Beyond First Order Model Theory, Volume II 0367208261, 9780367208264

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Beyond First Order Model Theory, Volume II Model theory is the meta-mathematical study of the concept of mathematical truth. After Alfred Tarski coined the term Theory of Models in the early 1950s, it rapidly became one of the central most active branches of mathematical logic. In the last few decades, ideas that originated within model theory have provided powerful tools to solve problems in a variety of areas of classical mathematics, including algebra, combinatorics, geometry, number theory, and Banach space theory and operator theory. The two volumes of Beyond First Order Model Theory present the reader with a fairly comprehensive vista, rich in width and depth, of some of the most active areas of contemporary research in model theory beyond the realm of the classical first-order viewpoint. Each chapter is intended to serve both as an introduction to a current direction in model theory and as a presentation of results that are not available elsewhere. All the articles are written so that they can be studied independently of one another. This second volume contains introductions to real-valued logic and applications, abstract elementary classes and applications, interconnections between model theory and function spaces, nonstucture theory, and model theory of second-order logic.

Features • A coherent introduction to current trends in model theory • Contains articles by some of the most influential logicians of the last hundred years. No other publication brings these distinguished authors together • Suitable as a reference for advanced undergraduates, postgraduates, and researchers • Material presented in the book (e.g, abstract elementary classes, first-order logics with dependent sorts, and applications of infinitary logics in set theory) is not easily accessible in the current literature • The various chapters in the book can be studied independently. José Iovino is a professor of Mathematics at The University of Texas at San Antonio. His research is in model theory and its applications. He is the author of the monograph Applications of Model Theory to Functional Analysis (Dover Publications, 2014), a co-author of Analysis and Logic (Cambridge University Press, 2003), and the editor of the first volume of Beyond First Order Model Theory (CRC Press, 2017).

Beyond First Order Model Theory, Volume II Edited by

José Iovino

The University of Texas at San Antonio, United States of America

Cover image design: Abigail Iovino First edition published 2023 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN CRC Press is an imprint of Taylor & Francis Group, LLC © 2023 Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright. com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Names: Iovino, José. Title: Beyond first order model theory / edited by José Iovino, The University of Texas at San Antonio, United States of America. Description: First edition. | Boca Raton : CRC Press, 2017-2023. | Includes bibliographical references and index. Identifiers: LCCN 2017005023| ISBN 9781498753975 (volume 1 : hardback) | ISBN 9781498754019 (volume 1 : ebook) | ISBN 9781315351094 (volume 1 : ebook) | ISBN 9781315332055 (volume 1 : ebook) | ISBN 9781315368078 (volume 1 : ebook) | ISBN 9781032516011 (volume 2 ; paperback) | ISBN 9780367208264 (volume 2 ; hardback) | ISBN 9780429263637 (volume 2 ; ebook) Subjects: LCSH: Model theory. | Logic, Symbolic and mathematical. Classification: LCC QA9.7 .I587 2017 | DDC 511.3/4--dc23 LC record available at https://lccn.loc.gov/2017005023 ISBN: 978-0-367-20826-4 (hbk) ISBN: 978-1-032-51601-1 (pbk) ISBN: 978-0-429-26363-7 (ebk) DOI: 10.1201/9780429263637 Typeset in Computer Modern by KnowledgeWorks Global Ltd. Publisher’s note: This book has been prepared from camera-ready copy provided by the authors

To H. Jerome Keisler and Saharon Shelah

H. Jerome Keisler Photograph by Steven Givant The Tarski Symposium, June 23–30, 1971

Saharon Shelah Abraham Robinson in the background Photograph by Steven Givant The Tarski Symposium, June 23–30, 1971

Contents

Foreword by H. Jerome Keisler

xiii

Preface

xvii

Contributors

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Real-Valued Structures and Applications

