Ordinal Analysis Without Proofs
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The Interplay Between Computability and Incomputability Draft 619.Tex
The Interplay Between Computability and Incomputability Draft 619.tex Robert I. Soare∗ January 7, 2008 Contents 1 Calculus, Continuity, and Computability 3 1.1 When to Introduce Relative Computability? . 4 1.2 Between Computability and Relative Computability? . 5 1.3 The Development of Relative Computability . 5 1.4 Turing Introduces Relative Computability . 6 1.5 Post Develops Relative Computability . 6 1.6 Relative Computability in Real World Computing . 6 2 Origins of Computability and Incomputability 6 2.1 G¨odel’s Incompleteness Theorem . 8 2.2 Incomputability and Undecidability . 9 2.3 Alonzo Church . 9 2.4 Herbrand-G¨odel Recursive Functions . 10 2.5 Stalemate at Princeton Over Church’s Thesis . 11 2.6 G¨odel’s Thoughts on Church’s Thesis . 11 ∗Parts of this paper were delivered in an address to the conference, Computation and Logic in the Real World, at Siena, Italy, June 18–23, 2007. Keywords: Turing ma- chine, automatic machine, a-machine, Turing oracle machine, o-machine, Alonzo Church, Stephen C. Kleene,klee Alan Turing, Kurt G¨odel, Emil Post, computability, incomputabil- ity, undecidability, Church-Turing Thesis (CTT), Post-Church Second Thesis on relative computability, computable approximations, Limit Lemma, effectively continuous func- tions, computability in analysis, strong reducibilities. 1 3 Turing Breaks the Stalemate 12 3.1 Turing’s Machines and Turing’s Thesis . 12 3.2 G¨odel’s Opinion of Turing’s Work . 13 3.3 Kleene Said About Turing . 14 3.4 Church Said About Turing . 15 3.5 Naming the Church-Turing Thesis . 15 4 Turing Defines Relative Computability 17 4.1 Turing’s Oracle Machines . -
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. -
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 . -
On Relating Theories: Proof-Theoretical Reduction
This is a repository copy of On Relating Theories: Proof-Theoretical Reduction. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/147184/ Version: Accepted Version Proceedings Paper: Rathjen, M and Toppel, M (2019) On Relating Theories: Proof-Theoretical Reduction. In: Centrone, S, Negri, S, Sarikaya, D and Schuster, PM, (eds.) Mathesis Universalis, Computability and Proof. Humboldt-Kolleg: Proof Theory as Mathesis Universalis, 24-28 Jul 2017, Loveno di Menaggio (Como), Italy. Springer , pp. 311-331. ISBN 978-3-030-20446-4 https://doi.org/10.1007/978-3-030-20447-1_16 © Springer Nature Switzerland AG 2019. This is an author produced version of a paper published in Mathesis Universalis, Computability and Proof. Uploaded in accordance with the publisher's self-archiving policy. Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ On relating theories: Proof-theoretical reduction Michael Rathjen and Michael Toppel Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom E-Mail: [email protected] Abstract The notion of proof-theoretical or finitistic reduction of one theory to another has a long tradition. -
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory Robert I. Soare∗ January 3, 2009 Contents 1 Introduction 4 1.1 Terminology: Incompleteness and Incomputability . 4 1.2 The Goal of Incomputability not Computability . 5 1.3 Computing Relative to an Oracle or Database . 5 1.4 Continuous Functions and Calculus . 6 2 Origins of Computability and Incomputability 7 2.1 G¨odel'sIncompleteness Theorem . 7 2.2 Alonzo Church . 8 2.3 Herbrand-G¨odelRecursive Functions . 9 2.4 Stalemate at Princeton Over Church's Thesis . 10 2.5 G¨odel'sThoughts on Church's Thesis . 11 3 Turing Breaks the Stalemate 11 3.1 Turing Machines and Turing's Thesis . 11 3.2 G¨odel'sOpinion of Turing's Work . 13 3.2.1 G¨odel[193?] Notes in Nachlass [1935] . 14 3.2.2 Princeton Bicentennial [1946] . 