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Investigation of Determinacy for Games of Variable Length UNLV Theses, Dissertations, Professional Papers, and Capstones May 2017 Investigation of Determinacy for Games of Variable Length Emi Ikeda University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/thesesdissertations Part of the Mathematics Commons Repository Citation Ikeda, Emi, "Investigation of Determinacy for Games of Variable Length" (2017). UNLV Theses, Dissertations, Professional Papers, and Capstones. 2987. http://dx.doi.org/10.34917/10985942 This Dissertation is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/or on the work itself. This Dissertation has been accepted for inclusion in UNLV Theses, Dissertations, Professional Papers, and Capstones by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. INVESTIGATION OF DETERMINACY FOR GAMES OF VARIABLE LENGTH Emi Ikeda Bachelor of Science - Mathematical Sciences Yamaguchi University, Japan 2004 Master of Science - Mathematical Sciences Yamaguchi University, Japan 2006 A dissertation submitted in partial fulfillment of the requirements for the Doctor of Philosophy - Mathematical Sciences Department of Mathematical Sciences College of Sciences The Graduate College University of Nevada, Las Vegas May 2017 Copyright 2017 by Emi Ikeda All Rights Reserved Dissertation Approval The Graduate College The University of Nevada, Las Vegas March 31, 2017 This dissertation prepared by Emi Ikeda entitled Investigation of Determinacy for Games of Variable Length is approved in partial fulfillment of the requirements for the degree of Doctor of Philosophy - Mathematical Sciences Department of Mathematical Sciences Derrick DuBose, Ph.D. Kathryn Hausbeck Korgan, Ph.D. Examination Committee Chair Graduate College Interim Dean Douglas Burke, Ph.D. Examination Committee Member Peter Shiue, Ph.D. Examination Committee Member Pushkin Kachroo, Ph.D. Graduate College Faculty Representative ii Abstract Many well-known determinacy results calibrate determinacy strength in terms of large cardi- nals (e.g., a measurable cardinal) or a "large cardinal type" property (e.g., zero sharp exists). Some of the other results are of the form that subsets of reals of a certain complexity will satisfy a well-known property when a certain amount of determinacy holds. The standard game tree considered in the study of determinacy involves games in which all moves are from omega and all plays have length omega (i.e. the game tree is !<! and the body of the game tree is !!). There are also many well-known results on the game trees !<α for α countable (all moves from omega and all paths are of fixed length α). However, one can easily construct a nondetermined open game on a game tree T , in which all moves are from !, but some paths of T have length omega while the others of length ! + 1. Many determinacy results consider games on a fixed game tree with each path having the same length. In this dissertation, we investigate the determinacy of games on game trees with variable length paths. Especially, we investigate two types of such game trees, which we named Type 1 and Type 2. The length of each path in a Type 1 tree is determined by its first ! moves. A Type 2 tree is generalization of a Type 1 tree. In other words, a Type 1 tree is a special case of a Type 2 tree. We shall consider collections C of such game trees, iii that will be defined from particular parameters ranging over certain sets. A T ree1 collection will be a collection of Type 1 trees. A T ree2 collection will be a collection of Type 2 trees. Given a T ree1 (respectively, T ree2) collection C and a fixed complexity (e.g., open, Borel, 1 Σ1), we calibrate the strength of the determinacy of games with that complexity on all trees in the collection C in terms of well-known determinacy. iv Acknowledgments I would like to take this time to thank the many people that have helped me achieving my goal of acquiring my Ph.D. First, I would like to thank my advisor Dr. Derrick DuBose for his help and guidance throughout my time here at UNLV and my dissertation process. He has taken many of days and long nights to help me gain a better understanding of Set Theory. I would also like to especially thank Dr. Douglas Burke for taking time out of his schedule to help me through my whole time at UNLV. He has been like a co-advisor to me. He has taken time out of his day to meet with me every week to help