1

1 Metastable convergence and logical compactness Xavier Caicedo, Eduardo Dueñez, and José Iovino 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . I Connectives and classical quantifiers . . . . . . . . . . II Metric structures and logics for metric structures . . . III Examples of logics for metric structures . . . . . . . . IV Relativizations . . . . . . . . . . . . . . . . . . . . . . V [κ, λ]-compactness and (κ, λ)-compactness . . . . . . . 1.2 [κ, κ]-compactness and cofinality . . . . . . . . . . . . . . . . 1.3 Metastability and uniform metastability . . . . . . . . . . . . I Metastability: Basic definitions and examples . . . . . II Uniform metastability from a topological viewpoint . . 1.4 The Main Theorem: Uniform metastability and logical compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Compactness and RPC∆ -characterizability of general structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Model theory for real-valued structures H. Jerome Keisler 2.1 Introduction . . . . . . . . . . . . . . . . . . . . 2.2 Basic model theory for general structures . . . . I General structures . . . . . . . . . . . . . II Ultraproducts . . . . . . . . . . . . . . . . III Definability and types . . . . . . . . . . . IV Saturated and special structures . . . . . V Pre-metric structures . . . . . . . . . . . . VI Some variants of continuous model theory

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Turning general structures into metric structures . I Definitional expansions . . . . . . . . . . . II Pre-metric expansions . . . . . . . . . . . . III The Expansion Theorem . . . . . . . . . . . IV Absoluteness . . . . . . . . . . . . . . . . . 2.4 Properties of general structures . . . . . . . . . . I Types in pre-metric expansions . . . . . . . II Definable predicates . . . . . . . . . . . . . III Topological and uniform properties . . . . . IV Infinitary continuous logic . . . . . . . . . . V Many-sorted metric structures . . . . . . . VI Bounded and unbounded metric structures VII Imaginaries . . . . . . . . . . . . . . . . . . VIII Definable sets . . . . . . . . . . . . . . . . . IX Stable theories . . . . . . . . . . . . . . . . X Building stable theories . . . . . . . . . . . XI Simple and rosy theories . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . .

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3 Spectral gap and definability Isaac Goldbring 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . I A crash course in continuous logic . . . . . . . . . 3.2 Definability in continuous logic . . . . . . . . . . . . . . . I Generalities on formulae . . . . . . . . . . . . . . . II Definability relative to a theory . . . . . . . . . . . III Definability in a structure . . . . . . . . . . . . . . 3.3 Spectral gap for unitary group representations . . . . . . I Generalities on unitary group representations . . . II Introducing spectral gap . . . . . . . . . . . . . . . III Spectral gap and definability . . . . . . . . . . . . IV Spectral gap and ergodic theory . . . . . . . . . . 3.4 Basic von Neumann algebra theory . . . . . . . . . . . . I Preliminaries . . . . . . . . . . . . . . . . . . . . . II Tracial von Neumann algebras as metric structures III Property Gamma and the McDuff property . . . . 3.5 Spectral gap subalgebras . . . . . . . . . . . . . . . . . . I Introducing spectral gap for subalgebras . . . . . . II Spectral gap and definability . . . . . . . . . . . . III Relative bicommutants and e.c. II1 factors . . . . . 3.6 Continuum many theories of II1 factors . . . . . . . . . . I The history and the main theorem . . . . . . . . . II The base case . . . . . . . . . . . . . . . . . . . . .

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Contents III

A digression on good factors . . . . . . . . IV The inductive step . Bibliography . . . . . . . . . . .

II

unitaries and . . . . . . . . . . . . . . . . . . . . . . . .

generalized McDuff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Abstract Elementary Classes and Applications

4 Lf groups, aec amalgamation, few Saharon Shelah 4.0 Introduction . . . . . . . . . . . I Review . . . . . . . . . . II Amalgamation spectrum . III Preliminaries on groups . 4.1 Amalgamation bases . . . . . . . 4.2 Definability . . . . . . . . . . . . 4.3 Density of being complete in Klfλ Bibliography . . . . . . . . . . . . . .

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automorphisms . . . . . . . .

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Model Theory and Topology of Function Spaces 175

5 Cp -Theory for model theorists 177 Clovis Hamel and Franklin D. Tall 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5.2 Preliminaries in Cp -theory . . . . . . . . . . . . . . . . . . . 179 I Some basic results . . . . . . . . . . . . . . . . . . . . 179 II Lindelöf Σ-spaces are Grothendieck . . . . . . . . . . . 188 III Grothendieck spaces and double limit conditions . . . 189 5.3 Stability, definability, and double (ultra)limit conditions . . . 191 5.4 Applications and examples concerning the undefinability of pathological Banach spaces . . . . . . . . . . . . . . . . . . . 198 5.5 The NIP and the Bourgain-Fremlin-Talagrand dichotomy . . 203 5.6 Appendix: Proof that all Lindelöf Σ-spaces are Grothendieck 205 5.7 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . 211 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

IV

Constructing Many Models

6 General non-structure theory Saharon Shelah 6.0 Introduction . . . . . . . . . I Preliminaries . . . . . 6.1 Models from indiscernibles . I Background . . . . . . II GEM models . . . . . III Finding templates . .

