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Results on Martin's Conjecture by Patrick Lutz a Dissertation Results on Martin’s Conjecture by Patrick Lutz 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 Theodore Slaman, Chair Professor Antonio Montalbán Professor John Steel Spring 2021 Results on Martin’s Conjecture Copyright 2021 by Patrick Lutz 1 Abstract Results on Martin’s Conjecture by Patrick Lutz Doctor of Philosophy in Mathematics University of California, Berkeley Professor Theodore Slaman, Chair Martin’s conjecture is an attempt to classify the behavior of all definable functions on the Turing degrees under strong set theoretic hypotheses. Very roughly it says that every such function is either eventually constant, eventually equal to the identity function or eventually equal to a transfinite iterate of the Turing jump. It is typically divided into two parts: the first part states that every function is either eventually constant or eventually above the identity function and the second part states that every function which is above the identity is eventually equal to a transfinite iterate of the jump. If true, it would provide an explanation for the unique role of the Turing jump in computability theory and rule out many types of constructions on the Turing degrees. In this thesis we will introduce a few tools which we use to prove several cases of Martin’s conjecture. It turns out that both these tools and these results on Martin’s conjecture have some interesting consequences both for Martin’s conjecture and for a few related topics. The main tool that we introduce is a basis theorem for perfect sets, improving a theorem due to Groszek and Slaman [GS98]. We also introduce a general framework for proving certain special cases of Martin’s conjecture which unifies a few pre-existing proofs. We will use these tools to prove three main results about Martin’s conjecture: that it holds for regressive functions on the hyperarithmetic degrees (answering a question of Slaman and Steel), that part 1 holds for order preserving functions on the Turing degrees, and that part 1 holds for a class of functions that we introduce, called measure preserving functions. This last result has several interesting consequences for the study of Martin’s conjecture. In particular, it shows that part 1 of Martin’s conjecture is equivalent to a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees. This suggests several possible strategies for working on part 1 of Martin’s conjecture, which we will discuss. The basis theorem that we use to prove these results also has some applications outside of Martin’s conjecture. We will use it to prove a few theorems related to Sacks’ question about 2 whether it is provable in ZFC that every locally countable partial order of size continuum embeds into the Turing degrees. We will show that in a certain extension of ZF (which is incompatible with ZFC), this holds for all partial orders of height two, but not for all partial orders of height three. We will also present an obstacle to embedding height three partial orders into the Turing degrees in ZFC which shows that one of the most natural ways of trying to do so cannot work. We will end the thesis with a list of open questions related to Martin’s conjecture, which we hope will stimulate further research. i Contents Contents i 1 Introduction 1 1.1 Motivation: Natural Functions on the Turing Degrees ............. 2 1.2 The Martin Order and the Martin Measure ................... 4 1.3 The Axiom of Determinacy ........................... 8 1.4 Statement of Martin’s Conjecture ........................ 14 1.5 Special Cases of Martin’s Conjecture ...................... 16 1.6 Martin’s Conjecture on Other Degree Structures . 19 1.7 Summary of What’s Known ........................... 24 2 Tools 25 2.1 A Framework for Proving Instances of Part 1 of Martin’s Conjecture . 26 2.2 Ordinal Invariants of Turing Degrees ...................... 31 2.3 A Basis Theorem for Perfect Sets ........................ 33 2.4 The Solecki Dichotomy .............................. 39 1 2.5 Σ1-bounding and Pseudo-Wellorders ...................... 43 2.6 Minimal Upper Bounds .............................. 44 3 Uniformly Invariant Functions 47 3.1 Local Martin’s Conjecture ............................ 49 3.2 A Local Solution to Sacks’ Problem for Uniformly Invariant R.E. Operators 51 3.3 A Local Version of Part 2? ............................ 54 4 Regressive Functions 57 4.1 Regressive Functions on the Hyperarithmetic Degrees: AD Case . 59 4.2 Regressive Functions on the Hyperarithmetic Degrees: Borel Case . 65 5 Measure Preserving Functions 68 5.1 Equivalent Definitions of Measure Preserving . 70 5.2 Warm-up: Regressive Measure Preserving Functions . 74 5.3 Part 1 of Martin’s Conjecture for Measure Preserving Functions . 