DOCSLIB.ORG

Modern Descriptive Set Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modern Descriptive Set Theory Modern descriptive set theory Jindˇrich Zapletal Czech Academy of Sciences University of Florida ii Contents 1 Introduction 1 2 Polish spaces 3 2.1 Basic definitions . 3 2.2 Production theorems . 5 2.3 Polish groups . 7 2.4 Universal objects . 9 2.5 Natural classes of mathematical objects form Polish spaces . 12 2.5.1 Countable groups . 12 2.5.2 Separable Banach spaces . 12 3 Borel sets, analytic sets 13 3.1 Borel hierarchy . 13 3.2 Projective hierarchy . 14 3.3 Wadge hierarchy . 16 3.4 Borel and analytic sets . 16 3.4.1 Separation theorems . 16 3.4.2 Uniformization . 17 3.4.3 Coding of Borel sets . 18 3.5 Examples . 18 3.5.1 Borel sets . 18 3.5.2 Analytic sets . 19 3.6 Effective theory . 20 4 Borel equivalence relations 21 4.1 Examples . 22 4.1.1 Ideal equivalences . 22 4.1.2 Isomorphisms of structures . 22 4.1.3 Group actions and orbit equivalences . 23 4.2 Constructing the map . 24 4.3 The map description . 25 4.3.1 id . 25 4.3.2 E0 ............................... 25 4.3.3 E1 ............................... 27 iii iv CONTENTS 4.3.4 Kσ ............................... 27 4.3.5 C . 28 4.3.6 ES∞ .............................. 28 4.3.7 EΣ ............................... 29 4.3.8 Gmax ............................. 30 4.4 Some proofs . 30 5 Determinacy 35 5.1 Games: basic definitions . 35 5.2 Basic determinacy results . 36 5.3 Applications to abstract analysis . 38 5.3.1 Perfect set property . 38 5.3.2 Baire category . 40 5.3.3 Lebesgue measure and capacities . 40 5.3.4 Superperfect set theorem . 41 5.3.5 Continuous reducibility . 42 5.3.6 Hausdorff measures . 43 5.4 Full determinacy . 43 5.4.1 Models of determinacy . 43 5.4.2 Well-ordered cardinals . 44 5.4.3 Non-well-ordered cardinals . 45 5.4.4 Periodicity theorems . 45 5.4.5 Inner models . 45 Chapter 1 Introduction I wrote these notes as the text for a topics course in set theory at University of Florida in Spring 2005. The intended audience is a mix of graduate students specializing in mathematical logic, topology and abstract analysis. The course seeks to expose them to the basic ideas behind infinitary games, determinacy and their uses in these parts of mathematics. I expected the students would have previous exposure to very basic topology and set theory; they should understand notions such as ”topological space” and ”cardinality”. Other than that, there are no prerequisites. A necessary part of a rigorous development of the theory of infinitary games is the study of hierarchies of complexity for subsets of Polish spaces. The first chapter is a very rudimentary introduction to Polish spaces, expected to take no longer than three weeks. The second chapter introduces the Borel, projective and Wadge hierarchies, again expected to take no longer than three weeks. The third chapter defines infinitary games and states the key determinacy theorems. I find that there is no time to prove Borel determinacy within the confines of a semester-long course, and I suspect the proof would go well beyond the attention span of my intended audience; therefore I treat it as a black box. A student interested in set theory should certainly go through the proof. The fourth chapter then introduces a number of infinitary games relevant to set theory, topology and analysis. Since this is the main topic of the course, I hope to reserve at least six weeks for this chapter. The notation follows the set theoretic standard. Ordinal numbers, including natural numbers, are treated as von Neumann ordinals, and so α ∈ β indi- cates that α is an ordinal smaller than β. The letter c denotes the size of the continuum. 1 2 CHAPTER 1. INTRODUCTION Chapter 2 Polish spaces 2.1 Basic definitions In this section, I will introduce the most basic concepts used in this textbook. The reader is assumed to be familiar with most of them. Definition 2.1.1. A topological space is a pair X, O, where X is a set and O is the topology, a subset of P(X) including 0,X, and closed under finite intersections and arbitrary unions. The topology will be often clear from the context and not mentioned at all. The sets in the topology are called open, their complements are closed, sets which are both open and closed are called clopen. Topologies are often generated by a collection Ogen of sets that we want to declare to be open. Just let O to be the closure of the