University of California Los Angeles Calibrating Determinacy Strength in Borel Hierarchies A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Sherwood Julius Hachtman 2015 c Copyright by Sherwood Julius Hachtman 2015 Abstract of the Dissertation Calibrating Determinacy Strength in Borel Hierarchies by Sherwood Julius Hachtman Doctor of Philosophy in Mathematics University of California, Los Angeles, 2015 Professor Itay Neeman, Chair We study the strength of determinacy hypotheses in levels of two hierarchies of subsets of 0 0 1 Baire space: the standard Borel hierarchy hΣαiα<ω1 , and the hierarchy of sets hΣα(Π1)iα<ω1 in the Borel σ-algebra generated by coanalytic sets. 0 We begin with Σ3, the lowest level at which the strength of determinacy had not yet been characterized in terms of a natural theory. Building on work of Philip Welch [Wel11], 0 [Wel12], we show that Σ3 determinacy is equivalent to the existence of a β-model of the 1 axiom of Π2 monotone induction. 0 For the levels Σ4 and above, we prove best-possible refinements of old bounds due to Harvey Friedman [Fri71] and Donald A. Martin [Mar85, Mar] on the strength of determinacy in terms of iterations of the Power Set axiom. We introduce a novel family of reflection 0 principles, Π1-RAPα, and prove a level-by-level equivalence between determinacy for Σ1+α+3 and existence of a wellfounded model of Π1-RAPα. For α = 0, we have the following concise 0 result: Σ4 determinacy is equivalent to the existence of an ordinal θ so that Lθ satisfies \P(!) exists, and all wellfounded trees are ranked." 0 We connect our result on Σ4 determinacy to work of Noah Schweber [Sch13] on higher order reverse mathematics. Schweber shows, using the method of forcing, that for games R with real number moves, clopen determinacy (∆1 -DET) does not imply open determinacy R 3 (Σ1 -DET) over the weak base theory RCA0. We show that the model Lθ is a witness to ii this separation result, and furthermore, that Lθ is (in the appropriate sense) the minimal (third-order) β-model of a natural theory of projective transfinite recursion. We obtain 0 R R that Σ4 determinacy falls strictly between the principles Σ1 -DET; ∆1 -DET in terms of β-consistency strength. Finally, we combine our methods with those of John Steel [Ste82] and Itay Neeman [Nee00], [Nee06] to characterize the strength of determinacy for sets in the pointclasses 0 1 Σ1+α+3(Π1). Granted that the reals are closed under the sharp function, we show this determinacy is equivalent to the existence of an iterable mouse with a measurable cardinal ++ κ of Mitchell order o(κ) = κ in which Π1-RAPκ+1+α holds. iii The dissertation of Sherwood Julius Hachtman is approved. Matthias J. Aschenbrenner John P. Carriero Donald A. Martin Yiannis N. Moschovakis Itay Neeman, Committee Chair University of California, Los Angeles 2015 iv For Myrna, who understands v Table of Contents 1 An overview of the results ::::::::::::::::::::::::::::: 1 1.1 Background and motivation . 1 1.2 Reverse mathematics and Borel determinacy . 4 1.3 Beyond Borel determinacy . 9 0 1 2 Σ3 determinacy and Π2 monotone induction :::::::::::::::::: 12 2.1 Proving determinacy . 15 1 2.2 Π2 monotone induction from infinite depth Σ2-nestings . 22 2.3 Infinite depth Σ2-nestings from determinacy . 29 0 3 Determinacy in Σ4 and higher :::::::::::::::::::::::::: 35 3.1 The Π1-Reflection to Admissibles Principle . 36 3.2 Proving determinacy . 43 3.3 The lower bound . 52 0 3.4 Generalizing to Σα+3-DET, for α > 1. 62 3.5 Borel determinacy and inductive definitions . 70 4 A connection with higher order reverse mathematics :::::::::::: 73 R R 4.1 Separating Σ1 -DET and ∆1 -DET . 75 4.2 Third order β-models and transfinite recursion . 78 4.3 From β-models to set models. 83 4.4 Higher levels . 87 4.5 Remarks: Provability versus β-consistency strength . 89 vi 1 5 Measurable cardinals and Borel-on-Π1 determinacy ::::::::::::: 91 0 1 5.1 Unraveling Σ1(Π1) sets and proving determinacy . 93 5.2 Inner models with sequences of measures . 94 5.3 The lower bound . 