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xii

Contents IV How forcing helps . . . . . . . . . . . . . . . . Models represented in free algebras and applications . I Representation, non-embeddability and bigness II Example: Unsuperstability . . . . . . . . . . . . III Example: Separable reduced Abelian p-groups ˙ . IV An example: rigid Boolean algebras . . . . . . V Closure sums . . . . . . . . . . . . . . . . . . . VI Back to linear orders . . . . . . . . . . . . . . . 6.3 Order implies many nonisomorphic models . . . . . . I Skeleton like sequence and invariants . . . . . . II Representing invariants . . . . . . . . . . . . . III Harder results . . . . . . . . . . . . . . . . . . . IV Using Infinitary sequences . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2

V

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231 237 237 242 245 248 249 255 260 261 268 271 276 283

Model Theory of Second Order Logic

289

7 Model theory of second order logic Jouko Väänänen 7.1 Introduction . . . . . . . . . . . . . . . . . . . 7.2 Second order characterizable structures . . . . 7.3 Weakly second order characterizable structures 7.4 Non second order characterizable structures . . 7.5 Categoricity of second order theories . . . . . . 7.6 What is left out? . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . .

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Foreword by H. Jerome Keisler

H. Jerome Keisler1 Much of model theory’s power comes from the fact that it can generalize a concept that appears in one area of mathematics and apply it across all of mathematics. A distinct feature of model-theoretic arguments is that they routinely treat formulas as mathematical objects. This has led to many spectacular contributions of model theory to other areas of mathematics. First order model theory began with the theorem of Löwenheim in 1915, the Gödel completeness theorem in 1930, and the compactness theorem, by Malcev in 1936. It emerged as a separate area of research around 1950, pioneered by Alfred Tarski and Abraham Robinson. It has remained the core of model theory since then, because all the basic tools, such as the compactness and Löenheim-Skolem theorem, are available. In the 1950’s, much of the work was motivated by applications to algebra, set theory, and arithmetic, and this work has continued. In the decades since 1950, first order model theory has spawned a number of successful programs. Among these are Shelah’s classification theory (including stable, simple, dependent, and o-minimal theories), finite model theory, recursive model theory, ultraproducts, and nonstandard analysis. Each of these programs is well-suited for applications to some area of classical mathematics. These applications are critical for the success of the program, because they attract researchers to the area. 1 Photograph

courtesy of the University of Wisconsin-Madison Archives.

xiii

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Foreword by H. Jerome Keisler

The basic framework for several extensions of first order model theory appeared soon after 1950, such as second order model theory, many-valued and continuous model theory, and model theory for infinitary logic, logic with additional quantifiers, and modal logic. Often, first order model theory was used as a guide, and substitutes for first order methods were found, such as weak models for logic with the uncountability quantifier, and the Barwise compactness theorem for infinitary logic. Lindström’s characterization of first order logic in 1969 spawned the area of abstract model theory. Continuous model theory has a long history with many twists and turns, and has finally emerged as an active area of research with applications to several areas in analysis. Some references can be found ny paper in this volume. In the 1966 monograph “Continuous Model Theory” ([CK1966]), C.C.Chang and I introduced the first version of continuous model theory, with truth values in a compact Hausdorff space X (which works for the ultraproduct construction) and an equality predicate. The connectives were continuous functions on X, and the quantifiers were continuous functions from the hyperspace of X into X. A featured special case had X = [0, 1], with the quantifiers sup, inf. Over several decades, beginning in 1972, Ward Henson developed a very different approach, called Banach space model theory, using approximate satisfaction of positive bounded formulas. Notably, it connected Shelah’s stability theory to Krivine’s notion of a stable Banach space. More recently, Henson and José Iovino generalized that approach to metric structures. In 2003, Itaï Ben Yaacov introduced another very different approach, compact abstract theories (cats). He presented this in two equivalent ways, one using positive formulas, and the other using a category of existentially closed structures. He used this to study continuous structures, including a continuous analogue of simple theories, with the case of Banach spaces as a motivating example. In 2008, the currently dominant version of continuous model theory, called the model theory of metric structures, was introduced in a paper by Ben Yaacov and Usvyatsov, and a long paper by Ben Yaacov, Berenstein, Henson, and Usvyatsov. They take the [0, 1]-valued case from [CK1966], and replace equality by a distinguished metric and a signature that specifies bounds of uniform convergence for each function and predicate symbol. Magically, a surprisingly large part of first order model theory, including much of classification theory, carries over in a natural way to metric structures. This approach is especially well-suited for applications to mathematical analysis, mostly because the structures and formulas are transparently analogous to classical first order structures and formulas. As a result, continuous model theory has recently seen an explosion of research activity. As mentioned above, [CK1966] allowed any continuous function from the hyperspace into the truth value space as a quantifier. One might ask whether there are interesting applications of continuous model theory to classical mathematics that use quantifiers other than sup and inf, or use compact Hausdorff spaces other than [0, 1] as the space of truth values.