set Ogen on finite intersections and arbitrary unions. The category of topological spaces comes equipped with continuous functions and homeomorphisms. A map f : X → Y is continuous if preimages of open sets are open. It is a homeomorphism if it is one to one, onto, and both it and its inverse are continuous. The continuity concept brings another common way of generating a topology O on a set X uses a collection F of maps into a topological space Y : O will be the smallest topology that makes all the functions in F continuous. In other word, O is generated by the preimages of open sets under functions in the collection F . Topological spaces come in many different flavors. In this book, we will be interested in Polish spaces, a very familiar kind of topological spaces that find uses in most parts of mathematics. Definition 2.1.2. A topological space X is separable if it has a countable dense set, i.e. a set intersecting every nonempty open set. Definition 2.1.3. A metric on a set X is a map d from X2 to nonnegative reals such that d(x, y) = d(y, x), d(x, y) = 0 ↔ x = y and d(x, y) ≤ d(x, z) + d(z, y). 3 4 CHAPTER 2. POLISH SPACES The metric is complete if every Cauchy sequence has a limit. The associated metric topology is generated by open balls, sets of the form B(x, ε) = {y ∈ X : d(x, y) < ε}, for x ∈ X and ε > 0. Definition 2.1.4. A topological space X is Polish if it is separable and com- pletely metrizable, i.e. there is a complete metric on X that generates the topology on X. The nature of the metric generating a given Polish topology is mostly irrele- vant in our considerations. There can be many very different metrics generating the same topology. One context in which the existence of a suitable metric be- comes relevant is the Polish groups. Example 2.1.5. A countable set with the discrete topology. Example 2.1.6. The Cantor space 2ω with the minimum difference metric. Just let d(x, y) = 2−n where n = ∆(x, y) is the least number such that x(n) 6= ω y(n). The topology is generated by sets of the form Ot = {x ∈ 2 : t ⊂ x}, where t ∈ 2ω ranges over all finite binary sequences. Note that if O ⊂ 2ω is open and x ∈ O, then there must be a number n ∈ ω such that O ⊂ O. xn The Cantor space possesses two important characteristics: it is zero-dimensional (this means that the topology is generated by clopen sets) and compact (every cover by open sets has a finite subcover). To prove compactness, assume for contradiction that C is an open cover of 2ω with no finite subcover. By induc- tion on n find bits bn ∈ 2 such that the sets Otn cannot be covered by finitely ω many elements of C, where tn = hb0, b1, . bn−1i. In the end, let x ∈ 2 be the sequence given by x(n) = bn and choose a set O ∈ C such that x ∈ O. Since the set O is open, there must be a number n ∈ ω such that O ⊂ O. This xn contradicts the induction hypothesis at n though. Example 2.1.7. The Baire space ωω with the minimum difference metric. The Baire space is again zero-dimensional, but it is not compact. For example, the cover C consisting of sets Ohni : n ∈ ω does not have a finite subcover. In fact, The Baire space cannot be covered by countably many compact subsets. Example 2.1.8. The real line with the Euclidean metric. The real line is not zero-dimensional; the only clopen subsets are 0 and R. It can be decomposed into two zero-dimensional subsets, such as the rationals and irrationals. An example of a Polish space that cannot be decomposed into countably many zero-dimensional subsets is the Hilbert cube, see below. The real line is not compact. However, it is locally compact: every point has a neigborhood whose closure is compact. Example 2.1.9. Every separable Banach space with the norm metric. 2.2. PRODUCTION THEOREMS 5 2.2 Production theorems Later on, I will have to verify that various sets form Polish spaces with a topology that is naturally derived from the context. This may not be quite easy. In this section, I will list a number of theorems that produce more complicated Polish spaces from simpler ones. Theorem 2.2.1. If X is a Polish space and Y ⊂ X is a Gδ set, then Y with the inherited topology forms a Polish space. Here, a Gδ set is one that is equal to the intersection of countably many open sets. Note that in Polish spaces, this includes all closed sets. If F ⊂ X T is closed then F = q Oq where q ranges over positive rationals and Oq is the open set of all points with distance < q from F .
