Towards a Descriptive Theory of Cb 0-Spaces
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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. -
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 . -
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. -
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. -
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. -
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. -
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. -
Wadge Degrees of Ω-Languages of Petri Nets Olivier Finkel
Wadge Degrees of !-Languages of Petri Nets Olivier Finkel To cite this version: Olivier Finkel. Wadge Degrees of !-Languages of Petri Nets. 2018. hal-01666992v2 HAL Id: hal-01666992 https://hal.archives-ouvertes.fr/hal-01666992v2 Preprint submitted on 26 Mar 2018 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 Degrees of ω-Languages of Petri Nets Olivier Finkel Equipe de Logique Mathématique Institut de Mathématiques de Jussieu - Paris Rive Gauche CNRS et Université Paris 7, France. [email protected] Abstract We prove that ω-languages of (non-deterministic) Petri nets and ω-languages of (non-deterministic) Tur- ing machines have the same topological complexity: the Borel and Wadge hierarchies of the class of ω-languages of (non-deterministic) Petri nets are equal to the Borel and Wadge hierarchies of the class of ω-languages of (non-deterministic) Turing machines which also form the class of effective analytic CK 0 sets. In particular, for each non-null recursive ordinal α < ω1 there exist some Σα-complete and some 0 Πα-complete ω-languages of Petri nets, and the supremum of the set of Borel ranks of ω-languages of 1 CK Petri nets is the ordinal γ2 , which is strictly greater than the first non-recursive ordinal ω1 . -
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. -
Effective Wadge Hierarchy in Computable Quasi-Polish Spaces
Effective Wadge hierarchy in computable quasi-Polish spaces Victor Selivanov1 A.P. Ershov IIS SB RAS (Novosibirsk) Mathematical Logic and its Applications Workshop March 22, 2021 Victor Selivanov Effective Wadge hierarchy in computable quasi-Polish spaces Contents 1. Introduction. 2. Effective spaces. 3. Iterated labeled trees. 4. Defining the (effective) WH. 5. The preservation property. 6. Hausdorff-Kuratowski-type results. 7. The non-collapse property. 8. Conclusion. Victor Selivanov Effective Wadge hierarchy in computable quasi-Polish spaces Introduction The classical Borel, Luzin, and Hausdorff hierarchies in Polish spaces, which are defined using set operations, play an important role in descriptive set theory (DST). In 2013 these hierarchies were extended and shown to have similar nice properties also in quasi-Polish spaces which include many non-Hausdorff spaces of interest for several branches of mathematics and theoretical computer science (M. de Brecht). The Wadge hierarchy is non-classical in the sense that it is based on a notion of reducibility that was not recognized in the classical DST, and on using ingenious versions of Gale-Stewart games rather than on set operations. For subsets A; B of the Baire space ! −1 N = ! , A is Wadge reducible to B (A ≤W B), if A = f (B) for some continuous function f on N . The quotient-poset of the preorder (P(N ); ≤W ) under the induced equivalence relation ≡W on the power-set of N is called the structure of Wadge degrees in N . Victor Selivanov Effective Wadge hierarchy in computable quasi-Polish spaces Introduction W. Wadge characterised the structure of Wadge degrees of Borel sets (i.e., the quotient-poset of (B(N ); ≤W )) up to isomorphism. -
Borel Hierarchy and Omega Context Free Languages Olivier Finkel
Borel Hierarchy and Omega Context Free Languages Olivier Finkel To cite this version: Olivier Finkel. Borel Hierarchy and Omega Context Free Languages. Theoretical Computer Science, Elsevier, 2003, 290 (3), pp.1385-1405. hal-00103679v2 HAL Id: hal-00103679 https://hal.archives-ouvertes.fr/hal-00103679v2 Submitted on 18 Jan 2011 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. BOREL HIERARCHY AND OMEGA CONTEXT FREE LANGUAGES Olivier Finkel ∗ Equipe de Logique Math´ematique CNRS et Universit´eParis 7, U.F.R. de Math´ematiques 2 Place Jussieu 75251 Paris cedex 05, France. Abstract We give in this paper additional answers to questions of Lescow and Thomas [Logi- cal Specifications of Infinite Computations, In:”A Decade of Concurrency”, Springer LNCS 803 (1994), 583-621], proving topological properties of omega context free lan- guages (ω-CFL) which extend those of [O. Finkel, Topological Properties of Omega Context Free Languages, Theoretical Computer Science, Vol. 262 (1-2), 2001, p. 669-697]: there exist some ω-CFL which are non Borel sets and one cannot decide whether an ω-CFL is a Borel set. We give also an answer to a question of Niwin- ski [Problem on ω-Powers Posed in the Proceedings of the 1990 Workshop ”Logics and Recognizable Sets”] and of Simonnet [Automates et Th´eorie Descriptive, Ph.D. -
Wadge Hierarchy on Second Countable Spaces Reduction Via Relatively Continuous Relations
Wadge Hierarchy on Second Countable Spaces Reduction via relatively continuous relations Yann Honza Pequignot Université de Lausanne Université Paris Diderot Winter School In Abstract analysis Set Theory & Topology Hejnice, 15th January – 1st February, 2014 Classify Definable subsets of topological spaces nd X a 2 countable T0 topological space: A countable basis of open sets, Two points which have same neighbourhoods are equal. Borel sets are naturally classified according to their definition 0 Σ1(X) = {O ⊆ X | X is open}, 0 [ Σ (X) = Bi Bi is a Boolean combination of open sets , 2 i∈ω 0 { 0 Πα(X) = A A ∈ Σα(X) , 0 \ [ 0 Σ (X) = Pi Pi ∈ Π (X) , for α > 2. α β i∈ω β<α [ 0 [ 0 Borel subsets of X = Σα(X) = Πα(X) α<ω1 α<ω1 Wadge reducibility Let X be a topological space, A, B ⊆ X. A is Wadge reducible to B, or A ≤W B, if there is a continuous function f : X → X that reduces A to B, i.e. such that f −1(B) = A or equivalently ∀x ∈ X x ∈ A ←→ f (x) ∈ B. Bill Wadge The idea is that the continuous function f reduces the membership question for A to the membership question for B. The identity on X is continous, and continuous functions compose, so Wadge reducibility is a quasi order on subsets of X. Is it useful? On non 0-dim metric spaces, and many other non metrisable spaces the relation ≤W yields no hierarchy at all, by results of Schlicht, Hertling, Ikegami, Tanaka, Grigorieff, Selivanov and others.