Model Theory, Volume 47, Number 11
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Mathematisches Forschungsinstitut Oberwolfach Computability Theory
Mathematisches Forschungsinstitut Oberwolfach Report No. 08/2012 DOI: 10.4171/OWR/2012/08 Computability Theory Organised by Klaus Ambos-Spies, Heidelberg Rodney G. Downey, Wellington Steffen Lempp, Madison Wolfgang Merkle, Heidelberg February 5th – February 11th, 2012 Abstract. Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from G¨odel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science. Mathematics Subject Classification (2000): 03Dxx, 68xx. Introduction by the Organisers The workshop Computability Theory, organized by Klaus Ambos-Spies and Wolf- gang Merkle (Heidelberg), Steffen Lempp (Madison) and Rodney G. Downey (Wellington) was held February 5th–February 11th, 2012. This meeting was well attended, with 53 participants covering a broad geographic representation from five continents and a nice blend of researchers with various backgrounds in clas- sical degree theory as well as algorithmic randomness, computable model theory and reverse mathematics, reaching into theoretical computer science, model the- ory and algebra, and proof theory, respectively. Several of the talks announced breakthroughs on long-standing open problems; others provided a great source of important open problems that will surely drive research for several years to come. Computability Theory 399 Workshop: Computability Theory Table of Contents Carl Jockusch (joint with Rod Downey and Paul Schupp) Generic computability and asymptotic density ....................... 401 Laurent Bienvenu (joint with Andrei Romashchenko, Alexander Shen, Antoine Taveneaux, and Stijn Vermeeren) Are random axioms useful? ....................................... 403 Adam R. Day (joint with Joseph S. Miller) Cupping with Random Sets ...................................... -
⊥N AS an ABSTRACT ELEMENTARY CLASS We Show
⊥N AS AN ABSTRACT ELEMENTARY CLASS JOHN T. BALDWIN, PAUL C. EKLOF, AND JAN TRLIFAJ We show that the concept of an Abstract Elementary Class (AEC) provides a unifying notion for several properties of classes of modules and discuss the stability class of these AEC. An abstract elementary class consists of a class of models K and a strengthening of the notion of submodel ≺K such that (K, ≺K) satisfies ⊥ the properties described below. Here we deal with various classes ( N, ≺N ); the precise definition of the class of modules ⊥N is given below. A key innovation is ⊥ that A≺N B means A ⊆ B and B/A ∈ N. We define in the text the main notions used here; important background defini- tions and proofs from the theory of modules can be found in [EM02] and [GT06]; concepts of AEC are due to Shelah (e.g. [She87]) but collected in [Bal]. The surprising fact is that some of the basic model theoretic properties of the ⊥ ⊥ class ( N, ≺N ) translate directly to algebraic properties of the class N and the module N over the ring R that have previously been studied in quite a different context (approximation theory of modules, infinite dimensional tilting theory etc.). Our main results, stated with increasingly strong conditions on the ring R, are: ⊥ Theorem 0.1. (1) Let R be any ring and N an R–module. If ( N, ≺N ) is an AEC then N is a cotorsion module. Conversely, if N is pure–injective, ⊥ then the class ( N, ≺N ) is an AEC. (2) Let R be a right noetherian ring and N be an R–module of injective di- ⊥ ⊥ mension ≤ 1. -
The Metamathematics of Putnam's Model-Theoretic Arguments
The Metamathematics of Putnam's Model-Theoretic Arguments Tim Button Abstract. Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted mod- els. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I exam- ine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to meta- mathematical challenges. Copyright notice. This paper is due to appear in Erkenntnis. This is a pre-print, and may be subject to minor changes. The authoritative version should be obtained from Erkenntnis, once it has been published. Hilary Putnam famously attempted to use model theory to draw metaphys- ical conclusions. Specifically, he attacked metaphysical realism, a position characterised by the following credo: [T]he world consists of a fixed totality of mind-independent objects. (Putnam 1981, p. 49; cf. 1978, p. 125). Truth involves some sort of correspondence relation between words or thought-signs and external things and sets of things. (1981, p. 49; cf. 1989, p. 214) [W]hat is epistemically most justifiable to believe may nonetheless be false. (1980, p. 473; cf. 1978, p. 125) To sum up these claims, Putnam characterised metaphysical realism as an \externalist perspective" whose \favorite point of view is a God's Eye point of view" (1981, p. 49). Putnam sought to show that this externalist perspective is deeply untenable. To this end, he treated correspondence in terms of model-theoretic satisfaction. -
Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness . -
FOUNDATIONS of RECURSIVE MODEL THEORY Mathematics Department, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive
Annals of Mathematical Logic 13 (1978) 45-72 © North-H011and Publishing Company FOUNDATIONS OF RECURSIVE MODEL THEORY Terrence S. MILLAR Mathematics Department, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706, U.S.A. Received 6 July 1976 A model is decidable if it has a decidable satisfaction predicate. To be more precise, let T be a decidable theory, let {0, I n < to} be an effective enumeration of all formula3 in L(T), and let 92 be a countable model of T. For any indexing E={a~ I i<to} of I~1, and any formula ~eL(T), let '~z' denote the result of substituting 'a{ for every free occurrence of 'x~' in q~, ~<o,. Then 92 is decidable just in case, for some indexing E of 192[, {n 192~0~ is a recursive set of integers. It is easy, to show that the decidability of a model does not depend on the choice of the effective enumeration of the formulas in L(T); we omit details. By a simple 'effectivizaton' of Henkin's proof of the completeness theorem [2] we have Fact 1. Every decidable theory has a decidable model. Assume next that T is a complete decidable theory and {On ln<to} is an effective enumeration of all formulas of L(T). A type F of T is recursive just in case {nlO, ~ F} is a recursive set of integers. Again, it is easy to see that the recursiveness of F does not depend .on which effective enumeration of L(T) is used. -
(Aka “Geometric”) Iff It Is Axiomatised by “Coherent Implications”
GEOMETRISATION OF FIRST-ORDER LOGIC ROY DYCKHOFF AND SARA NEGRI Abstract. That every first-order theory has a coherent conservative extension is regarded by some as obvious, even trivial, and by others as not at all obvious, but instead remarkable and valuable; the result is in any case neither sufficiently well-known nor easily found in the literature. Various approaches to the result are presented and discussed in detail, including one inspired by a problem in the proof theory of intermediate logics that led us to the proof of the present paper. It can be seen as a modification of Skolem's argument from 1920 for his \Normal Form" theorem. \Geometric" being the infinitary version of \coherent", it is further shown that every infinitary first-order theory, suitably restricted, has a geometric conservative extension, hence the title. The results are applied to simplify methods used in reasoning in and about modal and intermediate logics. We include also a new algorithm to generate special coherent implications from an axiom, designed to preserve the structure of formulae with relatively little use of normal forms. x1. Introduction. A theory is \coherent" (aka \geometric") iff it is axiomatised by \coherent implications", i.e. formulae of a certain simple syntactic form (given in Defini- tion 2.4). That every first-order theory has a coherent conservative extension is regarded by some as obvious (and trivial) and by others (including ourselves) as non- obvious, remarkable and valuable; it is neither well-enough known nor easily found in the literature. We came upon the result (and our first proof) while clarifying an argument from our paper [17]. -
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. -
Ultraproducts and Elementary Classes *
MATHEMATICS ULTRAPRODUCTS AND ELEMENTARY CLASSES * BY H. JEROME KEISLER (Communicated by Prof. A. HEYTING at the meeting of May 27, 1961 INTRODUCTION In the theory of models of the first order predicate logic with identity we study the relationship between notions from logic and purely mathe matical objects, called relational systems, or (first order) structures, m, consisting of a set A and a sequence of relations on A corresponding to the predicate symbols of the logic. In such studies some of the most useful methods are those which permit us to form structures which have certain semantical properties. For example, the completeness theorem permits us to form a structure which satisfies a given consistent set of sentences. A more recent tool is the reduced product operation, which is particularly simple and direct from the mathematical point of view. Speaking very roughly, a reduced product Iliei ill:i/D of the structures mi, i El, is formed by first taking their direct product and then forming the quotient system determined in a certain way by the filter D on the index set 1. If D is an ultrafilter we have an ultraproduct (or prime reduced product), and if each m:i coincides with a fixed system ill: we have a reduced power, or ultrapower, ill: I j D. Ultraproducts have recently been applied to obtain new, comparatively direct mathematical proofs and at the same time stronger forms of a number of theorems in model theory. An outstanding example is the proof of the compactness theorem outlined in [33]. They have also been used to obtain several completely new results in the theory of models (for example, see [35]). -
Integer-Valued Definable Functions
Integer-valued definable functions G. O. Jones, M. E. M. Thomas and A. J. Wilkie ABSTRACT We present a dichotomy, in terms of growth at infinity, of analytic functions definable in the real exponential field which take integer values at natural number inputs. Using a result concerning the density of rational points on curves definable in this structure, we show that if a definable, k analytic function 1: [0, oo)k_> lR is such that 1(N ) C;;; Z, then either sUPlxl(r 11(x)1 grows faster than exp(r"), for some 15 > 0, or 1 is a polynomial over <Ql. 1. Introduction The results presented in this note concern the growth at infinity of functions which are analytic and definable in the real field expanded by the exponential function and which take integer values on N. They are descendants of a theorem of P6lya [8J from 1915 concerning integer valued, entire functions on <C. This classic theorem tells us that 2z is, in some sense, the smallest non-polynomialfunction of this kind. More formally, letm(f,r) := sup{lf(z)1 : Izl ~ r}. P6lya's Theorem, refined in [3, 9J, states the following. THEOREM 1.1 [9, Theorem IJ. Iff: C --, C is an entire function which satisfies both f(N) <::: Z and . m(f, r) 11m sup < 1, r-->x 2r then f is a polynomial. This result cannot be directly transferred to the real analytic setting (for example, consider the function f (x) = sin( K::C)). In fact, it appears that there are very few results of this character known for such functions (some rare examples may be found in [1]). -