Load more

Recommended publications
  • Calibrating Determinacy Strength in Levels of the Borel Hierarchy
    CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY SHERWOOD J. HACHTMAN Abstract. We analyze the set-theoretic strength of determinacy for levels of the Borel 0 hierarchy of the form Σ1+α+3, for α < !1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α+1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of Π1-reflection principles, Π1-RAPα, whose consistency strength corresponds 0 CK exactly to that of Σ1+α+3-Determinacy, for α < !1 . This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: the least θ so that all winning strategies 0 in Σ4 games belong to Lθ+1 is the least so that Lθ j= \P(!) exists + all wellfounded trees are ranked". x1. Introduction. Given a set A ⊆ !! of sequences of natural numbers, consider a game, G(A), where two players, I and II, take turns picking elements of a sequence hx0; x1; x2;::: i of naturals. Player I wins the game if the sequence obtained belongs to A; otherwise, II wins. For a collection Γ of subsets of !!, Γ determinacy, which we abbreviate Γ-DET, is the statement that for every A 2 Γ, one of the players has a winning strategy in G(A). It is a much-studied phenomenon that Γ -DET has mathematical strength: the bigger the pointclass Γ, the stronger the theory required to prove Γ -DET.
    [Show full text]
  • Descriptive Set Theory of Complete Quasi-Metric Spaces
    数理解析研究所講究録 第 1790 巻 2012 年 16-30 16 Descriptive set theory of complete quasi-metric spaces Matthew de Brecht National Institute of Information and Communications Technology Kyoto, Japan Abstract We give a summary of results from our investigation into extending classical descriptive set theory to the entire class of countably based $T_{0}$ -spaces. Polish spaces play a central role in the descriptive set theory of metrizable spaces, and we suggest that countably based completely quasi-metrizable spaces, which we refer to as quasi-Polish spaces, play the central role in the extended theory. The class of quasi-Polish spaces is general enough to include both Polish spaces and $\omega$ -continuous domains, which have many applications in theoretical computer science. We show that quasi-Polish spaces are a very natural generalization of Polish spaces in terms of their topological characterizations and their complete- ness properties. In particular, a metrizable space is quasi-Polish if and only if it is Polish, and many classical theorems concerning Polish spaces, such as the Hausdorff-Kuratowski theorem, generalize to all quasi-Polish spaces. 1. Introduction Descriptive set theory has proven to be an invaluable tool for the study of separable metrizable spaces, and the techniques and results have been applied to many fields such as functional analysis, topological group theory, and math- ematical logic. Separable completely metrizable spaces, called Polish spaces, play a central role in classical descriptive set theory. These spaces include the space of natural numbers with the discrete topology, the real numbers with the Euclidean topology, as well as the separable Hilbert and Banach spaces.
    [Show full text]
  • Games in Descriptive Set Theory, Or: It's All Fun and Games Until Someone Loses the Axiom of Choice Hugo Nobrega
    Games in Descriptive Set Theory, or: it’s all fun and games until someone loses the axiom of choice Hugo Nobrega Cool Logic 22 May 2015 Descriptive set theory and the Baire space Presentation outline [0] 1 Descriptive set theory and the Baire space Why DST, why NN? The topology of NN and its many flavors 2 Gale-Stewart games and the Axiom of Determinacy 3 Games for classes of functions The classical games The tree game Games for finite Baire classes Descriptive set theory and the Baire space Why DST, why NN? Descriptive set theory The real line R can have some pathologies (in ZFC): for example, not every set of reals is Lebesgue measurable, there may be sets of reals of cardinality strictly between |N| and |R|, etc. Descriptive set theory, the theory of definable sets of real numbers, was developed in part to try to fill in the template “No definable set of reals of complexity c can have pathology P” Descriptive set theory and the Baire space Why DST, why NN? Baire space NN For a lot of questions which interest set theorists, working with R is unnecessarily clumsy. It is often better to work with other (Cauchy-)complete topological spaces of cardinality |R| which have bases of cardinality |N| (a.k.a. Polish spaces), and this is enough (in a technically precise way). The Baire space NN is especially nice, as I hope to show you, and set theorists often (usually?) mean this when they say “real numbers”. Descriptive set theory and the Baire space The topology of NN and its many flavors The topology of NN We consider NN with the product topology of discrete N.
