Reflections on Finite Model Theory
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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. -
Model Theory
Class Notes for Mathematics 571 Spring 2010 Model Theory written by C. Ward Henson Mathematics Department University of Illinois 1409 West Green Street Urbana, Illinois 61801 email: [email protected] www: http://www.math.uiuc.edu/~henson/ c Copyright by C. Ward Henson 2010; all rights reserved. Introduction The purpose of Math 571 is to give a thorough introduction to the methods of model theory for first order logic. Model theory is the branch of logic that deals with mathematical structures and the formal languages they interpret. First order logic is the most important formal language and its model theory is a rich and interesting subject with significant applications to the main body of mathematics. Model theory began as a serious subject in the 1950s with the work of Abraham Robinson and Alfred Tarski, and since then it has been an active and successful area of research. Beyond the core techniques and results of model theory, Math 571 places a lot of emphasis on examples and applications, in order to show clearly the variety of ways in which model theory can be useful in mathematics. For example, we give a thorough treatment of the model theory of the field of real numbers (real closed fields) and show how this can be used to obtain the characterization of positive semi-definite rational functions that gives a solution to Hilbert’s 17th Problem. A highlight of Math 571 is a proof of Morley’s Theorem: if T is a complete theory in a countable language, and T is κ-categorical for some uncountable κ, then T is categorical for all uncountable κ. -
The Dividing Line Methodology: Model Theory Motivating Set Theory
The dividing line methodology: Model theory motivating set theory John T. Baldwin University of Illinois at Chicago∗ January 6, 2020 The 1960’s produced technical revolutions in both set theory and model theory. Researchers such as Martin, Solovay, and Moschovakis kept the central philosophical importance of the set theoretic work for the foundations of mathematics in full view. In contrast the model theoretic shift is often seen as ‘ technical’ or at least ‘merely mathematical’. Although the shift is productive in multiple senses, is a rich mathematical subject that provides a metatheory in which to investigate many problems of traditional mathematics: the profound change in viewpoint of the nature of model theory is overlooked. We will discuss the effect of Shelah’s dividing line methodology in shaping the last half century of model theory. This description will provide some background, definitions, and context for [She19]. In this introduction we briefly describe the paradigm shift in first order1 model theory that is laid out in more detail in [Bal18]. We outline some of its philosophical consequences, in particular concerning the role of model theory in mathematics. We expound in Section 1 the classification of theories which is the heart of the shift and the effect of this division of all theories into a finite number of classes on the development of first order model theory, its role in other areas of mathematics, and on its connections with set theory in the last third of the 20th century. We emphasize that for most practitioners of late 20th century model theory and especially for applications in traditional mathematics the effect of this shift was to lessen the links with set theory that had seemed evident in the 1960’s. -
The Axiom of Choice
THE AXIOM OF CHOICE THOMAS J. JECH State University of New York at Bufalo and The Institute for Advanced Study Princeton, New Jersey 1973 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK 0 NORTH-HOLLAND PUBLISHING COMPANY - 1973 AN Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Copyright owner. Library of Congress Catalog Card Number 73-15535 North-Holland ISBN for the series 0 7204 2200 0 for this volume 0 1204 2215 2 American Elsevier ISBN 0 444 10484 4 Published by: North-Holland Publishing Company - Amsterdam North-Holland Publishing Company, Ltd. - London Sole distributors for the U.S.A. and Canada: American Elsevier Publishing Company, Inc. 52 Vanderbilt Avenue New York, N.Y. 10017 PRINTED IN THE NETHERLANDS To my parents PREFACE The book was written in the long Buffalo winter of 1971-72. It is an attempt to show the place of the Axiom of Choice in contemporary mathe- matics. Most of the material covered in the book deals with independence and relative strength of various weaker versions and consequences of the Axiom of Choice. Also included are some other results that I found relevant to the subject. The selection of the topics and results is fairly comprehensive, nevertheless it is a selection and as such reflects the personal taste of the author. So does the treatment of the subject. The main tool used throughout the text is Cohen’s method of forcing. -