Foreword by H. Jerome Keisler

xv

My own paper in this volume argues for a shift in viewpoint that is even closer to [CK1966]. Rather than replacing equality with a distinguished metric, one can simply remove equality. While a metric structure has formulas with truth values in [0, 1], a distinguished metric, and bounds of uniform continuity for each symbol, a general structure has the same formulas with truth values in [0, 1], but no distinguished metric or uniform continuity requirements. Thus the notion of a general structure is both more general and simpler than that of a metric structure. The paper shows that almost all model-theoretic results about metric structures automatically carry over to general structures. Roughly, that works because every general theory can be made into a metric theory in a definable way. This simpler approach makes it easier to apply continuous model theory to classical mathematics by removing the need to specify moduli of uniform continuity, and opens up the possibility of also looking at structures with discontinuous functions and predicates. The papers in this volume give a glimpse into possible future directions for research in model theory. But, as always, chances are that the next big thing will be a complete surprise. H. Jerome Keisler

Preface

The first volume of Beyond First Order Model Theory appeared in 2017. Our world has changed much since then. We have become increasingly interconnected, and access to information is more immediate than it ever was. However, we have been reminded that these privileges do not come without challenges. Immediacy has brought with it impulses toward simplism. Still, recalling the words of A. N. Whitehead in his classic Science and the Modern World [1925] may serve as an antidote for possible episodes of disillusion: The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit. Another claimant for this position is music. What better embodiment of Whitehead’s foresightful statement than investigations in model theory, the mathematical study of the notion of mathematical truth. The two volumes, taken as a unit, present the reader with a fairly comprehensive vista of some of the most active areas of contemporary research in model theory beyond the realm of the classical first-order point of view. Each article was prepared by a foremost expert in the field. As highlighted in the first volume, each article is intended to serve both as an invitation to a current direction in model theory and as a presentation of results that are not available elsewhere, and all the articles are written so that they can be studied independently of one another. The volumes evolved from a series of lectures and workshops that met at The University of Texas at San Antonio. I am thankful to the contributors for their enthusiasm, and to Taylor & Francis/CRC Press for its support. A special debt of gratitude is owed to Saf Khan and Mansi Kabra of CRC Press, and to the anonymous referees, without whose selfless work this book would not have been possible. The foreword of the first volume of the series was written by Saharon Shelah, and the foreword to this volume is presented by H. Jerome Keisler. It is not an exaggeration to say that Keisler and Shelah are among the most influential logicians of the last hundred years. This series has been graced with paradigm-shifting articles by them that became major references even before the volumes went to print.