15 ∗Parts of this paper were delivered in an address to the conference, Computation and Logic in the Real World, at Siena, Italy, June 18{23, 2007. Keywords: Turing ma- chine, automatic machine, a-machine, Turing oracle machine, o-machine, Alonzo Church, Stephen C. Kleene, Alan Turing, Kurt G¨odel, Emil Post, computability, incomputability, undecidability, Church-Turing Thesis, Post-Turing Thesis on relative computability, com- putable approximations, Limit Lemma, effectively continuous functions, computability in analysis, strong reducibilities. Thanks are due to C.G. Jockusch, Jr., P. Cholak, and T. Slaman for corrections and suggestions. 1 3.2.3 The Flaw in Church's Thesis . 16 3.2.4 G¨odelon Church's Thesis . 17 3.2.5 G¨odel'sLetter to Kreisel [1968] . -
An Ordinal Analysis of Admissible Set Theory Using Recursion on Ordinal Notations∗
An ordinal analysis of admissible set theory using recursion on ordinal notations¤ Jeremy Avigad May 23, 2001 Abstract The notion of a function from N to N defined by recursion on ordinal notations is fundamental in proof theory. Here this notion is generalized to functions on the universe of sets, using notations for well-orderings longer than the class of ordinals. The generalization is used to bound the rate of growth of any function on the universe of sets that is Σ1-definable in Kripke-Platek admissible set theory with an axiom of infinity. Formalizing the argument provides an ordinal analysis. 1 Introduction In informal proof-theoretic parlance, the definition of a set of objects is said to be impredicative if it makes reference to a collection of sets that includes the set being defined. A classic example arises if one takes the real numbers to be lower Dedekind cuts of rationals, and then defines the least upper bound of a bounded set of reals to be the intersection of all the upper bounds. A theory is said to be (prima facie) impredicative if its intended interpretation depends on such a definition. The circularity implicit in an impredicative theory poses problems for its ordinal analysis, since the goal of ordinal analysis is to measure the theory’s strength in terms of well-founded ordinal notations — that is, “from the bot- tom up.” For that reason, the first ordinal analyses of impredicative theories, due to Takeuti, Buchholz, and Pohlers were a landmark (see the discussion in the introduction to [7]). Another important step was the move to studying fragments of set theory instead of second-order arithmetic, carried out by J¨ager [12, 13, 14], providing a more natural framework for the analysis of impredica- tivity. -
Cardinals and Intro to Ordinals Notations
Cardinals and Intro to Ordinals Notations Pete Manolios Northeastern Formal Methods, Lecture 5 September 2008 1 Last Time: Axiomatic Set Theory Axiom 0. Set Existence: ⟨∃x :: x=x⟩ Axiom 1. Extensionality: ⟨∀x,y::⟨∀z :: z∈x ≡ z∈y⟩ ⇒ x=y⟩ Axiom 3. Comprehension Scheme: For each formula φ without y free, the universal closure of the following is an axiom: ⟨∃y :: ⟨∀x :: x∈y ≡ x∈z ∧ φ⟩⟩ Axiom 4. Pairing: ⟨∀x,y :: ⟨∃z :: x∈z ∧ y ∈z⟩⟩ Axiom 5. Union: ⟨∀F :: ⟨∃A :: ⟨∀x,Y :: (Y∈F ∧ x∈Y) ⇒ x∈A⟩⟩⟩ Axiom 6. Replacement Scheme. For each formula φ without Y free, the universal closure of the following is an axiom: ⟨∀x∈A :: ⟨∃!z:: φ(x,z)⟩⟩ ⇒ ⟨∃Y:: ⟨∀x∈A :: ⟨∃z∈Y :: φ(x,z)⟩⟩⟩ Axiom 7. Infinity: ⟨∃x:: 0∈x ∧ ⟨∀y∈x :: y+ ∈ x⟩⟩ (y+ = y ⋃ {y}) Axiom 9. Choice. ⟨∀A :: ⟨∃R :: R well-orders A⟩⟩ 2 Hey, What About Axiom 2? Axiom 2: Foundation. Sets are well-founded under ∈ ⟨∀x :: ⟨∃y∈x⟩ ⇒ ⟨∃y∈x :: ¬⟨∃z∈x :: z∈y⟩⟩⟩ y is a minimal element above See handout for an alternate definition of foundation There are set theories where foundation is omitted So, we can have x∈x The problem is: when are two sets equal? What would you do? 3 Cardinals (& Axiom 8) |A| is the least ordinal α s.t. α≈A (≈ means bijection) α is a cardinal iff α = |α| A cardinal number is an ordinal number whose size is > any of its elements A is finite iff |A| < ω A is infinite iff it is not finite A is countable iff |A| ≤ ω A is uncountable iff it is not countable We can’t prove that uncountable sets exist Axiom 8. -
