Highly Undecidable Problems About Recognizability by Tiling Systems Olivier Finkel
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Basic Computability Theory
Basic Computability Theory Jaap van Oosten Department of Mathematics Utrecht University 1993, revised 2013 ii Introduction The subject of this course is the theory of computable or recursive functions. Computability Theory and Recursion Theory are two names for it; the former being more popular nowadays. 0.1 Why Computability Theory? The two most important contributions Logic has made to Mathematics, are formal definitions of proof and of algorithmic computation. How useful is this? Mathematicians are trained in understanding and constructing proofs from their first steps in the subject, and surely they can recognize a valid proof when they see one; likewise, algorithms have been an essential part of Mathematics for as long as it exists, and at the most elementary levels (think of long division, or Euclid’s algorithm for the greatest common divisor function). So why do we need a formal definition here? The main use is twofold. Firstly, a formal definition of proof (or algorithm) leads to an analysis of proofs (programs) in elementary reasoning steps (computation steps), thereby opening up the possibility of numerical classification of proofs (programs)1. In Proof Theory, proofs are assigned ordinal numbers, and theorems are classified according to the complexity of the proofs they need. In Complexity Theory, programs are analyzed according to the number of elementary computation steps they require (as a function of the input), and programming problems are classified in terms of these complexity functions. Secondly, a formal definition is needed once one wishes to explore the boundaries of the topic: what can be proved at all? Which kind of problems can be settled by an algorithm? It is this aspect that we will focus on in this course. -
Computability Theory of Hyperarithmetical Sets
WESTFALISCHE¨ WILHELMS-UNIVERSITAT¨ MUNSTER¨ INSTITUT FUR¨ MATHEMATISCHE LOGIK UND GRUNDLAGENFORSCHUNG Computability Theory of Hyperarithmetical Sets Lecture by Wolfram Pohlers Summer–Term 1996 Typeset by Martina Pfeifer Preface This text contains the somewhat extended material of a series of lectures given at the University of M¨unster. The aim of the course is to give an introduction to “higher” computability theory and to provide background material for the following courses in proof theory. The prerequisites for the course are some basic facts about computable functions and mathe- matical logic. Some emphasis has been put on the notion of generalized inductive definitions. Whenever it seemed to be opportune we tried to obtain “classical” results by using generalized inductive definitions. I am indebted to Dipl. Math. INGO LEPPER for the revising and supplementing the original text. M¨unster, September 1999 WOLFRAM POHLERS 2 Contents Contents 1 Computable Functionals and Relations ....................... 5 1.1FunctionalsandRelations................................ 5 1.2TheNormal–formTheorem............................... 10 1.3 Computability relativized . ............................... 13 2 Degrees ......................................... 15 2.1 m–Degrees....................................... 15 2.2 TURING–Reducibility . ............................... 16 2.3 TURING–Degrees.................................... 18 3 The Arithmetical Hierarchy ............................. 25 3.1TheJumpoperatorrevisited.............................. 25 -
These Are First-Draft Lecture Notes Issued for an Optional PG Theory
DÖafØ ß ÒÓØ fÓÖ diרÖibÙØiÓÒ These are first-draft lecture notes issued for an optional PG theory course in Edinburgh. The course was concerned with descriptive hierarchies, specifically the arithmetical, analytic, and fixpoint hierarchies. The main aim was to go through the proof of the modal mu-calculus alternation hierarchy; on the way, a quick survey of the basics of effective descriptive set theory is given, together with a few of the more amusing set-theoretic facts. – JCB 1 Introduction Let us recall some basic facts about computable functions on the natural numbers, which were covered in earlier courses. We know that a (possibly partial) function is computable by a Turing machine iff it can be defined by means of a small set of operators, including in particular primitive recursion and minimalization. Accordingly we call a set of numbers ‘recursive’ if its characteristic function is computable. We know that there exist non-recursive sets and functions; indeed, we know that there are sets that are not even semi-recursive. (Recall that a set is semi-recursive if there is a program which confirms membership, even if it doesn’t necessarily terminate on a non-member. Equivalently, we can list all the members of the set; hence the equivalent term ‘recursively enumerable’, or r.e.) Given this, we might reasonably ask, what can we say about these ‘hard’ non-r.e. sets? How complicated can they be? And what is the right notion of complicated, anyway? One approach to answering these questions is to keep our focus on the notation of compu- tation, and start asking questions such as: if somebody gives me some non-recursive set as an oracle, what other sets can I then compute, using a standard Turing machine together with the oracle? If I then take a set I can’t even compute with my oracle, and then add that as an oracle, what happens? How far can I go on? Can I order sets in terms of the strength of the oracles needed to compute them? And so on. -