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This second volume is dedicated to these two much-admired researchers, with gratitude for their contributions to mathematical logic. They have inspired, and continue to inspire, generations of mathematicians through the indescribable insight, beauty, and universality of their work. José Iovino San Antonio, Winter solstice 2022

Contributors

Xavier Caicedo Universidad de los Andes Bogotá, Colombia

H. Jerome Keisler University of Wisconsin-Madison Madison, Wisconsin, USA

Eduardo Dueñez The University of Texas at San Antonio San Antonio, Texas, USA

Saharon Shelah The Hebrew University of Jerusalem Jerusalem, Israel

José Iovino The University of Texas at San Antonio San Antonio, Texas, USA Clovis Hamel University of Toronto Toronto, Ontario, Canada Isaac Goldbring University of California, Irvine Irvine, California, USA

Rutgers, The State University of New Jersey Piscataway, New Jersey, USA Franklin D. Tall University of Toronto Toronto, Ontario, Canada Jouko Väänänen University of Helsinki Helsinki, Finland

xix

Part I

Real-Valued Structures and Applications

Chapter 1 Metastable convergence and logical compactness Xavier Caicedo Universidad de los Andes Bogotá, Colombia Eduardo Dueñez The University of Texas at San Antonio San Antonio, Texas, USA José Iovino The University of Texas at San Antonio San Antonio, Texas, USA 1.1

1.2 1.3

1.4 1.5

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Connectives and classical quantifiers . . . . . . . . . . . . . . . . . . . . II Metric structures and logics for metric structures . . . . . . III Examples of logics for metric structures . . . . . . . . . . . . . . . . IV Relativizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V [κ, λ]-compactness and (κ, λ)-compactness . . . . . . . . . . . . . . [κ, κ]-compactness and cofinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metastability and uniform metastability . . . . . . . . . . . . . . . . . . . . . . . . I Metastability: Basic definitions and examples . . . . . . . . . . II Uniform metastability from a topological viewpoint . . . . The Main Theorem: Uniform metastability and logical compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compactness and RPC∆ -characterizability of general structures Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 8 10 11 12 14 16 18 19 22 27 32 37

Dueñez and Iovino were partially supported by NSF grant DMS-1500615. Caicedo was partially supported by a Universidad de los Andes Faculty of Science grant.

DOI: 10.1201/9780429263637-1

3

4

Beyond First Order Model Theory, Volume 2

Introduction The concept of metastable convergence was isolated by Tao. It played a crucial role in the proof of his remarkable result on the convergence of ergodic averages for polynomial abelian group actions [Tao08], and again in Walsh’s generalization of Tao’s theorem to polynomial nilpotent group actions [Wal12]. Metastability is a reformulation of the Cauchy property for sequences, i.e., a sequence in a metric space is metastable if and only if it is Cauchy. However, for a collection of sequences, being uniformly metastable is weaker than being uniformly Cauchy. In his 2008 paper [Tao08], Tao proved a metastable version of the classical dominated convergence theorem that he then used to obtain uniform metastability rates of convergence for ergodic averages. Tao remarked in his paper that metastability is connected to ideas from mathematical logic. He noted, thanking U. Kohlenbach for the observation, that metastability is an instance of Kreisel’s no-counterexample interpretation [Kre51, Kre52], which is in turn a particular case of Gödel’s Dialectica interpretation [Göd58]. In fact, before Tao’s paper, the concept had been used under different nomenclature by Avigad, Gerhardy, Kohlenbach, and Towsner in the context of proof mining. See [AGT10, KL04, Koh05, Koh08]. For a more up-to-date survey on metastability rates obtained by proof mining, see Kohlenbach’s lecture at the 2018 International Congress of Mathematicians [Koh, pp. 68–69]. A connection between uniform metastable convergence and modeltheoretic compactness was first exposed by Avigad and Iovino by using ultraproducts [AI13]. After this, Dueñez and Iovino proved a metatheorem called the Uniform Metastability Principle ([DnI17], Proposition 2.4), which roughly states the following: If a classical statement about convergence in metric structures is refined to a statement about metastable convergence with some uniform rate, and this latter refinement can be expressed in the language of continuous first-order logic, then the validity of the original statement implies the validity of its uniformly metastable version. The operative word above is uniformly: The striking fact about the Uniform Metastability Principle is that it allows one to convert a theorem about simple convergence into a stronger theorem about uniformly metastable convergence automatically, provided that in the statement of the theorem, one replaces convergence by the mathematically equivalent notion of metastability. Thus, for instance, Tao’s uniformly metastable dominated convergence theorem follows from the classical dominated convergence theorem as a particular application of this metatheorem. Also, as Tao pointed out, his proposed