To Infinity and Beyond
To Infinity and Beyond Some Notes on Set Theory and Transfinite Numbers Prerna Nadathur August 15, 2008 Contents 1 Introduction 1 2 The Axioms 2 2.1 The Axioms . 2 2.2 Some Consequences . 4 2.2.1 Intersections, Unions and Complements . 4 2.2.2 Ordered Pairs and Cartesian Products . 9 2.2.3 Relations and Functions . 11 2.3 An Axiom for the Natural Numbers . 13 3 Order 14 4 Ordinals and Cardinals 16 4.1 Ordinals . 16 4.1.1 Ordinal Numbers . 16 4.1.2 Ordinal Arithmetic . 18 4.1.3 Countability . 19 4.2 Cardinals . 22 4.3 Cofinality . 26 5 The Axiom of Choice, Zorn's Lemma, and the Well-Ordering Theorem 27 1 Introduction Perhaps the first problem we encounter in set theory is that of the definition of a set. It is a remarkable (although not altogether curious) fact that most books on set theory either actively decline to define a set, or else neglect to mention the problem of definition altogether. Intuitively, we know that sets contain things{elements, or objects, and so in some sense, we 1 can define a set as a collection of objects. Yet this can lead us into trouble: we shall see that there are collections of objects that are not sets, that are \too big" to be sets, that create paradoxes if we regard them as sets. For example, although Cantor's original development of set theory was not axiomatic, it is clear that he relied implicitly on three assumptions: the axiom of extension (more later), the axiom of abstraction, and the axiom of choice. -
Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions
This is a repository copy of Constructive zermelo-fraenkel set theory, power set, and the calculus of constructions. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/75182/ Book Section: Rathjen, M (2012) Constructive zermelo-fraenkel set theory, power set, and the calculus of constructions. In: Dybjer, P, Lindström, S, Palmgren, E and Sundholm, G, (eds.) Epistemology versus ontology: Essays on the philosophy and foundations of mathematics in honour of Per Martin-Löf. Logic, Epistemology, and the Unity of Science, 27 . Springer , Dordrecht, Netherlands , 313 - 349. ISBN 9789400744356 https://doi.org/10.1007/978-94-007-4435-6 Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions Michael Rathjen∗ Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom [email protected] May 4, 2012 Abstract Full intuitionistic Zermelo-Fraenkel set theory, IZF, is obtained from constructive Zermelo- Fraenkel set theory, CZF, by adding the full separation axiom scheme and the power set axiom. -
Proof Theory of Martin-Löf Type Theory – an Overview
Proof Theory of Martin-L¨of Type Theory { An Overview Anton Setzer∗ February 22, 2004 Abstract We give an overview over the historic development of proof theory and the main techniques used in ordinal theoretic proof theory. We argue, that in a revised Hilbert's programme, ordinal theoretic proof theory has to be supplemented by a second step, namely the development of strong equiconsis- tent constructive theories. Then we show, how, as part of such a programme, the proof theoretic analysis of Martin-L¨of type theory with W-type and one microscopic universe containing only two finite sets is carried out. Then we look at the analysis of Martin-L¨of type theory with W-type and a universe closed under the W-type, and consider the extension of type theory by one Mahlo universe and its proof-theoretic analysis. Finally we repeat the con- cept of inductive-recursive definitions, which extends the notion of inductive definitions substantially. We introduce a closed formalisation, which can be used in generic programming, and explain, what is known about its strength. Keywords: Martin-L¨of type theory, proof theory, Kripke-Platek set theory, W-type, well-founded trees, Kleene's O, Mahlo universe, inductive-recursive definitions, generic programming. 1 Introduction The goal of this article is to introduce the reader, who is not necessarily an expert in proof theory, into the current state of the art of proof theory of Martin-L¨of type theory and the techniques used there. We start by giving a brief overview over the contents of this article. -
