The Determinacy of Context-Free Games Olivier Finkel
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Set Theory, Including the Axiom of Choice) Plus the Negation of CH
ANNALS OF ~,IATltEMATICAL LOGIC - Volume 2, No. 2 (1970) pp. 143--178 INTERNAL COHEN EXTENSIONS D.A.MARTIN and R.M.SOLOVAY ;Tte RockeJ~'ller University and University (.~t CatiJbrnia. Berkeley Received 2 l)ecemt)er 1969 Introduction Cohen [ !, 2] has shown that tile continuum hypothesis (CH) cannot be proved in Zermelo-Fraenkel set theory. Levy and Solovay [9] have subsequently shown that CH cannot be proved even if one assumes the existence of a measurable cardinal. Their argument in tact shows that no large cardinal axiom of the kind present;y being considered by set theorists can yield a proof of CH (or of its negation, of course). Indeed, many set theorists - including the authors - suspect that C1t is false. But if we reject CH we admit Gurselves to be in a state of ignorance about a great many questions which CH resolves. While CH is a power- full assertion, its negation is in many ways quite weak. Sierpinski [ 1 5 ] deduces propcsitions there called C l - C82 from CH. We know of none of these propositions which is decided by the negation of CH and only one of them (C78) which is decided if one assumes in addition that a measurable cardinal exists. Among the many simple questions easily decided by CH and which cannot be decided in ZF (Zerme!o-Fraenkel set theory, including the axiom of choice) plus the negation of CH are tile following: Is every set of real numbers of cardinality less than tha't of the continuum of Lebesgue measure zero'? Is 2 ~0 < 2 ~ 1 ? Is there a non-trivial measure defined on all sets of real numbers? CIhis third question could be decided in ZF + not CH only in the unlikely event t Tile second author received support from a Sloan Foundation fellowship and tile National Science Foundation Grant (GP-8746). -
COMPSCI 501: Formal Language Theory Insights on Computability Turing Machines Are a Model of Computation Two (No Longer) Surpris
Insights on Computability Turing machines are a model of computation COMPSCI 501: Formal Language Theory Lecture 11: Turing Machines Two (no longer) surprising facts: Marius Minea Although simple, can describe everything [email protected] a (real) computer can do. University of Massachusetts Amherst Although computers are powerful, not everything is computable! Plus: “play” / program with Turing machines! 13 February 2019 Why should we formally define computation? Must indeed an algorithm exist? Back to 1900: David Hilbert’s 23 open problems Increasingly a realization that sometimes this may not be the case. Tenth problem: “Occasionally it happens that we seek the solution under insufficient Given a Diophantine equation with any number of un- hypotheses or in an incorrect sense, and for this reason do not succeed. known quantities and with rational integral numerical The problem then arises: to show the impossibility of the solution under coefficients: To devise a process according to which the given hypotheses or in the sense contemplated.” it can be determined in a finite number of operations Hilbert, 1900 whether the equation is solvable in rational integers. This asks, in effect, for an algorithm. Hilbert’s Entscheidungsproblem (1928): Is there an algorithm that And “to devise” suggests there should be one. decides whether a statement in first-order logic is valid? Church and Turing A Turing machine, informally Church and Turing both showed in 1936 that a solution to the Entscheidungsproblem is impossible for the theory of arithmetic. control To make and prove such a statement, one needs to define computability. In a recent paper Alonzo Church has introduced an idea of “effective calculability”, read/write head which is equivalent to my “computability”, but is very differently defined. -
Lecture Notes
