DOCSLIB.ORG

Lecture Notes on Descriptive Set Theory Notation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture Notes on Descriptive Set Theory Notation Lecture Notes on Descriptive Set Theory Jan Reimann∗ Department of Mathematics Pennsylvania State University Notation U"(x) Ball of radius " about x U Topological closure of U 2<N Set of finite binary strings σ, τ, . Finite binary strings 2N Cantor space, set of all infinite binary sequences NN Baire space, set of all infinite sequences of natural numbers α n Length n initial segment of sequence α, α(0) ... α(n 1) j − Nσ Open cylinder defined by σ LipL(X ) Set of L-Lipschitz functions on X diam(X ) Diameter of a set X in a metric space, diam(X ) = sup d(x, y): x, y X f 2 g ∗[email protected] Lecture 1: Perfect Subsets of the Real Line Descriptive set theory nowadays is understood as the study of definable subsets of Polish Spaces. Many of its problems and techniques arose out of efforts to answer basic questions about the real numbers. A prominent example is the Continuum Hypothesis (CH): If A R is uncountable, does there exist a bijection between A and ⊆ R? That is, is every uncountable subset of R of the same cardinality as R? [Cantor, 1890’s] Early approaches to this problem tried to show that CH holds for a number of sets with an easy topological structure. It is a standard exercise of analysis to show that every open set satisfies CH. (An open set contains an interval, which maps bijectively to R.) For closed sets, the situation is less clear. Given a set A R, we call x R an accumulation point of A if ⊆ 2 ε > 0 z A [z = x & z U"(x)], 8 9 2 6 2 1 where U"(x) denotes the standard "-neighborhood of x in R . Call a non-empty set P R perfect if it is closed and every point of P is an accumulation point. In other⊆ words, a perfect set is a closed set that has no isolated points. It is not hard to see that for a perfect set P, every neighborhood of a point p P contains infinitely many points. 2 Obviously, R itself is perfect, as is any closed interval in R. There are totally disconnected perfect sets, such as the middle-third Cantor set in [0, 1] Theorem 1.1: A perfect subset of R has the same cardinality as R. Proof. Let P R be perfect. We construct an injection from the set 2N of all ⊆ infinite binary sequences into P. An infinite binary sequence ξ = ξ0ξ1ξ2 ... can be identified with a real number [0, 1] via the mapping 2 X i 1 ξ ξi2− − . 7! i 0 ≥ Note that this mapping is onto. Hence the cardinality of P is at least as large as the cardinality of [0, 1]. The Cantor-Schröder-Bernstein Theorem (for a proof 0 see e.g. (author?) [Jec03]) implies that P = 2@ . j j 1There are some divergences in terminology. Some authors call an accumulation point a limit point. We reserve the latter term for any point that is the limit of a sequence of points from a given set. Hence every member of a set is a limit point of that set. In particular, isolated members of a set are limit points. 1 – 1 Choose x P, and let 1 20. Since P is perfect, P U x . Let x x "0 = = "0 ( ) 0 = 1 be two points2 in P U x , distinct from x. Let be\ such that 6 1 2, "0 ( ) "1 "1 = U x , U x U\ x , and U x U x , where U denotes≤ the "1 ( 0) "1 ( 1) "0 ( ) "1 ( 0) "1 ( 1) = closure of U. ⊆ \ ; We can iterate this procedure recursively with smaller and smaller diameters, using the fact that P is perfect. This gives rise to a so-called Cantor scheme, a family of open balls (Uσ). Here the index σ is a finite binary sequence, also called a string. The scheme has the following properties. σ C1) diam(Uσ) 2−| j, where σ denotes the length of σ. ≤ j j C2) If τ is a proper extension of σ, then Uτ Uσ. ⊂ C3) If τ and σ are incompatible (i.e. neither extends the other), then Uτ Uσ = . \ ; C4) The center of each Uσ, call it xσ, is in P. U U0 U1 U00 U01 U10 U11 Figure 1: Cantor Scheme Let ξ be an infinite binary sequence. Given n 0, we denote by ξ n the string formed by the first n bits of ξ, i.e. ≥ j ξ n= ξ0ξ1 ... ξn 1. j − 1 – 2 The finite initial segments give rise to a sequence x of centers. By (C1) and ξ n (C2), this is a Cauchy sequence. By (C4), the sequencej lies in P. Since P is closed, the limit xξ is in P. By (C3), the mapping ξ xξ is well-defined and injective. 7! Theorem 1.2: Every uncountable closed subset of R contains a perfect subset. Proof. Let C R be uncountable and closed. We say z R is a condensation point of C if ⊆ 2 " > 0 [U"(z) C uncountable]. 8 \ Let D be the set of all condensation points of C. Note that D C, since every condensation point is clearly an accumulation point and C is closed.