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Abstract Determinacy in the Low Levels of The ABSTRACT DETERMINACY IN THE LOW LEVELS OF THE PROJECTIVE HIERARCHY by Michael R. Cotton We give an expository introduction to determinacy for sets of reals. If we are to assume the Axiom of Choice, then not all sets of reals can be determined. So we instead establish determinacy for sets according to their levels in the usual hierarchies of complexity. At a certain point ZFC is no longer strong enough, and we must begin to assume large cardinals. The primary results of this paper are Martin's theorems that Borel sets are determined, homogeneously Suslin sets are determined, and if a measurable cardinal κ exists then the complements of analytic sets are κ-homogeneously Suslin. We provide the necessary preliminaries, prove Borel determinacy, introduce measurable cardinals and ultrapowers, and then prove analytic determinacy from the existence of a measurable cardinal. Finally, we give a very brief overview of some further results which provide higher levels of determinacy relative to larger cardinals. DETERMINACY IN THE LOW LEVELS OF THE PROJECTIVE HIERARCHY A Thesis Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Arts Department of Mathematics by Michael R. Cotton Miami University Oxford, Ohio 2012 Advisor: Dr. Paul Larson Reader: Dr. Dennis K. Burke Reader: Dr. Tetsuya Ishiu Contents Introduction 1 0.1 Some notation and conventions . 2 1 Reals, trees, and determinacy 3 1.1 The reals as !! .............................. 3 1.2 Determinacy . 5 1.3 Borel sets . 8 1.4 Projective sets . 10 1.5 Tree representations . 12 2 Borel determinacy 14 2.1 Games with a tree of legal positions . 14 2.2 Coverings of trees and Borel determinacy . 15 2.3 Proofs of the lemmas . 17 3 Measurable cardinals 24 3.1 Filters, ultrapowers, and elementary embeddings . 24 3.2 Normal measures . 30 1 4 Det(Π1) given a measurable cardinal 34 4.1 Towers of measures and homogeneously Suslin sets . 34 4.2 Determinacy of homogeneously Suslin sets . 35 4.3 Effect of a measurable cardinal . 37 5 Remarks and further results 42 Bibliography 44 ii Acknowledgments The author would like to thank the Miami University Department of Mathematics for financial support and quality education. Also, Paul Larson gave many hours of an already busy year to our independent meetings and should be thanked for an excellent introduction to the world of set theory. The author thanks the readers Dennis Burke and Tetsuya Ishiu for their interest and suggestions. Also to Patrick Dowling and Dennis Burke again for a great deal of guidance and support. Most importantly, the author is grateful to his parents William and Roxanne Cotton. It has been an eventful six years, and this would never have been possible without them. iii Introduction In descriptive set theory, we study the definable sets in spaces that are nice enough to behave like the real numbers, and one area of investigation is the structural con- sequences of determinacy. However, we cannot always be certain which sets are determined. So then the study of determinacy becomes two-sided and leads to many connections between descriptive set theory, large cardinals, combinatorial set theory, inner models, and even forcing. The first side is exactly how things began, meaning the exploration of the consequences of certain sets being determined (for example, 1 1 If Πn sets are determined, then all Σn+1 sets of reals are Lebegue measurable, have the Baire property, and have the perfect set property). The second side is to see how much determinacy is actually derivable. This paper is an expository introduction to the second. It is easily shown that, if we are to assume the Axiom of Choice, then not all sets of reals can be determined. So the natural direction then is to try to work our way up the sets in the usual hierarchies of complexity. At a certain point ZFC is no longer strong enough, and we must begin to assume large cardinals. In Section 1 we introduce the concepts necessary to understand the determinacy proofs presented, and in Section 2 we prove D.A. Martin's theorem that in ZFC all Borel sets are determined. But this is as far as ZFC can take us. So in Section 3 we introduce measurable cardinals, and since measurable cardinals are important for further studies in determinacy, we provide a more thorough introduction than is immediately necessary. Finally, Section 4 is devoted to Martin's theorem that analytic sets are determined if a measurable cardinal exists. The important results which we explore in detail are: · Borel sets are