Metastable convergence and logical compactness

5

abstract version of Walsh’s ergodic theorem [Tao] follows from the original version [Wal12] and the aforementioned Avigad-Iovino paper [AI13].) It is natural to ask if the Uniform Metastability Principle holds with logics more expressive than continuous first-order. The more expressive the logic, the more powerful the metatheorem. On the other hand, the proof of the Uniform Metastability Principle uses the fact that continuous first-order logic is compact, and there is a delicate balance between compactness of a logic and its expressive power. In this paper we show that the Uniform Metastability Principle is in fact equivalent to compactness for logics. More precisely, we prove the following theorem: Theorem. Let L be a logic for metric structures. Then L is compact if and only if every theory of convergence expressible in L is a theory of uniformly metastable convergence. Our main results, Theorems 1.4.5 and 1.4.6, establish a fine correspondence between the many forms of compactness arising in logic (for theories or families of theories — see V of the Preliminaries section) and natural forms of the Uniform Metastability Principle (for sequences, for nets, etc.) Among the many forms of compactness that are studied in logic, the strongest is compactness for arbitrary theories, while the weakest is countable compactness for theories (i.e., any countable finitely satisfiable theory is satisfiable). At the full compactness end, the correspondence with uniform metastable convergence takes the form quoted above; at the countable compactness end, it takes the following form: Theorem. Let L be a logic for metric structures. Then L is countably compact if and only if every countable theory of convergence for sequences expressible in L is a theory of uniformly metastable convergence. These results provide mathematicians with a “black box” to convert theorems about convergence into theorems about uniform (metastable) convergence: If a convergence theorem can be written in a logic that is compact, then its uniform metastable version is automatically true; if not, then the automatic conversion is impossible. A given theorem in analysis may not be expressible in first-order logic, which is a fully compact logic, but the natural framework for the theorem may be a stronger logic that admits a weaker degree of compactness, say, countable compactness. We deal with metric structures, but even in the discrete case, i.e., for two-valued logics in discrete structures, the information given by these results appears to be new. There are many examples of countably compact logics (typically, extensions of first-order by generalized quantifiers — see Examples 1.1.28). Therefore, the forward implications of Theorem 1 give a vast generalization of the Uniform Metastability Principle, and they extend the scope of this metatheorem to contexts where proof-theoretic methods may be unavailable.

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We obtain the forward implications from purely topological considerations (see Section 1.3-II, where we give a topological version of the Uniform Metastability Principle). To prove the reverse implications, we extend to the setting of metric structures a characterization of [κ, κ]-compactness originally proved by Makowsky and Shelah [MS79], and we adapt it to characterize (κ, κ)-compactness. In the last section of the paper, we obtain new general results about the Uniform Metastability Principle (namely, descent of the Uniform Metastability Principle from convergence of nets nets to convergence of sequences — see Theorem 1.5.12) by combining the Main Theorem with classical facts that connect compactness with RPC∆ -characterizability of structures. We do not presuppose expertise in mathematical logic from the reader. However, we assume familiarity with the concept of structure, as is defined in any model theory textbook, and the concepts of language or vocabulary of a structure. The authors are grateful to Clovis Hamel, Ulrich Kohlenbach, and Frank Tall for invaluable comments on earlier versions of the manuscript.

1.1

Preliminaries

The formal definition of model-theoretic logic was given by P. Lindström in his celebrated 1969 paper [Lin69]. In this chapter, we will study logics with nonclassical truth values. However, as motivation, we start by recalling Lindström’s classical definition. Definition 1.1.1. A Lindström logic L is a triple (C , SentL , |=L ), where C is a class of first-order structures that is closed under isomorphisms, renamings and reducts, SentL is a function that assigns to every first-order vocabulary L a set SentL (L) called the set of L-sentences of L , and |=L is a binary relation between structures and sentences, such that the following conditions hold: 1. If L ⊆ L0 , then SentL (L) ⊆ SentL (L0 ). 2. If M |=L ϕ (i.e., if M and ϕ are related under |=L ), then there is a vocabulary L such that M is an L-structure in C and ϕ an L-sentence. If M |=L ϕ, we say that M satisfies ϕ, or that M is a model of ϕ. 3. Isomorphism Property. If M, N are isomorphic structures in C , then M |=L ϕ if and only if N |=L ϕ. 4. Reduct Property. If L ⊆ L0 , ϕ is an L-sentence, and M an L0 -structure in C , then M |=L ϕ

if and only if

(M  L) |=L ϕ.