⊆ Furthermore, we claim that D is perfect. Clearly D is closed. Suppose z D and " > 0. Then 2 U"(z) C is uncountable. We would like to conclude that U"(z) D is uncountable, \ \ too, since this would mean in particular that U"(z) D is infinite. The conclusion holds if C D is countable. To show that C D is countable,\ we use the fact that n n every open interval in R is the union of countably many open intervals with rational endpoints. Note that there are only countably many such intervals. If y C D, then for some δ > 0, Uδ(y) C is countable. y is contained in some 2 n \ subinterval Uy Uδ(y) with rational endpoints. Thus, we have ⊆ [ C D Uy C, n ⊆ y C D \ 2 n and the right hand side is a countable union of countable sets, hence countable. We will later encounter an alternative (more constructive) proof that gives additional information about the complexity of the closed set C. For now we conclude with the fact we started out to prove. Corollary 1.3: Every closed subset of R is either countable or of the cardinality of the continuum. 1 – 3 Lecture 2: Polish Spaces The proofs in the previous lecture are quite general, that is, they make little use of specific properties of R. If we scan the arguments carefully, we see that we can replace R by any metric space that is complete and contains a countable basis of the topology. Review of some concepts from topology Let (X , O) be a topological space. A family B O of subsets if X is a basis for the topology if every open set from O is the⊆ union of elements of B. For example, the open intervals with rational endpoints form a basis of the standard topology of R. (We used this fact in Lecture1.) S O is a subbasis if the set of finite intersections of sets in S is a basis for the topology.⊆ Finally, if S is any family of subset of X , the topology generated by S is the smallest topology on X containing S. It consists of all unions of finite intersections of sets in S X , . [f ;g A set D X is dense if for open U = there exists z D U. If a topological space (X⊂, O) has a countable dense6 subset,; the space is2 called\ separable. If (Xi)i I is a family of topological spaces, one defines the product topology on 2 1 Πi I Xi to be the topology generated by the sets π−i (U), where i I, U Xi is 2 2 ⊆ open, and πi : Πi I Xi Xi is the ith projection. 2 ! Now suppose (X , d) is a metric space. With each point x X and every " > 0 we associate an "-neighborhood or "-ball 2 U"(x) = y X : d(x, y) < " . f 2 g The "-neighborhoods form the basis of a topology, called the topology of the metric space (X , d). If this topology agrees with a given topology O on X , we say the metric d is compatible with the topology O. If for a topological space 2 (X , O) there exists a compatible metric, (X , O) is called metrizable . If a topological space (X , O) is separable and metrizable, then the balls with center in a countable dense subset D and rational radius form a countable base of the topology. 2Note that a compatible metric is not necessarily unique. 2 – 1 Polish spaces Definition 2.1: A Polish space is a separable topological space X for which exists a compatible metric d such that (X , d) is a complete metric space. As mentioned before, there may be many different compatible metrics that make X complete. If X is already given as a complete metric space with countable dense subset, then we call X a Polish metric space. The standard example is, of course, R, the set of real numbers. One can obtain other Polish spaces using the following basic observations. Proposition 2.2: 1) A closed subset of a Polish space is Polish. 2) The product of a countable (in particular, finite) sequence of Polish spaces is Polish. n n Hence we can conclude that R , C, C , the unit interval [0, 1], the unit circle T = z C: z = 1 , and the infinite dimensional spaces RN and [0, 1]N (the Hilbertf 2 cube)j arej Polishg spaces.
Load more

Recommended publications
  • 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]
  • 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]
  • 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]
  • The Realm of Ordinal Analysis
    The Realm of Ordinal Analysis Michael Rathjen Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom Denn die Pioniere der Mathematik hatten sich von gewissen Grundlagen brauchbare Vorstellungen gemacht, aus denen sich Schl¨usse,Rechnungsarten, Resultate ergaben, deren bem¨achtigten sich die Physiker, um neue Ergebnisse zu erhalten, und endlich kamen die Techniker, nahmen oft bloß die Resultate, setzten neue Rechnungen darauf und es entstanden Maschinen. Und pl¨otzlich, nachdem alles in sch¨onste Existenz gebracht war, kamen die Mathematiker - jene, die ganz innen herumgr¨ubeln, - darauf, daß etwas in den Grundlagen der ganzen Sache absolut nicht in Ordnung zu bringen sei; tats¨achlich, sie sahen zuunterst nach und fanden, daß das ganze Geb¨audein der Luft stehe. Aber die Maschinen liefen! Man muß daraufhin annehmen, daß unser Dasein bleicher Spuk ist; wir leben es, aber eigentlich nur auf Grund eines Irrtums, ohne den es nicht entstanden w¨are. ROBERT MUSIL: Der mathematische Mensch (1913) 1 Introduction A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their ‘rank’ or ‘complexity’ in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of ‘proof the- oretic ordinals’ to theories, gauging their ‘consistency strength’ and ‘computational power’. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen’s head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories.
