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Interactions Between Computability Theory and Set Theory by Noah Interactions between computability theory and set theory by Noah Schweber A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Antonio Montalban, Chair Professor Leo Harrington Professor Wesley Holliday Spring 2016 Interactions between computability theory and set theory Copyright 2016 by Noah Schweber i To Emma ii Contents Contents ii 1 Introduction1 1.1 Higher reverse mathematics ........................... 1 1.2 Computability in generic extensions....................... 3 1.3 Further results .................................. 5 2 Higher reverse mathematics, 1/28 2.1 Introduction.................................... 8 2.2 Reverse mathematics beyond type 1....................... 11 2.3 Separating clopen and open determinacy.................... 25 2.4 Conclusion..................................... 38 3 Higher reverse mathematics, 2/2 41 3 3.1 The strength of RCA0 ............................... 41 3.2 Choice principles ................................. 50 4 Computable structures in generic extensions 54 4.1 Introduction.................................... 54 4.2 Generic reducibility................................ 58 4.3 Generic presentability and !2 .......................... 64 4.4 Generically presentable rigid structures..................... 71 5 Expansions and reducts of R 74 5.1 Introduction.................................... 74 ∗ 5.2 Rexp ≡w R ..................................... 77 5.3 Generalizing.................................... 84 5.4 The structure Rint ................................ 86 5.5 Applying the general results........................... 87 5.6 Arbitrary continuous f .............................. 88 6 Notes on structures computing every real 90 6.1 Introduction.................................... 90 iii 6.2 Functions on Cantor space............................ 90 6.3 Ultrafilters..................................... 92 6.4 Degrees of structures computing all reals.................... 94 7 Limit computability and ultrafilters 97 7.1 Introduction.................................... 97 7.2 Basic Properties of δU .............................. 100 7.3 Building Scott sets ................................ 103 7.4 Lowness notions.................................. 105 7.5 Comparing ultrafilters .............................. 110 7.6 Further directions................................. 113 8 Computability theoretic aspects of ordinals 117 8.1 Introduction.................................... 117 8.2 Copying ordinals ................................. 118 8.3 Medvedev degrees of ordinals .......................... 120 Bibliography 126 iv Acknowledgments I am extremely grateful to my advisor, Antonio Montalban. In addition to the excellent supervision he provided to me here in Berkeley | including excellent choices of problems to work on, extensive feedback, and advice about all aspects of mathematical life | he was my teacher in college, and taught my first logic class. I could not ask for nine years of better mentorship. It is of course true that without his help, none of this would have been possible. I am also very grateful to Leo Harrington, Wesley Holliday, Ted Slaman, and John Steel for many insightful conversations and help throughout this process. I also want to thank my co-authors, Uri Andrews, Mingzhong Cai, David Diamondstone, Greg Igusa, Julia Knight, and Antonio Montalban, both for working with me on a variety of interesting problems and for allowing me to include the work we did together in this thesis. I have deeply enjoyed working with all of them, and hope to continue to in the future. There is not enough room here to list all the people who have supported me throughout my time in Berkeley, but I will try to thank a few in particular: Alex Cottrill, Alex Kruckman, Alla Hoffman, Andy Voellmer, Clare Stinchcombe, Dan Ioppolo, Haruka Cowhey, James Walsh, Matthew Harrison-Trainor, Nick Ramsey, Peter Borah. I would also like to thank everyone in the logic community who has helped me reach this point. And I am deeply grateful to my family for supporting me through this process. Finally, I would like to give special thanks to Damir Dzhafarov for addicting me to mathematical logic; without Damir, none of this would have been necessary. 