Metastable convergence and logical compactness

7

5. Renaming Property. If ρ : L → L0 is a renaming (i.e., a bijection r : L → L0 that respects symbol type and and arity), then for each L-sentence ϕ there is an L0 -sentence ϕρ such that M |=L ϕ if and only if Mρ |=L ϕρ for every L-structure M in C . (Here, Mρ denotes the structure that results from converting M into an L0 -structure through ρ.) A classical first-order structure M consists of a nonempty universe M together with finitary functions and relations (or “predicates”) on M . If n is a nonnegative integer, any n-ary relation on M can be seen as a function of M n into {0, 1}. In this chapter we will deal with the more general concept of [0, 1]-valued structure, which is defined as follows: A [0, 1]-valued structure M consists of a nonempty set M called the universe of M, together with finitary functions and predicates on M ; but in this case, the predicates are [0, 1]valued, rather than {0, 1}-valued. A simple example of [0, 1]-valued structure is a pseudometric space (M, d) of diameter bounded by 1. The universe of the structure is M and the only predicate of the structure is d. The following extension of Definition 1.1.1 was introduced by Caicedo and Iovino [CI14]: Definition 1.1.2. A [0, 1]-valued logic is a triple L = (C , SentL , V), where C is a class of [0, 1]-valued structures that is closed under under isomorphisms, renamings and reducts, SentL is a function that assigns to every first-order vocabulary L a set SentL (L) called the set of L-sentences of L , and V is a real-valued partial function on C × SentL such that the following conditions hold: 1. If L ⊆ L0 , then SentL (L) ⊆ SentL (L0 ). 2. For every L, the function V assigns to every pair (M, ϕ), where M is an L-structure in C and ϕ is an L-sentence of L , a real number V(M, ϕ) = ϕM ∈ [0, 1] called the truth value of ϕ in M. 3. Isomorphism Property for [0, 1]-valued logics. If M, N are isomorphic structures in C and ϕ is an L-sentence of L , then ϕM = ϕN . 4. Reduct Property for [0, 1]-valued logics. If L ⊆ L0 , ϕ is an L-sentence of L , and M an L0 -structure in C , then ϕM = ϕML . 5. Renaming Property for [0, 1]-valued logics. If ρ : L → L0 is a renaming, then for each L-sentence ϕ of L there is an L0 -sentence ϕρ such that ρ ϕM = (ϕρ )M for every L-structure M in C . If L is a [0, 1]-valued logic, L is a vocabulary ϕ is an L-sentence of L and M is an L-structure in C such that ϕM = 1, we say that M satisfies ϕ, or that M is a model of ϕ, and write M |=L ϕ. If L is a [0, 1]-valued logic such that ϕM ∈ {0, 1} for every sentence ϕ of L and every structure M, we say that L is a two-valued logic, or a discrete logic.

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Definition 1.1.3. Let L be a a [0, 1]-valued logic and let L be a vocabulary. 1. An L-theory (or simply a theory if the vocabulary is given by the context) of L is a set of L-sentences of L . 2. Let T be an L-theory of L . If M is an L-structure such that M |=L ϕ for each ϕ ∈ T , we say that M is a model of T and write M |=L T . 3. A theory is satisfiable if it has a model. 4. If M is structure of L , the complete L -theory of M, denoted ThL (M), is the set { ϕ : M |=L ϕ }. If L is a vocabulary, x ¯ = x1 , . . . , xn is a finite list of constant symbols not in L, and ϕ is an (L ∪ {¯ x})-sentence, we emphasize this by writing ϕ as ϕ(¯ x). In this case we may say that ϕ(¯ x) is an L-formula. If M is an L-structure and a ¯ = a1 , . . . , an is a list of elements of M, we write (M, a1 , . . . , an ) |=L ϕ(x1 , . . . , xn ), or M |=L ϕ[¯ a], if the L ∪ {¯ x} expansion of M that results from interpreting xi as ai (for i = 1, . . . , n) satisfies ϕ(¯ x). Definition 1.1.4. Let M, N be L-structures. We say that M and N are equivalent in L , and write M ≡L N, if for every L-sentence ϕ we have φM = φN . If M is an L-structure and A is a subset of the universe of M, we denote by L[A] the expansion of the vocabulary L obtained by adding distinct new constant symbols ca , one for each a ∈ A. We also denote by (M, a)a∈A the expansion of M to an L[A]-structure obtained by interpreting each ca as a. The structure (M, a)a∈A is said to be an expansion of M by constants. Definition 1.1.5. Let L be a [0, 1]-valued logic and let M, N be L-structures with M a substructure of N. We say that M is an L -substructure of N or that N is an L -extension of M, and we write M L N, if (M, a)a∈M ≡L (N, a)a∈M . Convention 1.1.6. In order to avoid clutter in the notation, if a logic L is fixed, we may suppress the subindex in symbols like ≡L , ≺L , ThL (·), etc.