Geometry and Categoricity∗
Geometry and Categoricity∗ John T. Baldwiny University of Illinois at Chicago July 5, 2010 1 Introduction The introduction of strongly minimal sets [BL71, Mar66] began the idea of the analysis of models of categorical first order theories in terms of combinatorial geometries. This analysis was made much more precise in Zilber’s early work (collected in [Zil91]). Shelah introduced the idea of studying certain classes of stable theories by a notion of independence, which generalizes a combinatorial geometry, and characterizes models as being prime over certain independent trees of elements. Zilber’s work on finite axiomatizability of totally categorical first order theories led to the development of geometric stability theory. We discuss some of the many applications of stability theory to algebraic geometry (focusing on the role of infinitary logic). And we conclude by noting the connections with non-commutative geometry. This paper is a kind of Whig history- tying into a (I hope) coherent and apparently forward moving narrative what were in fact a number of independent and sometimes conflicting themes. This paper developed from talk at the Boris-fest in 2010. But I have tried here to show how the ideas of Shelah and Zilber complement each other in the development of model theory. Their analysis led to frameworks which generalize first order logic in several ways. First they are led to consider more powerful logics and then to more ‘mathematical’ investigations of classes of structures satisfying appropriate properties. We refer to [Bal09] for expositions of many of the results; that’s why that book was written. It contains full historical references. -
Notes on Mathematical Logic David W. Kueker
Notes on Mathematical Logic David W. Kueker University of Maryland, College Park E-mail address: [email protected] URL: http://www-users.math.umd.edu/~dwk/ Contents Chapter 0. Introduction: What Is Logic? 1 Part 1. Elementary Logic 5 Chapter 1. Sentential Logic 7 0. Introduction 7 1. Sentences of Sentential Logic 8 2. Truth Assignments 11 3. Logical Consequence 13 4. Compactness 17 5. Formal Deductions 19 6. Exercises 20 20 Chapter 2. First-Order Logic 23 0. Introduction 23 1. Formulas of First Order Logic 24 2. Structures for First Order Logic 28 3. Logical Consequence and Validity 33 4. Formal Deductions 37 5. Theories and Their Models 42 6. Exercises 46 46 Chapter 3. The Completeness Theorem 49 0. Introduction 49 1. Henkin Sets and Their Models 49 2. Constructing Henkin Sets 52 3. Consequences of the Completeness Theorem 54 4. Completeness Categoricity, Quantifier Elimination 57 5. Exercises 58 58 Part 2. Model Theory 59 Chapter 4. Some Methods in Model Theory 61 0. Introduction 61 1. Realizing and Omitting Types 61 2. Elementary Extensions and Chains 66 3. The Back-and-Forth Method 69 i ii CONTENTS 4. Exercises 71 71 Chapter 5. Countable Models of Complete Theories 73 0. Introduction 73 1. Prime Models 73 2. Universal and Saturated Models 75 3. Theories with Just Finitely Many Countable Models 77 4. Exercises 79 79 Chapter 6. Further Topics in Model Theory 81 0. Introduction 81 1. Interpolation and Definability 81 2. Saturated Models 84 3. Skolem Functions and Indescernables 87 4. Some Applications 91 5. -
Arxiv:1205.6044V1 [Math.HO] 28 May 2012 Prescriptive/Prohibitive Context
FOUNDATIONS AS SUPERSTRUCTURE1 (Reflections of a practicing mathematician) Yuri I. Manin Max–Planck–Institut f¨ur Mathematik, Bonn, Germany, ABSTRACT. This talk presents foundations of mathematics as a historically variable set of principles appealing to various modes of human intuition and de- void of any prescriptive/prohibitive power. At each turn of history, foundations crystallize the accepted norms of interpersonal and intergenerational transfer and justification of mathematical knowledge. Introduction Foundations vs Metamathematics. In this talk, I will interpret the idea of Foundations in the wide sense. For me, Foundations at each turn of history embody currently recognized, but historically variable, principles of organization of mathematical knowledge and of the interpersonal/transgenerational transferral of this knowledge. When these principles are studied using the tools of mathematics itself, we get a new chapter of mathematics, metamathematics. Modern philosophy of mathematics is often preoccupied with informal interpre- tations of theorems, proved in metamathematics of the XX–th century, of which the most influential was probably G¨odel’s incompleteness theorem that aroused considerable existential anxiety. In metamathematics, G¨odel’s theorem is a discovery that a certain class of finitely arXiv:1205.6044v1 [math.HO] 28 May 2012 generated structures (statements in a formal language) contains substructures that are not finitely generated (those statements that are true in a standard interpreta- tion). It is no big deal for an algebraist, but certainly interesting thanks to a new context. Existential anxiety can be alleviated if one strips “Foundations” from their rigid prescriptive/prohibitive, or normative functions and considers various foundational 1Talk at the international interdisciplinary conference “Philosophy, Mathematics, Linguistics: Aspects of Interaction” , Euler Mathematical Institute, May 22–25, 2012.