding lattices into hyperdegrees | 183 The rigidity of hyperdegrees | 186 Biinterpretability | 188

Contents | XIII

11 11.1 11.2 11.3

Basis theorems | 195 A basis theorem for Δ11 -sets of reals | 195 An antibasis theorem for Π10 -sets | 199 Perfect sets in L | 207

Part IV: Higher randomness theory 12 12.1 12.2 12.3

Review of classical algorithmic randomness | 213 Randomness via measure theory | 213 Randomness via complexity theory | 216 Lowness for randomness | 220

13 13.1 13.2 13.3

More on hyperarithmetic theory | 223 Hyperarithmetic measure theory | 223 Coding sets above Kleene’s O | 229 Hyperarithmetic computation | 236

14 14.1 14.2 14.3 14.4 14.5 14.6

The theory of higher randomness | 245 Higher Kurtz randomness | 245 Δ11 and Π11 -Martin-Löf randomness | 248 Π11 -randomness | 254 Δ12 and Σ21 -randomness | 259 Kolmogorov complexity and randomness | 262 Lowness for randomness | 263

A A.1 A.2 A.3

Open problems | 273 Hyperarithmetic theory | 273 Set-theoretic problems in recursion theory | 273 Higher randomness theory | 274

B

An interview with Gerald E. Sacks | 275

C

Notations and symbols | 293

Bibliography | 295 Index | 303

| Part I: Fundamental theory

1 An introduction to higher recursion theory 1.1 Projective predicates 1.1.1 Recursive and arithmetical predicates A recursive predicate R on ω is a total recursive function Φ : ω → 2 such that R(n) ⇔ Φ(n) = 1. With a standard coding of ω