me study and present me with a lot of ideas to help me in understanding Set Theory. He has helped me solve many of the problems I had concerning my studies. He guided me to the kind of game trees to look for in my dissertation. He has been a great influence for me. I would remise if I did not thank him for his guidance and patience. Without these two I do not know if I could had made it to the position I am in today. I would like to thank my other committee members Dr. Peter Shiue and Graduate College Representative Dr. Pushkin Kachroo. They have generously taken time out of their schedules to listen to my comprehensive exam and dissertation defense. I would also like to thank the Mathematical Sciences Department at UNLV for affording me the opportunity to study and achieve my dream of obtaining my v Ph.D. My original area of focus was topology while I was working on my Master's. I was planning on continuing to study Topology when I came to UNLV. Sadly, I was not able to find a professor to work with me in topology. When I was taking MAT 701 and MAT 702, Dr. DuBose suggested to me to study Set Theory. I learned Set Theory from scratch. He gave me the opportunity to pursue a new field of study and he invited me to the Set Theory Seminars. For that I am grateful for the opportunity to learn this new and exciting field of which I have been studying for the last few years. I would also like to thank a Ph.D. student Josh Reagan and a Master's student Katherine Yost for studying with me in the Set Theory Student Workshop here on campus. Finally, I would like to give a special acknowledgment to my family and friends for their loving support throughout the years of my studies as a Ph.D. student. vi Table of Contents Abstract .......................................... iii Acknowledgments .................................... v Table of Contents .................................... xii List of Figures ...................................... xiii Chapter 1 Preliminaries and Introduction ..................... 1 1.1 General notations for a product space . 4 1.2 Definition of a game . 7 1.2.1 Definitions related to a game tree . 7 1.2.2 Definitions related to a game on a tree . 9 1.3 Definition of complexities . 13 1.3.1 Definitions related to topologies . 13 1.3.2 Open sets . 15 1.3.3 Borel hierarchy . 16 1.3.4 Projective hierarchy . 20 1.3.5 Difference hierarchy . 23 vii 1.4 Well-known determinacy results . 27 1.4.1 Determinacy results from ZFC . 28 1.4.2 Determinacy results from large cardinals . 29 1.5 Introduction to this dissertation . 38 1.5.1 Motivation to study long trees . 38 1.5.2 Difference from usual determinacy results . 40 1.5.3 Notations for this dissertation . 41 Ψ;B Chapter 2 Type 1 Tree : TX;Y ............................. 45 2.1 Definition of a Type 1 tree . 47 2.2 Definition of a T ree1 collection and a collection of games on a T ree1 collection with complexity Ξ . 50 0 0 2.3 Equivalence between Σα and Πα determinacy on a T ree1 collection and equiv- 1 1 alence between Σn and Πn determinacy on a T ree1 collection . 55 2.4 Using the determinacy of games on a T ree1 collection to obtain the determi- nacy of games on X<! .............................. 82 0 2.4.1 (ZF-P) Using ∆1 determinacy on a T ree1 collection to obtain finite Borel determinacy on X<! ........................ 83 0 2.4.2 Using Σ1 determinacy on a T ree1 collection to obtain the determinacy of games on X<! ............................. 106 1 1 2.4.3 Using α-Π1 determinacy on T ree1 collection to obtain α + 1-Π1 de- ! terminacy on X for even α 2 !1 . 123 viii 2.5 Getting the determinacy of the games on a T ree1 collection from the deter- minacy of the games on X<! (Reversed direction of section 2.4) . 135 2.5.1 Getting the determinacy of games on a T ree1 collection with countable Y from the determinacy of games on X<! . 137 2.5.2 Obtaining the determinacy of open games on a T ree1 collection with countable Y from the determinacy of games on X<! . 156 2.5.3 Obtaining the determinacy of Borel games on a T ree1 collection with countable Y from the determinacy of Borel games on X<! . 167 2.5.4 Obtaining the determinacy of projective games on a T ree1 collection with countable Y from the determinacy of projective games on X<! . 183 2.5.5 Well-known results about uncountable Y = N . 194 <! 2.6 Determinacy equivalences between games on X and games on T ree1 collec- tions .
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