The Recent History of Model Theory
The recent history of model theory Enrique Casanovas Universidad de Barcelona The history of Model Theory can be traced back to the work of Charles Sanders Peirce and Ernst Schr¨oder, when semantics started playing a role in Logic. But the first outcome dates from 1915. It appears in the paper Uber¨ M¨oglichkeiten im Relativkalk¨ul (Math. Ann. 76, 445-470) by Leopold L¨owenheim. The period from 1915 to 1935 is, in words of R.L. Vaught, extraordinary. The method of elimination of quantifiers is developed and applied to give decision methods for the theories of (Q, <) (C.H. Langford), of (ω, +) and (Z, +, <) (M. Presburger) and, finally, of the field of complex numbers and of the ordered field of real numbers (A. Tarski). Also the completeness theorem of Kurt G¨odel (1930) has to be considered among the earliest results of Model Theory. The influence of Alfred Tarski was decisive in this early stage and in the successive years. This is due not only to his discovery of unquestionable definitions of the notions of truth and definability in a structure, but also to his founding of the basic notions of the theory, such as elementary equivalence and elementary extension. In the fifties and six- ties JerryLoˇsintroduced the ultraproducts, Ronald Fra¨ıss´edeveloped the back-and-forth methods and investigated amalgamation properties, and Abraham Robinson started his voluminous contribution to Model Theory, including his celebrated non-standard analysis. Robinson’s non-standard analysis attracted the attention of mathematicians and philoso- phers. But at that time a feeling of exhaustion started pervading the whole theory. -
Model Theory on Finite Structures∗
Model Theory on Finite Structures∗ Anuj Dawar Department of Computer Science University of Wales Swansea Swansea, SA2 8PP, U.K. e-mail: [email protected] 1 Introduction In mathematical logic, the notions of mathematical structure, language and proof themselves become the subject of mathematical investigation, and are treated as first class mathematical objects in their own right. Model theory is the branch of mathematical logic that is particularly concerned with the relationship between structure and language. It seeks to establish the limits of the expressive power of our formal language (usually the first order predicate calculus) by investigating what can or cannot be expressed in the language. The kinds of questions that are asked are: What properties can or cannot be formulated in first order logic? What structures or relations can or cannot be defined in first order logic? How does the expressive power of different logical languages compare? Model Theory arose in the context of classical logic, which was concerned with resolving the paradoxes of infinity and elucidating the nature of the infinite. The main constructions of model theory yield infinite structures and the methods and results assume that structures are, in general, infinite. As we shall see later, many, if not most, of these methods fail when we confine ourselves to finite structures Many questions that arise in computer science can be seen as being model theoretic in nature, in that they investigate the relationship between a formal language and a structure. However, most structures of interest in computer science are finite. The interest in finite model theory grew out of questions in theoretical computer science (particularly, database theory and complexity theory). -
Topics in the Theory of Recursive Functions
SETS, MODELS, AND PROOFS: TOPICS IN THE THEORY OF RECURSIVE FUNCTIONS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by David R. Belanger August 2015 This document is in the public domain. SETS, MODELS, AND PROOFS: TOPICS IN THE THEORY OF RECURSIVE FUNCTIONS David R. Belanger, Ph.D. Cornell University We prove results in various areas of recursion theory. First, in joint work with Richard Shore, we prove a new jump-inversion result for ideals of recursively enu- merable (r.e.) degrees; this defeats what had seemed to be a promising tack on the automorphism problem for the semilattice R of r.e. degrees. Second, in work spanning two chapters, we calibrate the reverse-mathematical strength of a number of theorems of basic model theory, such as the Ryll-Nardzewski atomic-model theorem, Vaught's no-two-model theorem, Ehrenfeucht's three-model theorem, and the existence theorems for homogeneous and saturated models. Whereas most of these are equivalent over RCA0 to one of RCA0, WKL0, ACA0, as usual, we also uncover model-theoretic statements with exotic complexities such as :WKL0 _ 0 ACA0 and WKL0 _ IΣ2. Third, we examine the possible weak truth table (wtt) degree spectra of count- able first-order structures. We find several points at which the wtt- and Turing- degree cases differ, notably that the most direct wtt analogue of Knight's di- chotomy theorem does not hold. Yet we find weaker analogies between the two, including a new trichotomy theorem for wtt degree spectra in the spirit of Knight's. -