determined. · Homogeneously Suslin sets are determined. · If there is a measurable cardinal κ, then complements of analytic sets are κ-homogeneously Suslin. None of the theorems here are due to the author. The results in Sections 2 and 4 are Martin's, with the proofs in Section 4 being the author's solutions to a sequence of exercises given in Neeman [5]. Also, the work in Section 3 should be credited to Scott, Keisler, and Tarski. Finally, while some preliminary materials are provided, we assume a beginning to 1 intermediate level of set theory as well as basic topology, logic, and perhaps a bit of measure theory. However, no specific background in descriptive set theory or large cardinals is needed. When large cardinal hypotheses are mentioned that we do not define, this is merely to help provide some context, and it is not necessary to know what they are in order to continue reading. 0.1 Some notation and conventions Since we assume the Axiom of Choice throughout the paper (Although it is not nec- essary for many of the results), we use ordinal notation rather than alephs. So when @0 ! referring to cardinality, @0 = !, @1 = !1, 2 = 2 , etc. Given sets X and Y , X Y is the set of functions from X into Y . In the case of or- dinals, we use αX interchangeably to mean the set of functions f : α ! X or the set of ordered subsets of X of length α, and where both sets are ordinals the usual superscipt notation is reserved for arithmetic. For example, 2! is the cardinality of the reals while !2 is the set of infinite binary sequences. The usual interval notation for ordinals is used (i.e. if α < β then (α; β] = fζ : α < ζ ≤ βg,[α; β) = fζ : α ≤ ζ < βg, etc.) as well as interval notation on R. And to avoid confusion between intervals and ordered pairs we usually write ordered pairs and sequences with the angled brackets h i. If κ is a cardinal and X is a set, then [X]κ denotes the set of unordered subsets of X of size κ, and [X]<κ denotes the subsets of X of size less than κ. However, in the ordered case we continue to use the left superscripts so that <!X means the set finite sequences from X. Also, the usual jXj is used to mean the cardinality of a set X. So since a finite sequence <! s 2 X may be equivalently thought of as a function fh0; x0i; h1; x1i;:::; hn; xnig, its length (in this case n + 1) is also denoted jsj. While working with ultrapowers and embeddings in Section 3, we often quantify over proper classes (for example, \there exists an elementary embedding j : V ≺ M") which is not formally proper since it cannot be a theorem of ZFC ( VM is not a set to consider existence from). So such a thing should technically be considered a schema of theorems by isolating each embedding or restricting to initial segments Vα, Mα of the models. However, we make no effort to do so, and it should just be understood that formalization is possible whenever these issues might be considered. 2 1 Reals, trees, and determinacy In this section we give a brief introduction to the the space !! and its topology, determined sets, the Borel and projective hierarchies, and the use of tree structures to describe certain subsets of !!. 1.1 The reals as !! From a course in general topology, one might already be familiar with the fact that, when ! is given the discrete topology, then the Tychonoff product space !!, consist- ing of all sequences of natural numbers, is homeomorphic to the irrationals. So it is a good set-theoretic representation of the real numbers. We usually consider the base B for a topology on !! where ! <! B = fB ⊆ ! : 9s 2 ! 8x 2 B (x jsj = s)g In other words, we want a basic open set to be all extensions of some fixed ini- tial segment (or cones if we picture a tree structure), which produces precisely the same topology as the base obtained by fixing finitely many coordinates. To see this, notice that the base B is contained in the usual base. So it suffices for us to show that any basic open set in the Tychonoff topology can be generated by elements of B. <! ! For each s 2 !, we let Bs be the basic open set fx 2 ! :(x jsj) = sg. Then if F is some finite subset of ! and f : F ! ! is a function giving the coordinates at each point of F . Then the basic open set U = fx 2 !! : 8k 2 F (x(k) = f(k))g is equal to the set: [ fBs : jsj > supfk : k 2 F g ^ 8k 2 F (s(k) = f(k))g which is a union of sets from B, and since any basic open set of the Tychonoff topol- ogy can be expressed in this way, this shows that our base generated by fixing initial segments is sufficient for handling !!.
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