    [Show full text]
  • Descriptive Set Theory and the Ergodic Theory of Countable Groups
    DESCRIPTIVE SET THEORY AND THE ERGODIC THEORY OF COUNTABLE GROUPS Thesis by Robin Daniel Tucker-Drob In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2013 (Defended April 17, 2013) ii c 2013 Robin Daniel Tucker-Drob All Rights Reserved iii Acknowledgements I would like to thank my advisor Alexander Kechris for his invaluable guidance and support, for his generous feedback, and for many (many!) discussions. In addition I would like to thank Miklos Abert,´ Lewis Bowen, Clinton Conley, Darren Creutz, Ilijas Farah, Adrian Ioana, David Kerr, Andrew Marks, Benjamin Miller, Jesse Peterson, Ernest Schimmerling, Miodrag Sokic, Simon Thomas, Asger Tornquist,¨ Todor Tsankov, Anush Tserunyan, and Benjy Weiss for many valuable conversations over the past few years. iv Abstract The primary focus of this thesis is on the interplay of descriptive set theory and the ergodic theory of group actions. This incorporates the study of turbulence and Borel re- ducibility on the one hand, and the theory of orbit equivalence and weak equivalence on the other. Chapter 2 is joint work with Clinton Conley and Alexander Kechris; we study measurable graph combinatorial invariants of group actions and employ the ultraproduct construction as a way of constructing various measure preserving actions with desirable properties. Chapter 3 is joint work with Lewis Bowen; we study the property MD of resid- ually finite groups, and we prove a conjecture of Kechris by showing that under general hypotheses property MD is inherited by a group from one of its co-amenable subgroups. Chapter 4 is a study of weak equivalence.
    [Show full text]
  • Determinacy and Large Cardinals
    Determinacy and Large Cardinals Itay Neeman∗ Abstract. The principle of determinacy has been crucial to the study of definable sets of real numbers. This paper surveys some of the uses of determinacy, concentrating specifically on the connection between determinacy and large cardinals, and takes this connection further, to the level of games of length ω1. Mathematics Subject Classification (2000). 03E55; 03E60; 03E45; 03E15. Keywords. Determinacy, iteration trees, large cardinals, long games, Woodin cardinals. 1. Determinacy Let ωω denote the set of infinite sequences of natural numbers. For A ⊂ ωω let Gω(A) denote the length ω game with payoff A. The format of Gω(A) is displayed in Diagram 1. Two players, denoted I and II, alternate playing natural numbers forming together a sequence x = hx(n) | n < ωi in ωω called a run of the game. The run is won by player I if x ∈ A, and otherwise the run is won by player II. I x(0) x(2) ...... II x(1) x(3) ...... Diagram 1. The game Gω(A). A game is determined if one of the players has a winning strategy. The set A is ω determined if Gω(A) is determined. For Γ ⊂ P(ω ), det(Γ) denotes the statement that all sets in Γ are determined. Using the axiom of choice, or more specifically using a wellordering of the reals, it is easy to construct a non-determined set A. det(P(ωω)) is therefore false. On the other hand it has become clear through research over the years that det(Γ) is true if all the sets in Γ are definable by some concrete means.
    [Show full text]
  • Set Ideal Topological Spaces
    Set Ideal Topological Spaces W. B. Vasantha Kandasamy Florentin Smarandache ZIP PUBLISHING Ohio 2012 This book can be ordered from: Zip Publishing 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: (614) 485-0721 E-mail: [email protected] Website: www.zippublishing.com Copyright 2012 by Zip Publishing and the Authors Peer reviewers: Prof. Catalin Barbu, V. Alecsandri National College, Mathematics Department, Bacau, Romania. Prof. Valeri Kroumov, Okayama Univ. of Science, Japan. Dr. Sebastian Nicolaescu, 2 Terrace Ave., West Orange, NJ 07052, USA. Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-13: 978-1-59973-193-3 EAN: 9781599731933 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two SET IDEALS IN RINGS 9 Chapter Three SET IDEAL TOPOLOGICAL SPACES 35 Chapter Four NEW CLASSES OF SET IDEAL TOPOLOGICAL SPACES AND APPLICATIONS 93 3 FURTHER READING 109 INDEX 111 ABOUT THE AUTHORS 114 4 PREFACE In this book the authors for the first time introduce a new type of topological spaces called the set ideal topological spaces using rings or semigroups, or used in the mutually exclusive sense. This type of topological spaces use the class of set ideals of a ring (semigroups). The rings or semigroups can be finite or infinite order. By this method we get complex modulo finite integer set ideal topological spaces using finite complex modulo integer rings or finite complex modulo integer semigroups. Also authors construct neutrosophic set ideal toplogical spaces of both finite and infinite order as well as complex neutrosophic set ideal topological spaces.