On Finite Domains in First-Order Linear Temporal Logic Denis Kuperberg, Julien Brunel, David Chemouil
On Finite Domains in First-Order Linear Temporal Logic Denis Kuperberg, Julien Brunel, David Chemouil To cite this version: Denis Kuperberg, Julien Brunel, David Chemouil. On Finite Domains in First-Order Linear Temporal Logic. 14th International Symposium on Automated Technology for Verification and Analysis, Oct 2016, Chiba, Japan. 10.1007/978-3-319-46520-3_14. hal-01343197 HAL Id: hal-01343197 https://hal.archives-ouvertes.fr/hal-01343197 Submitted on 7 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On Finite Domains in First-Order Linear Temporal Logic? Denis Kuperberg1, Julien Brunel2, and David Chemouil2 1 TU Munich, Germany 2 DTIM, Université fédérale de Toulouse, ONERA, France Abstract. We consider First-Order Linear Temporal Logic (FO-LTL) over linear time. Inspired by the success of formal approaches based upon finite-model finders, such as Alloy, we focus on finding models with finite first-order domains for FO-LTL formulas, while retaining an infinite time domain. More precisely, we investigate the complexity of the following problem: given a formula ' and an integer n, is there a model of ' with domain of cardinality at most n? We show that depending on the logic considered (FO or FO-LTL) and on the precise encoding of the problem, the problem is either NP-complete, NEXPTIME-complete, PSPACE- complete or EXPSPACE-complete. -
A Finitist's Manifesto: Do We Need to Reformulate the Foundations Of
A Finitist’s Manifesto: Do we need to Reformulate the Foundations of Mathematics? Jonathan Lenchner 1 The Problem There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been sleep-walking in their infinitary mathematical paradise to take heed. Much of mathematics relies upon either (i) the “existence” of objects that contain an infinite number of elements, (ii) our ability, “in theory”, to compute with an arbitrary level of precision, or (iii) our ability, “in theory”, to compute for an arbitrarily large number of time steps. All of calculus relies on the notion of a limit. The monumental results of real and complex analysis rely on a seamless notion of the “continuum” of real numbers, which extends in the plane to the complex numbers and gives us, among other things, “rigorous” definitions of continuity, the derivative, various different integrals, as well as the fundamental theorems of calculus and of algebra – the former of which says that the derivative and integral can be viewed as inverse operations, and the latter of which says that every polynomial over C has a complex root. This essay is an inquiry into whether there is any way to assign meaning to the notions of “existence” and “in theory” in (i) to (iii) above. On the one hand, we know from quantum mechanics that making arbitrarily precise measure- ments of objects is impossible. By the Heisenberg Uncertainty Principle the moment we pin down an object, typically an elementary particle, in space, thereby bringing its velocity, and hence mo- mentum, down to near 0, there is a limit to how precisely we can measure its spatial coordinates. -
Finite Model Theory and Finite Variable Logics
University of Pennsylvania ScholarlyCommons IRCS Technical Reports Series Institute for Research in Cognitive Science October 1995 Finite Model Theory and Finite Variable Logics Eric Rosen University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/ircs_reports Rosen, Eric, "Finite Model Theory and Finite Variable Logics " (1995). IRCS Technical Reports Series. 139. https://repository.upenn.edu/ircs_reports/139 University of Pennsylvania Institute for Research in Cognitive Science Technical Report No. IRCS-95-28. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/ircs_reports/139 For more information, please contact [email protected]. Finite Model Theory and Finite Variable Logics Abstract In this dissertation, I investigate some questions about the model theory of finite structures. One goal is to better understand the expressive power of various logical languages, including first order logic (FO), over this class. A second, related, goal is to determine which results from classical model theory remain true when relativized to the class, F, of finite structures. As it is well known that many such results become false, I also consider certain weakened generalizations of classical results. k k I prove some basic results about the languages L ∃ and L ∞ω∃, the existential fragments of the finite k k k variable logics L and L ∞ω. I show that there are finite models whose L (∃)-theories are not finitely k axiomatizable. I also establish the optimality of a normal form for L ∞ω∃, and separate certain fragments of this logic. I introduce a notion of a "generalized preservation theorem", and establish certain partial k positive results.