MEASURE THEORY AND STOCHASTIC PROCESSES Lecture Notes José Melo, Susana Cruz Faculdade de Engenharia da Universidade do Porto Programa Doutoral em Engenharia Electrotécnica e de Computadores March 2011 Contents 1 Probability Space3 1 Sample space Ω .....................................4 2 σ-eld F .........................................4 3 Probability Measure P .................................6 3.1 Measure µ ....................................6 3.2 Probability Measure P .............................7 4 Learning Objectives..................................7 5 Appendix........................................8 2 Chapter 1 Probability Space Let's consider the experience of throwing a dart on a circular target with radius r (assuming the dart always hits the target), divided in 4 dierent areas as illustrated in Figure 1.1. 4 3 2 1 Figure 1.1: Circular Target The circles that bound the regions 1, 2, 3, and 4, have radius of, respectively, r , r , 3r , and . 4 2 4 r Therefore, the probability that a dart lands in each region is: 1 , 3 , 5 , P (1) = 16 P (2) = 16 P (3) = 16 7 . P (4) = 16 For this kind of problems, the theory of discrete probability spaces suces. However, when it comes to problems involving either an innitely repeated operation or an innitely ne op- eration, this mathematical framework does not apply. This motivates the introduction of a measure-theoretic probability approach to correctly describe those cases. We dene the proba- bility space (Ω; F;P ), where Ω is the sample space, F is the event space, and P is the probability 3 measure. Each of them will be described in the following subsections. 1 Sample space Ω The sample space Ω is the set of all the possible results or outcomes ! of an experiment or observation. -
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. -
Borel Structure in Groups and Their Dualso
BOREL STRUCTURE IN GROUPS AND THEIR DUALSO GEORGE W. MACKEY Introduction. In the past decade or so much work has been done toward extending the classical theory of finite dimensional representations of com- pact groups to a theory of (not necessarily finite dimensional) unitary repre- sentations of locally compact groups. Among the obstacles interfering with various aspects of this program is the lack of a suitable natural topology in the "dual object"; that is in the set of equivalence classes of irreducible representations. One can introduce natural topologies but none of them seem to have reasonable properties except in extremely special cases. When the group is abelian for example the dual object itself is a locally compact abelian group. This paper is based on the observation that for certain purposes one can dispense with a topology in the dual object in favor of a "weaker struc- ture" and that there is a wide class of groups for which this weaker structure has very regular properties. If .S is any topological space one defines a Borel (or Baire) subset of 5 to be a member of the smallest family of subsets of 5 which includes the open sets and is closed with respect to the formation of complements and countable unions. The structure defined in 5 by its Borel sets we may call the Borel structure of 5. It is weaker than the topological structure in the sense that any one-to-one transformation of S onto 5 which preserves the topological structure also preserves the Borel structure whereas the converse is usually false. -
Axiomatic Set Teory P.D.Welch
Axiomatic Set Teory P.D.Welch. August 16, 2020 Contents Page 1 Axioms and Formal Systems 1 1.1 Introduction 1 1.2 Preliminaries: axioms and formal systems. 3 1.2.1 The formal language of ZF set theory; terms 4 1.2.2 The Zermelo-Fraenkel Axioms 7 1.3 Transfinite Recursion 9 1.4 Relativisation of terms and formulae 11 2 Initial segments of the Universe 17 2.1 Singular ordinals: cofinality 17 2.1.1 Cofinality 17 2.1.2 Normal Functions and closed and unbounded classes 19 2.1.3 Stationary Sets 22 2.2 Some further cardinal arithmetic 24 2.3 Transitive Models 25 2.4 The H sets 27 2.4.1 H - the hereditarily finite sets 28 2.4.2 H - the hereditarily countable sets 29 2.5 The Montague-Levy Reflection theorem 30 2.5.1 Absoluteness 30 2.5.2 Reflection Theorems 32 2.6 Inaccessible Cardinals 34 2.6.1 Inaccessible cardinals 35 2.6.2 A menagerie of other large cardinals 36 3 Formalising semantics within ZF 39 3.1 Definite terms and formulae 39 3.1.1 The non-finite axiomatisability of ZF 44 3.2 Formalising syntax 45 3.3 Formalising the satisfaction relation 46 3.4 Formalising definability: the function Def. 47 3.5 More on correctness and consistency 48 ii iii 3.5.1 Incompleteness and Consistency Arguments 50 4 The Constructible Hierarchy 53 4.1 The L -hierarchy 53 4.2 The Axiom of Choice in L 56 4.3 The Axiom of Constructibility 57 4.4 The Generalised Continuum Hypothesis in L. -