104 References ::::::::::::::::::::::::::::::::::::::::: 108 vii Acknowledgments I have had the unreasonable good fortune to meet and work under the auspices of many brilliant and giving mathematicians and friends. Without them this work would not have been possible. Above all I owe an enormous debt of gratitude to Itay Neeman. If he had merely suggested interesting problems, it would have been enough; if he had merely met with me once a week to discuss mathematics, it would have been enough. But he has gone above and beyond, providing me with every possible opportunity for growth and experience throughout my stay in graduate school. He has been persistently generous with his time, supportive of my endeavors, reassuring in times of distress, understandingly admonishing in the many times of sloth or stagnation, and always remarkably prudent in his advice and assistance. I have been truly lucky to have him as my advisor. It should be obvious from the title of my thesis alone that I am likewise forever indebted to Donald A. Martin. Even had I never met the man, his many ideas and writings, in their depth, clarity, and beauty, would have been a guiding light. As it stands, Tony was my first logic teacher at UCLA, an encouraging presence throughout my stay, frequently a lively sounding board, and always a generous source of good ideas. He is, with astonishing constancy, good-humored, modest, patient and forthright. To talk with and learn from him has been a great pleasure. I should also thank Simon Thomas, who took me under his wing at Rutgers College. Before meeting Simon, I had never considered a career in mathematics as something available to me. His intervention at a pivotal moment set me down the path I'm on today. I also thank Martin Zeman and Yiannis Moschovakis, each of whom provided clarity to material that was to become basic to my thesis. There are many young set theorists who I have been privileged to know and learn from, and whose many accomplishments serve as an inspiration. Among them are Grigor Sargsyan, Daisuke Ikegami, Nam Trang, Spencer Unger, and Noah Schweber; I know that I will continue viii to learn from them as time goes on, and will consider it a great achievement if I can teach them something in return. My time at UCLA would have been quite a different experience if not for the company and support of many wonderful friends. Without their camaraderie, there would really have been little point in doing any mathematics. Among them I would like to thank Justin Palumbo, Anush Tserunyan, William Chen, Jason Murphy, and Siddharth Bhaskar; each with their own brand of enthusiasm and discerning aesthetic and mathematical taste, they have helped me to inform and maintain my own. I thank C. Thi Nguyen, Melissa Hughs, Jonathan Gingerich, and Tamar Weber, for guidance in the art of living, and for being the best climbing partners one could ask for. Finally, I thank my family: Joey, Tom, Myrna, April, and Ilse. They have been a ceaseless source of love, support, and encouragement since before I can remember. Without them this work would not be possible, nor would I. ix Vita 2009 B.S., Mathematics, Rutgers, the State University of New Jersey. 2011 M.A., Mathematics, University of California, Los Angeles. 2014 Teaching Assistant Instructor, Integration and Infinite Series, Winter Quarter (UCLA). 2014 Instructor: Forcing and Independence in Set Theory. Undergraduate Sum- mer School, UCLA Logic Center. 2014-2015 Teaching Assistant Consultant. Office of Instructional Development, UCLA. Publications Calibrating determinacy strength in levels of the Borel hierarchy. Submitted. Determinacy in third order arithmetic. Submitted. x CHAPTER 1 An overview of the results 1.1 Background and motivation Let us define a game between two players, call them Player I and Player II, who take turns choosing natural numbers, x0; x1; x2;::: , in sequence: I x0 x2 : : : x2n ::: II x1 : : : x2n+1 ::: Supposing that the players have survived through infinitely many rounds of the game, they have cooperated to produce an infinite sequence of naturals, x = hx0; x1;:::i. A winner of the play x is then decided based on a priorly fixed winning condition: Namely, a set A ⊆ !! of sequences of naturals is fixed beforehand, called the payoff set. If x 2 A, then Player I wins; if x2 = A, then Player II wins. We denote the game with payoff A by G(A). A strategy for Player I is a partial function σ : !<! *! that is defined on finite sequences of even length. Intuitively, σ tells I what move to make whenever it is his turn. A strategy is winning for I in the game G(A) if whenever x is compatible with σ, in the sense that x2n+1 = σ(hx0; : : : ; x2ni) for all n, then we have x 2 A, that is, I wins the play x. Analogous definitions are made for Player II. If either player has a winning strategy in the game with payoff A, we say G(A) is determined. Such infinite two-person games of perfect information (henceforth simply infinite games) had been considered by mathematicians of the Lw´owschool [Mau81] and were introduced to the literature by Gale and Stewart [GS53] as a generalization of finite perfect information games, already an object of intense study in the burgeoning field of economic game theory 1 [NM44]. The stated hope of [GS53] was that study of infinite games would lead to the dis- covery of methods applicable to the study of games of finite length.