Math 117C Notes: Effective
Math 117c notes: Effective DST Forte Shinko May 27, 2019 1 Overview of effective DST Effective descriptive set theory is a field at the intersection of computability theory and descriptive set theory. It can often help us prove results in descriptive set theory which are not easily handled with the classical machinery. First of all, recall the arithmetical hierarchy on N: 0 0 0 Σ1 Σ2 Σ3 0 0 0 ∆1 ∆2 ∆3 ··· 0 0 0 Π1 Π2 Π3 The insight of effective DST is relate this hierarchy to the Borel hierarchy: 0 0 0 Σ1 Σ2 Σ3 0 0 0 ∆1 ∆2 ∆3 ··· 0 0 0 Π1 Π2 Π3 These pointclasses are defined as follows (on any topological space): 0 1. A set is Σ1 if it is open. 0 0 2. A set is Πn if its complement is Σn. 0 0 3. A set is Σn+1 if it is the union of countably many Πn sets. 0 0 0 4. A set is ∆n if it is both Σn and Πn. Here is an outline of the corresponding notions between the fields: Computability theory Effective DST Classical DST N Baire space N := N! arithmetical hierarchy first ! levels of Borel hierarchy 0 0 0 0 0 0 Σα; Πα; ∆α Σα; Πα; ∆α computably enumerable set effectively open set open set computable function continuous function hyperarithmetical set Borel set computable ordinal countable ordinal CK !1 !1 analytical hierarchy projective hierarchy 1 1 1 1 1 1 Σn; Πn; ∆n Σn; Πn; ∆n 1 2 Effective open sets ! <! Recall that the Baire space N := N has a basic open set Ns for every s 2 N , defined by Ns := fx 2 N : x sg The basic open sets in N are of the form fng for n 2 N. -
Matheology.Pdf
Matheology § 001 A matheologian is a man, or, in rare cases, a woman, who believes in thoughts that nobody can think, except, perhaps, a God, or, in rare cases, a Goddess. § 002 Can the existence of God be proved from mathematics? Gödel proved the existence of God in a relatively complicated way using the positive and negative properties introduced by Leibniz and the axiomatic method ("the axiomatic method is very powerful", he said with a faint smile). http://www.stats.uwaterloo.ca/~cgsmall/ontology.html http://userpages.uni-koblenz.de/~beckert/Lehre/Seminar-LogikaufAbwegen/graf_folien.pdf Couldn't the following simple way be more effective? 1) The set of real numbers is uncountable. 2) Humans can only identify countably many words. 3) Humans cannot distinguish what they cannot identify. 4) Humans cannot well-order what they cannot distinguish. 5) The real numbers can be well-ordered. 6) If this is true, then there must be a being with higher capacities than any human. QED [I K Rus: "Can the existence of god be proved from mathematics?", philosophy.stackexchange, May 1, 2012] http://philosophy.stackexchange.com/questions/2702/can-the-existence-of-god-be-proved-from- mathematics The appending discussion is not electrifying for mathematicians. But a similar question had been asked by I K Rus in MathOverflow. There the following more educational discussion occurred (unfortunately it is no longer accessible there). (3) breaks down, because although I can't identify (i.e. literally "list") every real number between 0 and 1, if I am given any two real numbers in that interval then I can distinguish them. -
The Theory of Countable Analytical Sets Author(S): Alexander S
The Theory of Countable Analytical Sets Author(s): Alexander S. Kechris Source: Transactions of the American Mathematical Society, Vol. 202 (Feb., 1975), pp. 259-297 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1997311 . Accessed: 22/05/2013 14:36 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . 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]. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society. http://www.jstor.org This content downloaded from 131.215.71.79 on Wed, 22 May 2013 14:36:53 PM All use subject to JSTOR Terms and Conditions TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 202, 1975 THE THEORYOF COUNTABLEANALYTICAL SETS(1) BY ALEXANDER S. KECHRIS ABSTRACT. The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable n set of reals, en (this is also true for n even, replacing 1n by En and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). -
Welch, P. D. (2018). the Ramified Analytical Hierarchy Using Extended Logics
Welch, P. D. (2018). The Ramified Analytical Hierarchy Using Extended Logics. Bulletin of Symbolic Logic, 24(3), 306-318 . https://doi.org/10.1017/bsl.2018.69 Peer reviewed version Link to published version (if available): 10.1017/bsl.2018.69 Link to publication record in Explore Bristol Research PDF-document This is the author accepted manuscript (AAM). The final published version (version of record) is available online via Cambridge University Press at https://doi.org/10.1017/bsl.2018.69 . Please refer to any applicable terms of use of the publisher. University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ THE RAMIFIED ANALYTICAL HIERARCHY USING EXTENDED LOGICS P.D. WELCH Abstract. The use of Extended Logics to replace ordinary second order definability in Kleene’s Ramified Analytical Hierarchy is investigated. This mirrors a similar investigation of Kennedy, Magidor and Va¨an¨ anen¨ [11] where Godel’s¨ universe L of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain minimal correct models of analysis. A wide spectrum of models can be so generated from abstract definability notions: one may take an ab- stract Spector Class and extract an extended logic for it. -