    [Show full text]
  • Topology and Descriptive Set Theory
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector TOPOLOGY AND ITS APPLICATIONS ELSEVIER Topology and its Applications 58 (1994) 195-222 Topology and descriptive set theory Alexander S. Kechris ’ Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA Received 28 March 1994 Abstract This paper consists essentially of the text of a series of four lectures given by the author in the Summer Conference on General Topology and Applications, Amsterdam, August 1994. Instead of attempting to give a general survey of the interrelationships between the two subjects mentioned in the title, which would be an enormous and hopeless task, we chose to illustrate them in a specific context, that of the study of Bore1 actions of Polish groups and Bore1 equivalence relations. This is a rapidly growing area of research of much current interest, which has interesting connections not only with topology and set theory (which are emphasized here), but also to ergodic theory, group representations, operator algebras and logic (particularly model theory and recursion theory). There are four parts, corresponding roughly to each one of the lectures. The first contains a brief review of some fundamental facts from descriptive set theory. In the second we discuss Polish groups, and in the third the basic theory of their Bore1 actions. The last part concentrates on Bore1 equivalence relations. The exposition is essentially self-contained, but proofs, when included at all, are often given in the barest outline. Keywords: Polish spaces; Bore1 sets; Analytic sets; Polish groups; Bore1 actions; Bore1 equivalence relations 1.
    [Show full text]
  • Polish Spaces and Baire Spaces
    Polish spaces and Baire spaces Jordan Bell [email protected] Department of Mathematics, University of Toronto June 27, 2014 1 Introduction These notes consist of me working through those parts of the first chapter of Alexander S. Kechris, Classical Descriptive Set Theory, that I think are impor- tant in analysis. Denote by N the set of positive integers. I do not talk about universal spaces like the Cantor space 2N, the Baire space NN, and the Hilbert cube [0; 1]N, or \localization", or about Polish groups. If (X; τ) is a topological space, the Borel σ-algebra of X, denoted by BX , is the smallest σ-algebra of subsets of X that contains τ. BX contains τ, and is closed under complements and countable unions, and rather than talking merely about Borel sets (elements of the Borel σ-algebra), we can be more specific by talking about open sets, closed sets, and sets that are obtained by taking countable unions and complements. Definition 1. An Fσ set is a countable union of closed sets. A Gδ set is a complement of an Fσ set. Equivalently, it is a countable intersection of open sets. If (X; d) is a metric space, the topology induced by the metric d is the topology generated by the collection of open balls. If (X; τ) is a topological space, a metric d on the set X is said to be compatible with τ if τ is the topology induced by d.A metrizable space is a topological space whose topology is induced by some metric, and a completely metrizable space is a topological space whose topology is induced by some complete metric.
    [Show full text]
  • The Wadge Hierarchy : Beyond Borel Sets
    Unicentre CH-1015 Lausanne http://serval.unil.ch Year : 2016 The Wadge Hierarchy : Beyond Borel Sets FOURNIER Kevin FOURNIER Kevin, 2016, The Wadge Hierarchy : Beyond Borel Sets Originally published at : Thesis, University of Lausanne Posted at the University of Lausanne Open Archive http://serval.unil.ch Document URN : urn:nbn:ch:serval-BIB_B205A3947C037 Droits d’auteur L'Université de Lausanne attire expressément l'attention des utilisateurs sur le fait que tous les documents publiés dans l'Archive SERVAL sont protégés par le droit d'auteur, conformément à la loi fédérale sur le droit d'auteur et les droits voisins (LDA). A ce titre, il est indispensable d'obtenir le consentement préalable de l'auteur et/ou de l’éditeur avant toute utilisation d'une oeuvre ou d'une partie d'une oeuvre ne relevant pas d'une utilisation à des fins personnelles au sens de la LDA (art. 19, al. 1 lettre a). A défaut, tout contrevenant s'expose aux sanctions prévues par cette loi. Nous déclinons toute responsabilité en la matière. Copyright The University of Lausanne expressly draws the attention of users to the fact that all documents published in the SERVAL Archive are protected by copyright in accordance with federal law on copyright and similar rights (LDA). Accordingly it is indispensable to obtain prior consent from the author and/or publisher before any use of a work or part of a work for purposes other than personal use within the meaning of LDA (art. 19, para. 1 letter a). Failure to do so will expose offenders to the sanctions laid down by this law.