75 ii 5.4 Part 1 of Martin’s Conjecture for Measure Preserving Functions: AD Proof . 78 5.5 Other Degree Structures ............................. 81 5.6 Non-invariant Functions ............................. 83 5.7 ZFC Counterexample ............................... 84 5.8 Application to Part 2 of Martin’s Conjecture . 86 5.9 Martin’s Conjecture and the Rudin-Keisler Order . 89 5.10 The Lebesgue and Baire Ultrafilters ....................... 92 5.11 Generic Ultrapowers and Forcing with Positive Sets . 96 6 Order Preserving Functions 100 6.1 Order Preserving Functions are Measure Preserving . 101 6.2 Three Proofs of Part 1 of Martin’s Conjecture for Order Preserving Functions 102 6.3 Ultrapowers by the Martin Measure . 106 6.4 ZFC Counterexample ............................... 108 7 Application 1: Sacks’ Question on Embedding Partial Orders Into the Turing Degrees 110 7.1 Embedding Height Two Partial Orders: ZFC Case . 111 7.2 Embedding Height Two Partial Orders: ADR Case . 115 7.3 Nonembedding for Height Three Partial Orders: ADR Case . 116 7.4 Obstacles to Embedding Partial Orders in the Turing Degrees . 118 8 Application 2: The Theory of Locally Countable Borel Quasi Orders 120 8.1 Turing Reducibility is not a Universal Locally Countable Borel Quasi Order 122 8.2 Borel Quasi Orders of Finite Height . 125 9 Open Questions 130 Bibliography 135 iii Acknowledgments I have always been a fan of expansive acknowledgments sections. I like the brief glimpse it gives you into the author’s personal life and the sense it gives of the network of human relationships in which research typically takes place. I also like the idea that some of the people thanked in the acknowledgments may stumble onto the work months or years later and feel happy to see their name. Still, a long acknowledgments section carries a risk. Paradoxically, the more people you thank, the greater the risk that you have left someone out. When you thank only your advisor and your parents, there is no chance that you have forgotten anyone. Not so when you thank dozens of people. So though I will thank many people in this section, I am sure I will forget many others. If you come across this thesis months or years from now and feel that your name deserves to be included in these acknowledgments, you are probably right and I’m sorry for leaving you out. I believe that the two most important factors in the graduate school experience are one’s advisor and one’s fellow graduate students. I was lucky on both accounts. My advisor, Ted Slaman, has always been incredibly kind to me. He has also been incredibly patient with me (maybe too patient) as I took time in grad school to pursue all my intellectual interests, both mathematical and non-mathematical. He has also given me many valuable insights into computability theory and mathematical logic. I have learned a tremendous amount, both about mathematics and about life, from my fellow grad students. I am grateful for Benny Siskind for taking me to a coffee shop on my first day of grad school and telling me about the existence of research seminars and for being a good friend and research collaborator throughout grad school. Izaak Meckler and German Stefanich have both greatly influenced my perspective on mathematics and I have been inspired by their enthusiasm and fearlessness when pursuing mathematical ideas. Chris and Wendy have been dear friends and great office-mates. James Walsh was a good friend and a great research collaborator. Rahul and Ravi have always been sources of thought-provoking conversations. I enjoyed working on stuff in Lean with Thomas and Jordan and I am sorry I had to take a break from it this semester (in part to write this thesis). I am proud to be in the same cohort as Milind, Ritvik, Mario, Yanhe, Mariel, Joe, Leon, Nic, Mariana and too many others to mention. I am also grateful to all the other logic students at Berkeley: James, Reid, Kentaro, Benny, Yifeng, Ben, Mariana, Adele, David Casey, Jacopo, Matthew, Diego, Guillaume, Scott, David Gonzalez, Ahmee, Jordan, Ronan and Shoshana. They have been good friends, teachers and sources of interesting ideas and discussions. I would also like to thank all the grad students from outside Berkeley who befriended me and taught me many things about math and logic, especially Jiachen, Wei Quan, Chieu-Minh and Jun Le. This thesis wouldn’t exist without my research collaborators—James, Benny, Vittorio and Kojiro. A few other people have also played a key role in shaping the research in this thesis. Ted, of course, but also especially Gabe Goldberg, Takayuki Kihara, Andrew Marks iv and Steffen Lempp. I would like to thank Gabe and Benny especially for putting up with dozens of hours of phone calls from me throughout the pandemic. I have been lucky to have had many good teachers and mentors throughout my life.
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