Ordinal Notation
Dmytro Taranovsky Last Update: December 31, 2018 Ordinal Notation Abstract: We introduce a framework for ordinal notation systems, present a family of strong yet simple systems, and give many examples of ordinals in these systems. While much of the material is conjectural, we include systems with conjectured strength beyond second order arithmetic (and plausibly beyond ZFC), and prove well-foundedness for some weakened versions. Contents: 1 Introduction 1.1 Introduction 1.2 Prior Work 1.3 Goals of Ordinal Analysis 2 A Framework for Ordinal Notations — defines the general structure for the notations. 2.1 Definition of the General Notation 2.2 Basic Properties 2.3 Bachmann-Howard Ordinal 2.4 One Variable C 2.5 Reflection Configurations 3 Degrees of Recursive Inaccessibility — presents a notation system for ATR0 plus "for every ordinal a, there is recursively a-inaccessible ordinal". 3.1 Definition of the Notation 3.2 Comparison Relation 3.3 Examples 4 Degrees of Reflection — presents a notation system for KP + Πn reflection and somewhat stronger systems, with the canonical assignment for nonrecursive ordinals. 4.1 Definition of Degrees of Reflection 4.2 Examples and Additional Properties 4.3 Assignment of Degrees 4.4 A Step towards Second Order Arithmetic 5 Main Ordinal Notation System — presents the main notation system, which plausibly reaches second order arithmetic (Z2) or higher. 5.1 Definition and Basic Properties 5.2 Strength of the Main System 5.3 Old Analysis and Examples 6 Beyond Second Order Arithmetic — gives insights for going beyond second order arithmetic and ZFC 6.1 Old Analysis 6.2 New Analysis 7 Iteration of n-built from below — proposes a strengthening of the main system. -
Replacement Versus Collection and Related Topics in Constructive Zermelo-Fraenkel Set Theory
Replacement versus Collection and related topics in Constructive Zermelo-Fraenkel Set Theory Michael Rathjen¤ Department of Mathematics, Ohio State University Columbus, OH 43210, U.S.A. [email protected] May 29, 2005 Abstract While it is known that intuitionistic ZF set theory formulated with Replacement, IZFR, does not prove Collection it is a longstanding open problem whether IZFR and intuitionistic set theory ZF formu- lated with Collection, IZF, have the same proof-theoretic strength. It has been conjectured that IZF proves the consistency of IZFR. This paper addresses similar questions but in respect of constructive Zermelo-Fraenkel set theory, CZF. It is shown that in the latter con- text the proof-theoretic strength of Replacement is the same as that of Strong Collection and also that the functional version of the Regular Extension Axiom is as strong as its relational version. Moreover, it is proved that, contrary to IZF, the strength of CZF increases if one adds an axiom asserting that the trichotomous ordinals form a set. Unlike IZF, constructive Zermelo-Fraenkel set theory is amenable to ordinal analysis and the proofs in this paper make pivotal use thereof in the guise of well-ordering proofs for ordinal representation systems. MSC: 03F50, 03F35 Keywords: Constructive set theory, replacement, strong collection, proof-theoretic strength, trichotomous ordinals 1 Introduction Zermelo-Fraenkel set theory, ZF, is formulated with Replacement rather than Collection, but it is readily seen that Collection is deducible in ZF. The proof, though, makes essential use of classical logic. On the other hand, it was shown in [12] that intuitionistic ZF set theory formulated ¤This material is based upon work supported by the National Science Foundation under Award No.