    [Show full text]
  • Wadge Hierarchy and Veblen Hierarchy Part I: Borel Sets of Finite Rank Author(S): J
    Wadge Hierarchy and Veblen Hierarchy Part I: Borel Sets of Finite Rank Author(s): J. Duparc Source: The Journal of Symbolic Logic, Vol. 66, No. 1 (Mar., 2001), pp. 56-86 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2694911 Accessed: 04/01/2010 12:38 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=asl. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. http://www.jstor.org THE JOURNAL OF SYMBOLICLOGIC Voluime 66.
    [Show full text]
  • Descriptive Set Theory
    Descriptive Set Theory David Marker Fall 2002 Contents I Classical Descriptive Set Theory 2 1 Polish Spaces 2 2 Borel Sets 14 3 E®ective Descriptive Set Theory: The Arithmetic Hierarchy 27 4 Analytic Sets 34 5 Coanalytic Sets 43 6 Determinacy 54 7 Hyperarithmetic Sets 62 II Borel Equivalence Relations 73 1 8 ¦1-Equivalence Relations 73 9 Tame Borel Equivalence Relations 82 10 Countable Borel Equivalence Relations 87 11 Hyper¯nite Equivalence Relations 92 1 These are informal notes for a course in Descriptive Set Theory given at the University of Illinois at Chicago in Fall 2002. While I hope to give a fairly broad survey of the subject we will be concentrating on problems about group actions, particularly those motivated by Vaught's conjecture. Kechris' Classical Descriptive Set Theory is the main reference for these notes. Notation: If A is a set, A<! is the set of all ¯nite sequences from A. Suppose <! σ = (a0; : : : ; am) 2 A and b 2 A. Then σ b is the sequence (a0; : : : ; am; b). We let ; denote the empty sequence. If σ 2 A<!, then jσj is the length of σ. If f : N ! A, then fjn is the sequence (f(0); : : :b; f(n ¡ 1)). If X is any set, P(X), the power set of X is the set of all subsets X. If X is a metric space, x 2 X and ² > 0, then B²(x) = fy 2 X : d(x; y) < ²g is the open ball of radius ² around x. Part I Classical Descriptive Set Theory 1 Polish Spaces De¯nition 1.1 Let X be a topological space.
    [Show full text]
  • Wadge Hierarchy of Omega Context Free Languages Olivier Finkel
    Wadge Hierarchy of Omega Context Free Languages Olivier Finkel To cite this version: Olivier Finkel. Wadge Hierarchy of Omega Context Free Languages. Theoretical Computer Science, Elsevier, 2001, 269 (1-2), pp.283-315. hal-00102489 HAL Id: hal-00102489 https://hal.archives-ouvertes.fr/hal-00102489 Submitted on 1 Oct 2006 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. WADGE HIERARCHY OF OMEGA CONTEXT FREE LANGUAGES Olivier Finkel Equipe de Logique Math´ematique CNRS URA 753 et Universit´eParis 7 U.F.R. de Math´ematiques 2 Place Jussieu 75251 Paris cedex 05, France. E Mail: fi[email protected] Abstract The main result of this paper is that the length of the Wadge hierarchy of omega context free languages is greater than the Cantor ordinal ε0, and the same result holds for the conciliating Wadge hierarchy, defined in [Dup99], of infinitary context free languages, studied in [Bea84a]. In the course of our proof, we get results on the Wadge hierarchy of iterated counter ω-languages, which we define as an extension of classical (finitary) iterated counter languages to ω-languages. Keywords: omega context free languages; topological properties; Wadge hier- archy; conciliating Wadge hierarchy; infinitary context free languages; iterated counter ω-languages.
    [Show full text]
  • The Envelope of a Pointclass Under a Local Determinacy Hypothesis
    THE ENVELOPE OF A POINTCLASS UNDER A LOCAL DETERMINACY HYPOTHESIS TREVOR M. WILSON Abstract. Given an inductive-like pointclass Γ and assuming the Ax- iom of Determinacy, Martin identified and analyzede the pointclass con- taining the norm relations of the next semiscale beyond Γ, if one exists. We show that much of Martin's analysis can be carriede out assuming only ZF + DCR + Det(∆Γ). This generalization requires arguments from Kechris{Woodin [10] ande e Martin [13]. The results of [10] and [13] can then be recovered as immediate corollaries of the general analysis. We also obtain a new proof of a theorem of Woodin on divergent models of AD+, as well as a new result regarding the derived model at an inde- structibly weakly compact limit of Woodin cardinals. Introduction Given an inductive-like pointclass Γ, Martin introduced a pointclass (which we call the envelope of Γ, following [22])e whose main feature is that it con- tains the prewellorderingse of the next scale (or semiscale) beyond Γ, if such a (semi)scale exists. This work was distributed as \Notes on the nexte Suslin cardinal," as cited in Jackson [3]. It is unpublished, so we use [3] as our reference instead for several of Martin's arguments. Martin's analysis used the assumption of the Axiom of Determinacy. We reformulate the notion of the envelope in such a way that many of its essen- tial properties can by derived without assuming AD. Instead we will work in the base theory ZF+DCR for the duration of the paper, stating determinacy hypotheses only when necessary.