    [Show full text]
  • ON BERNSTEIN SETS Let Us Recall That a Subset of a Topological Space
    ON BERNSTEIN SETS JACEK CICHON´ ABSTRACT. In this note I show a construction of Bernstein subsets of the real line which gives much more information about the structure of the real line than the classical one. 1. BASIC DEFINITIONS Let us recall that a subset of a topological space is a perfect set if is closed set and contains no isolated points. Perfect subsets of real line R have cardinality continuum. In fact every perfect set contains a copy of a Cantor set. This can be proved rather easilly by constructing a binary tree of decreasing small closed sets. There are continnum many perfect sets. This follows from a more general result: there are continuum many Borel subsets of real. We denote by R the real line. We will work in the theory ZFC. Definition 1. A subset B of the real line R is a Bernstein set if for every perfect subset P of R we have (P \ B 6= ;) ^ (P n B 6= ;) : Bernstein sets are interesting object of study because they are Lebesgue nonmeasurable and they do not have the property of Baire. The classical construction of a Berstein set goes as follows: we fix an enumeration (Pα)α<c of all perfect sets, we define a transfinite sequences (pα), (qα) of pairwise different points such that fpα; qαg ⊆ Pα, we put B = fpα : α < cg and we show that B is a Bernstein set. This construction can be found in many classical books. The of this note is to give slightly different construction, which will generate simultanously a big family of Bermstein sets.We start with one simple but beautifull result: Lemma 1.
    [Show full text]
  • Effective Descriptive Set Theory
    Effective Descriptive Set Theory Andrew Marks December 14, 2019 1 1 These notes introduce the effective (lightface) Borel, Σ1 and Π1 sets. This study uses ideas and tools from descriptive set theory and computability theory. Our central motivation is in applications of the effective theory to theorems of classical (boldface) descriptive set theory, especially techniques which have no classical analogues. These notes have many errors and are very incomplete. Some important topics not covered include: • The Harrington-Shore-Slaman theorem [HSS] which implies many of the theorems of Section 3. • Steel forcing (see [BD, N, Mo, St78]) • Nonstandard model arguments • Barwise compactness, Jensen's model existence theorem • α-recursion theory • Recent beautiful work of the \French School": Debs, Saint-Raymond, Lecompte, Louveau, etc. These notes are from a class I taught in spring 2019. Thanks to Adam Day, Thomas Gilton, Kirill Gura, Alexander Kastner, Alexander Kechris, Derek Levinson, Antonio Montalb´an,Dean Menezes and Riley Thornton, for helpful conversations and comments on earlier versions of these notes. 1 Contents 1 1 1 1 Characterizing Σ1, ∆1, and Π1 sets 4 1 1.1 Σn formulas, closure properties, and universal sets . .4 1.2 Boldface vs lightface sets and relativization . .5 1 1.3 Normal forms for Σ1 formulas . .5 1.4 Ranking trees and Spector boundedness . .7 1 1.5 ∆1 = effectively Borel . .9 1.6 Computable ordinals, hyperarithmetic sets . 11 1 1.7 ∆1 = hyperarithmetic . 14 x 1 1.8 The hyperjump, !1 , and the analogy between c.e. and Π1 .... 15 2 Basic tools 18 2.1 Existence proofs via completeness results .
    [Show full text]
  • Lecture 8: the Real Line, Part II February 11, 2009
    Lecture 8: The Real Line, Part II February 11, 2009 Definition 6.13. a ∈ X is isolated in X iff there is an open interval I for which X ∩ I = {a}. Otherwise, a is a limit point. Remark. Another way to state this is that a is isolated if it is not a limit point of X. Definition 6.14. X is a perfect set iff X is closed and has no isolated points. Remark. This definition sounds nice and tidy, but there are some very strange perfect sets. For example, the Cantor set is perfect, despite being nowhere dense! Our goal will be to prove the Cantor-Bendixson theorem, i.e. the perfect set theorem for closed sets, that every closed uncountable set has a perfect subset. Lemma 6.15. If P is a perfect set and I is an open interval on R such that I ∩ P 6= ∅, then there exist disjoint closed intervals J0,J1 ⊂ I such that int[J0] ∩ P 6= ∅ and int[J1] ∩ P 6= ∅. Moreover, we can pick J0 and J1 such that their lengths are both less than any ² > 0. Proof. Since P has no isolated points, there must be at least two points a0, a1 ∈ I ∩ P . Then just pick J0 3 a0 and J1 3 a1 to be small enough. SDG Lemma 6.16. If P is a nonempty perfect set, then P ∼ R. Proof. We exhibit a one-to-one mapping G : 2ω → P . Note that 2ω can be viewed as the set of all infinite paths in a full, infinite binary tree with each edge labeled by 0 or 1.
    [Show full text]