Measure and Category in Effective Descriptive Set Theory
ANNALS OF MATHEMATICAL LOGIC 5 (1973) 337-384. North-Holland Publishing Company MEASURE AND CATEGORY IN EFFECTIVE DESCRIPTIVE SET THEORY Alexander S. KECHRIS * Massachusetts Institl~te of Technology, Cambridge, Mass., USA Received 14 February 1973 O, Introduction We are concerned in this paper with some definability aspects of the theory of measure and category on the continuum. Our objects of study are the projective subsets of the reals and their structure from a measure theoretic and topological point of view. Typical problems we attack here and some of the results we prove are the following: (A) Computation of the complexity (in terms of the analytical hierar- chy) of the measure of an analytical set of reals. We prove that the mea- sure of a Z l set of reals is a Z~ real. We assume here the existence of a measurable cardi~,~al if n -- 2 and Projective Determinacy if n > 2. This applies also to the theorems stated below. (B) Effective approximations~ A representative result here is that a ~nl set (n even) or I1~ set (n odd) can be approximated 'arbitrarily close' by a A]a subset of it. (C) Existence of nice bases. The most significant fact we establish here is that the Anl reals (n odd) are a basis for the IIn1 sets of reals which have positive measure (or are not meager). * Some of the main results in this paper have been announced in [ 1 ! ] (The final statement ~n this abstract about ~n + l reals is wtong~) Part of Section 4 has been also included in the au- thor's Ph.D, thesis (UCLA, June 1972) written under the supervision of Professor Y.N. -
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 . -
Introduction to Classical Descriptive Set Theory
Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Introduction to Classical Descriptive Set Theory J.T. Chipman Logic Seminar Stanford University 13 October 2015 J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References \The efforts that I exerted led me to the following totally unexpected discovery: there exists a family [...] consisting of effective [i.e., definable] sets, such that one does not know and one will never know whether every set from this family, if uncountable, has the cardinality of the continuum, nor whether it [has the property of Baire], nor whether it is measurable. [...] This is the family of the projective sets of Mr. H. Lebesgue. It remains but to recognize the nature of this new development." (Luzin (1925)) J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Table of contents 1 Early Development 2 Definable Sets of Reals Borel Sets Projective Sets L(R) 3 Regularity Properties Lebesgue Measurability Property of Baire Perfect Set Property Distribution of Regularity Properties 4 Metamathematical Issues 5 References J.T. Chipman Logic Seminar Stanford University Introduction to Classical Descriptive Set Theory Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References Kanamori summarizes the subject matter of Descriptive Set Theory succinctly: \Descriptive Set Theory is the definability theory of the continuum." Concerning the origins of the theory, Moschovakis writes in the opening lines of his textbook: \The roots of Descriptive Set Theory go back to the work of Borel, Baire, and Lebesgue around the turn of the 20th century, when the young French analysts were trying to come to grips with the abstract notion of a function. -
Arxiv:1308.6318V1 [Math.LO] 28 Aug 2013 Date .Tecascleetv Orsodne27 26 Theory Set Descriptive Invariant 11 Correspondence Pointclasses Classical-Effective 6
ASPECTS OF CLASSICAL DESCRIPTIVE SET THEORY ALEX GALICKI SUPERVISORS: PROF. DAVID GAULD, PROF. ANDRE´ NIES Abstract. This report consists of two parts. The first part is a brief exposi- tion of classical descriptive set theory. This part introduces some fundamental concepts, motivations and results from the classical theory and ends with a section on the important result of Addison that established the correspondence between classical and effective notions. The second part is about invariant descriptive set theory. It consists of a brief introduction to notions relevant to this branch of descriptive set theory and then describes in details some of the relatively recent results concerning equivalence relations between Polish metric spaces. Contents 0.1. What is Descriptive Set Theory? 3 0.2. Historical background 3 0.3. Acknowledgements 4 0.4. Notation index 4 1. Regularity properties of sets of reals 5 1.1. The Baire property 6 1.2. Lebesgue measurability 7 1.3. The perfect set property 9 2. Polish spaces 10 2.1. Why Polish spaces? 10 2.2. Basic properties 11 2.3. TheBairespaceandtheCantorspace 11 2.4. Trees 12 2.5. Binary expansions 13 arXiv:1308.6318v1 [math.LO] 28 Aug 2013 2.6. Universal Polish spaces 13 2.7. Choquet spaces 14 3. Borel sets 15 3.1. Borel hierarchy 15 3.2. Borel spaces 21 4. Projective sets 23 4.1. Analytic sets 23 4.2. The Suslin operation A 24 4.3. The Projective Hierarchy 26 4.4. Regularity properties of the projective pointclasses 26 5. The classical-effective correspondence 27 6. Invariant Descriptive Set Theory 29 Date: 30th August 2013.