1 Chapter 1 Introduction This thesis explores connections between computability theory and set theory. The bulk of this thesis focuses on extending ideas and techniques from computability theory to higher set-theoretic levels | higher reverse mathematics (chapters 2 and 3) and uncountable com- putable structure theory (chapters 4, 5, and 6). We also look at applications of set theory to computability theory, either by directly answering computability-theoretic questions via set- theoretic considerations (chapter 8) or by bringing set-theoretic constructions into contact with classical computability theory (chapter 7). Finally, we also look at results which stretch from the computability-theoretic to the set-theoretic | specifically, determinacy principles (chapter 2). Below we give a summary of the results in this thesis. Each individual chapter is self- contained, but background knowledge in set theory and computability theory is helpful; we recommend [35] and [13] respectively, whose notation we follow. Small amounts of proof theory (conservative extensions) and basic model theory (o-minimality) appear in chapters 2 and 4, but are covered as needed, and we assume no background outside of a standard introduction to mathematical logic such as [50]. 1.1 Higher reverse mathematics In the first part of this thesis, we look at higher reverse mathematics | roughly speaking, the study of the effective content of theorems of mathematics which cannot easily be expressed in the language of second-order arithmetic. Chapter 1 | which consists of work published as \Transfinite recursion in higher reverse mathematics" [68] | gives an introduction to 3 the subject, introduces a base theory RCA0 for third-order reverse mathematics, and studies analogues of the system ATR0 at higher types. We show, for example, that the comparability 2 of well-orderings of sets of reals is a very weak principle, relatively speaking, and that Σ1- separation for functionals implies clopen determinacy for reals. The main result of chapter 2 is the separation of two determinacy principles. In 1977, John Steel [73] showed that clopen and open determinacy are equivalent over RCA0, despite CHAPTER 1. INTRODUCTION 2 their different computability-theoretic properties (e.g. clopen games have relatively hyper- arithmetic winning strategies); the culprit is the high complexity of the predicate \clopen," 1 which is Π1 complete. We show that once we pass to a context where clopen games are relatively easier to identify, the principles separate: we define clopen and open determinacy principles for games played on R, and construct a model M separating them. 3 Theorem 1.1.1. Over RCA0, clopen determinacy for reals is strictly weaker than open determinacy for reals. The construction of M uses a variation of a notion of forcing with tagged trees introduced by Steel [74]. Leaving aside the technical details, we let G be a certain generic tree, labelled with ordinals. Elements of M are given by names which depend on G in a \bounded" way: for a functional to be in M, we demand that it be the evaluation of some name which respects one of a prescribed family of equivalence relations on the forcing, P, used to produce G. A name ν for a functional respects an equivalence relation ≈ if whenever p ≈ q, r 2 R, and k 2 !, we have p ν(r) = k () q ν(r) = k: In classical Steel forcing, we instead look at functions which are hyperarithmetic relative to G; this has roughly the same effect, but the higher-type analogue of \hyperarithmetic" is not well-behaved, hence our more abstract approach. Of crucial importance is the countable closure of P, which is used to control the second-order part of the model and in the verification that clopen determinacy holds in M, to show that no clopen games of high rank enter M; this has no analogue in classical Steel-forcing arguments. From the proof of Borel determinacy (see [52]), we should expect a connection between Theorem 1.1.1 and determinacy principles for higher Borel levels of games on !. Indeed, Sherwood Hachtman [25] answered a question posed in an early draft of [68] by construct- ing a canonical separating model. Let θ be the least ordinal such that Lθ satisfies \P(!) exists and for every well-founded tree T of height !, there is a map ρ: T ! ON with ρ(x) < ρ(y) whenever x ) y;" Hachtman showed (in the course of his broader analysis of θ) that the structure (!; RLθ ; (!R)Lθ ) also satisfies clopen determinacy for reals but not open determinacy for reals. In chapter e, we present some further results in higher reverse mathematics of a more 3 technical nature. First, we show that RCA0 is a conservative subtheory of Kohlenbach's ! RCA0 . The main technical obstacle in this proof is that the desired term model has to be defined in a slightly subtle way; however, no major difficulties emerge. We then move on to choice principles in the context of higher reverse mathematics. Looking back at the results of chapter 1, two applications of the axiom of choice were relevant: that the reals are well- orderable (to get the Kleene-Brouwer order of a tree ⊆ R<!, and that real-indexed families of nonempty sets of reals have choice functions (to pass from
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