    [Show full text]
  • Descriptive Set Theory and Ω-Powers of Finitary Languages Olivier Finkel, Dominique Lecomte
    Descriptive Set Theory and !-Powers of Finitary Languages Olivier Finkel, Dominique Lecomte To cite this version: Olivier Finkel, Dominique Lecomte. Descriptive Set Theory and !-Powers of Finitary Languages. Adrian Rezus. Contemporary Logic and Computing, 1, College Publications, pp.518-541, 2020, Land- scapes in Logic. hal-02898919 HAL Id: hal-02898919 https://hal.archives-ouvertes.fr/hal-02898919 Submitted on 14 Jul 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Descriptive Set Theory and ω-Powers of Finitary Languages Olivier FINKEL and Dominique LECOMTE1 March 18, 2020 • CNRS, Universit´ede Paris, Sorbonne Universit´e, Institut de Math´ematiques de Jussieu-Paris Rive Gauche, Equipe de Logique Math´ematique Campus des Grands Moulins, bˆatiment Sophie-Germain, case 7012, 75205 Paris cedex 13, France fi[email protected] •1 Sorbonne Universit´e, Universit´ede Paris, CNRS, Institut de Math´ematiques de Jussieu-Paris Rive Gauche, Equipe d’Analyse Fonctionnelle Campus Pierre et Marie Curie, case 247, 4, place Jussieu, 75 252 Paris cedex 5, France [email protected] •1 Universit´ede Picardie, I.U.T. de l’Oise, site de Creil, 13, all´ee de la fa¨ıencerie, 60 107 Creil, France Abstract.
    [Show full text]
  • Regularity Properties and Determinacy
    Regularity Properties and Determinacy MSc Thesis (Afstudeerscriptie) written by Yurii Khomskii (born September 5, 1980 in Moscow, Russia) under the supervision of Dr. Benedikt L¨owe, and submitted to the Board of Examiners in partial fulfillment of the requirements for the degree of MSc in Logic at the Universiteit van Amsterdam. Date of the public defense: Members of the Thesis Committee: August 14, 2007 Dr. Benedikt L¨owe Prof. Dr. Jouko V¨a¨an¨anen Prof. Dr. Joel David Hamkins Prof. Dr. Peter van Emde Boas Brian Semmes i Contents 0. Introduction............................ 1 1. Preliminaries ........................... 4 1.1 Notation. ........................... 4 1.2 The Real Numbers. ...................... 5 1.3 Trees. ............................. 6 1.4 The Forcing Notions. ..................... 7 2. ClasswiseConsequencesofDeterminacy . 11 2.1 Regularity Properties. .................... 11 2.2 Infinite Games. ........................ 14 2.3 Classwise Implications. .................... 16 3. The Marczewski-Burstin Algebra and the Baire Property . 20 3.1 MB and BP. ......................... 20 3.2 Fusion Sequences. ...................... 23 3.3 Counter-examples. ...................... 26 4. DeterminacyandtheBaireProperty.. 29 4.1 Generalized MB-algebras. .................. 29 4.2 Determinacy and BP(P). ................... 31 4.3 Determinacy and wBP(P). .................. 34 5. Determinacy andAsymmetric Properties. 39 5.1 The Asymmetric Properties. ................. 39 5.2 The General Definition of Asym(P). ............. 43 5.3 Determinacy and Asym(P). ................. 46 ii iii 0. Introduction One of the most intriguing developments of modern set theory is the investi- gation of two-player infinite games of perfect information. Of course, it is clear that applied game theory, as any other branch of mathematics, can be modeled in set theory. But we are talking about the converse: the use of infinite games as a tool to study fundamental set theoretic questions.
    [Show full text]