The Art of Ordinal Analysis
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Intrinsic Bounds on Complexity and Definability at Limit Levels*
Intrinsic bounds on complexity and definability at limit levels∗ John Chisholm Ekaterina Fokina Department of Mathematics Department of Mathematics Western Illinois University Novosibirsk State University Macomb, IL 61455 630090 Novosibirsk, Russia [email protected] [email protected] Sergey S. Goncharov Academy of Sciences, Siberian Branch Mathematical Institute 630090 Novosibirsk, Russia [email protected] Valentina S. Harizanov Julia F. Knight Department of Mathematics Department of Mathematics George Washington University University of Notre Dame [email protected] [email protected] Sara Miller Department of Mathematics University of Notre Dame [email protected] May 31, 2007 Abstract We show that for every computable limit ordinal α,thereisacom- 0 0 putable structure that is ∆α categorical, but not relatively ∆α categor- A 0 ical, i.e., does not have a formally Σα Scott family. We also show that for every computable limit ordinal α, there is a computable structure with 0 A an additional relation R that is intrinsically Σα on , but not relatively 0 A intrinsically Σ on , i.e., not definable by a computable Σα formula with α A ∗The authors gratefully acknowledge support of the Charles H. Husking endowment at the University of Notre Dame. The last five authors were also partially supported by the NSF binational grant DMS-0554841, and the fourth author by the NSF grant DMS-0704256. In addition, the second author was supported by a grant of the Russian Federation as a 2006-07 research visitor to the University of Notre Dame. 1 finitely many parameters. Earlier results in [7], [10], [8] establish the same facts for computable successor ordinals α. -
Arxiv:2010.12452V1 [Math.LO]
ORDINAL ANALYSIS OF PARTIAL COMBINATORY ALGEBRAS PAUL SHAFER AND SEBASTIAAN A. TERWIJN Abstract. For every partial combinatory algebra (pca), we define a hierarchy of extensionality rela- tions using ordinals. We investigate the closure ordinals of pca’s, i.e. the smallest ordinals where these CK relations become equal. We show that the closure ordinal of Kleene’s first model is ω1 and that the closure ordinal of Kleene’s second model is ω1. We calculate the exact complexities of the extensionality relations in Kleene’s first model, showing that they exhaust the hyperarithmetical hierarchy. We also discuss embeddings of pca’s. 1. Introduction Partial combinatory algebras (pca’s) were introduced by Feferman [9] in connection with the study of predicative systems of mathematics. Since then, they have been studied as abstract models of computation, in the same spirit as combinatory algebras (defined and studied long before, in the 1920’s, by Sch¨onfinkel and Curry) and the closely related lambda calculus, cf. Barendregt [2]. As such, pca’s figure prominently in the literature on constructive mathematics, see e.g. Beeson [5] and Troelstra and van Dalen [22]. This holds in particular for the theory of realizability, see van Oosten [24]. For example, pca’s serve as the basis of various models of constructive set theory, first defined by McCarty [14]. See Rathjen [18] for further developments using this construction. A pca is a set A equipped with a partial application operator · that has the same properties as a classical (total) combinatory algebra (see Section 2 below for precise definitions). In particular, it has the combinators K and S. -
Computable Structure Theory on Ω 1 Using Admissibility
COMPUTABLE STRUCTURE THEORY USING ADMISSIBLE RECURSION THEORY ON !1 NOAM GREENBERG AND JULIA F. KNIGHT Abstract. We use the theory of recursion on admissible ordinals to develop an analogue of classical computable model theory and effective algebra for structures of size @1, which, under our assumptions, is equal to the contin- uum. We discuss both general concepts, such as computable categoricity, and particular classes of examples, such as fields and linear orderings. 1. Introduction Our aim is to develop computable structure theory for uncountable structures. In this paper we focus on structures of size @1. The fundamental decision to be made, when trying to formulate such a theory, is the choice of computability tools that we intend to use. To discover which structures are computable, we need to first describe which subsets of the domain are computable, and which functions are computable. In this paper, we use admissible recursion theory (also known as α-recursion theory) over the domain !1. We believe that this choice yields an interesting computable structure theory. It also illuminates the concepts and techniques of classical computable structure theory by observing similarities and differences between the countable and uncountable settings. In particular, it seems that as is the case for degree theory and for the study of the lattice of c.e. sets, the difference between true finiteness and its analogue in the generalised case, namely countability in our case, is fundamental to some constructions and reveals a deep gap between classical computability and attempts to generalise it to the realm of the uncountable. In this paper, we make a sweeping simplifying assumption about our set-theoretic universe. -
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 . -
Measures and Their Random Reals
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 7, July 2015, Pages 5081–5097 S 0002-9947(2015)06184-4 Article electronically published on January 30, 2015 MEASURES AND THEIR RANDOM REALS JAN REIMANN AND THEODORE A. SLAMAN Abstract. We study the randomness properties of reals with respect to ar- bitrary probability measures on Cantor space. We show that every non- computable real is non-trivially random with respect to some measure. The probability measures constructed in the proof may have atoms. If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that every non-hyperarithmetical real is random for a continuous measure. On the other hand, examples of reals not random for any continuous measure can be found throughout the hyperarithmetical Turing degrees. 1. Introduction Over the past decade, the study of algorithmic randomness has produced an impressive number of results. The theory of Martin-L¨of random reals, with all its ramifications (e.g. computable or Schnorr randomness, lowness and triviality) has found deep and significant applications in computability theory, many of which are covered in recent books by Downey and Hirschfeldt [5] and Nies [23]. Usually, the measure for which randomness is considered in these studies is the uniform (1/2, 1/2)-measure on Cantor space, which is measure theoretically isomorphic to Lebesgue measure on the unit interval. However, one may ask what happens if one changes the underlying measure. It is easy to define a generalization of Martin-L¨of tests which allows for a definition of randomness with respect to arbitrary computable measures. -
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.