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. -
Algorithms, Turing Machines and Algorithmic Undecidability
U.U.D.M. Project Report 2021:7 Algorithms, Turing machines and algorithmic undecidability Agnes Davidsdottir Examensarbete i matematik, 15 hp Handledare: Vera Koponen Examinator: Martin Herschend April 2021 Department of Mathematics Uppsala University Contents 1 Introduction 1 1.1 Algorithms . .1 1.2 Formalisation of the concept of algorithms . .1 2 Turing machines 3 2.1 Coding of machines . .4 2.2 Unbounded and bounded machines . .6 2.3 Binary sequences representing real numbers . .6 2.4 Examples of Turing machines . .7 3 Undecidability 9 i 1 Introduction This paper is about Alan Turing's paper On Computable Numbers, with an Application to the Entscheidungsproblem, which was published in 1936. In his paper, he introduced what later has been called Turing machines as well as a few examples of undecidable problems. A few of these will be brought up here along with Turing's arguments in the proofs but using a more modern terminology. To begin with, there will be some background on the history of why this breakthrough happened at that given time. 1.1 Algorithms The concept of an algorithm has always existed within the world of mathematics. It refers to a process meant to solve a problem in a certain number of steps. It is often repetitive, with only a few rules to follow. In more recent years, the term also has been used to refer to the rules a computer follows to operate in a certain way. Thereby, an algorithm can be used in a plethora of circumstances. The word might describe anything from the process of solving a Rubik's cube to how search engines like Google work [4]. -
STAT 571 Assignment 1 Solutions 1. If Ω Is a Set and C a Collection Of
STAT 571 Assignment 1 solutions 1. If Ω is a set and a collection of subsets of Ω, let be the intersection of all σ-algebras that contain . ProveC that is the σ-algebra generatedA by . C A C Solution: Let α α A be the collection of all σ-algebras that contain , and fA j 2 g C set = α. We first show that is a σ-algebra. There are three things to A \ A A α A prove. 2 (a) For every α A, α is a σ-algebra, so Ω α, and hence Ω α A α = . 2 A 2 A 2 \ 2 A A (b) If B , then B α for every α A. Since α is a σ-algebra, we have c 2 A 2 A 2 A c B α. But this is true for every α A, so we have B . 2 A 2 2 A (c) If B1;B2;::: are sets in , then B1;B2;::: belong to α for each α A. A A 2 Since α is a σ-algebra, we have 1 Bn α. But this is true for every A [n=1 2 A α A, so we have 1 Bn . 2 [n=1 2 A Thus is a σ-algebra that contains , and it must be the smallest one since α for everyA α A. C A ⊆ A 2 2. Prove that the set of rational numbers Q is a Borel set in R. Solution: For every x R, the set x is the complement of an open set, and hence Borel. -
(Measure Theory for Dummies) UWEE Technical Report Number UWEETR-2006-0008
A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 UWEE Technical Report Number UWEETR-2006-0008 May 2006 Department of Electrical Engineering University of Washington Box 352500 Seattle, Washington 98195-2500 PHN: (206) 543-2150 FAX: (206) 543-3842 URL: http://www.ee.washington.edu A Measure Theory Tutorial (Measure Theory for Dummies) Maya R. Gupta {gupta}@ee.washington.edu Dept of EE, University of Washington Seattle WA, 98195-2500 University of Washington, Dept. of EE, UWEETR-2006-0008 May 2006 Abstract This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like “closed” and “open.” The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Measure theory is a bit like grammar, many people communicate clearly without worrying about all the details, but the details do exist and for good reasons. There are a number of great texts that do measure theory justice. This is not one of them. Rather this is a hack way to get the basic ideas down so you can read through research papers and follow what’s going on. Hopefully, you’ll get curious and excited enough about the details to check out some of the references for a deeper understanding. -
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.