I

Connectives and classical quantifiers

Definition 1.1.7. The Łukasiewicz implication is the function → from [0, 1]2 into [0, 1] defined by x → y = min{1 − x + y, 1} for all (x, y) ∈ [0, 1]2 . Note that x → y has the value 1 if and only if x ≤ y.

Metastable convergence and logical compactness

9

Definition 1.1.8. We will say that a [0, 1]-valued logic L is closed under the basic connectives if the following conditions hold for every vocabulary L: 1. If ϕ, ψ ∈ SentL (L), then there exists a sentence ϕ → ψ in SentL (L) such that (ϕ → ψ)M = ϕM → ψ M for every L-structure M. 2. For each rational r ∈ [0, 1], the set SentL (L) contains a sentence with constant truth value r. These sentences are called the Pavelka constants of L ; these are not needed, but they simplify the exposition (see [Cai17]). Notation 1.1.9. If L is a [0, 1]-valued logic that is closed under the basic connectives, ϕ is a sentence of L , and r is a Pavelka constant of L , we will write ϕ ≤ r, ϕ ≥ r and ϕ = r as abbreviations, respectively, of ϕ → r, r → ϕ and (ϕ → r) ∧ (r → ϕ). Remark 1.1.10. Let L be a [0, 1]-valued logic and let L be a vocabulary. If M is an L-structure of L , ϕ is an L-sentence of L , and r is a Pavelka constant of L , then M |=L ϕ ≤ r if and only if ϕM ≤ r, and M |=L ϕ ≥ r if and only if ϕM ≥ r; thus, the truth value ϕM is determined by either of the sets { r ∈ Q ∩ [0, 1] : M |=L ϕ ≤ r },

{ r ∈ Q ∩ [0, 1] : M |=L ϕ ≥ r }.

Notation 1.1.11. If L is a [0, 1]-valued logic that is closed under the basic connectives and ϕ, ψ are sentences of L , we write ¬ϕ and ϕ ∨ ψ as abbreviations, respectively, of ϕ → 0 and (ϕ → ψ) → ψ, and ϕ ∧ ψ as an abbreviation of ¬(¬ϕ ∨ ¬ψ). Note that for every L-structure M, one has (ϕ ≤ 0)M = 1 − ϕM , (ϕ ∨ ψ)M = max{ϕM , ψ M }, (ϕ ∧ ψ)M = min{ϕM , ψ M }. In particular, every [0, 1]-valued logic that is closed under the basic connectives is closed under conjunctions and disjunctions. On the other hand, M |= ϕ implies M 6|= ¬ϕ, but not conversely. We call ¬ϕ the Łukasiewicz negation or weak negation of ϕ. We will refer to any function from [0, 1]n into [0, 1], where n is a nonnegative integer, as an n-ary connective. The Łukasiewicz implication and the Pavelka constants are continuous connectives, as are all the projections (x1 , . . . , xn ) 7→ xi . The following proposition states that any other other continuous connective can be approximated by finite combinations of these. Proposition 1.1.12. Let C be the class of connectives generated by composing the Łukasiewicz implication, the Pavelka constants, and the projections. Then every continuous connective is a uniform limit of connectives in C .

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Beyond First Order Model Theory, Volume 2

Proof. Since C is closed under the connectives max{x, y} and min{x, y}, by the Stone-Weierstrass theorem for lattices [GJ76, pp. 241-242], we only need to show that the connectives rx, where r is a dyadic rational, can be approximated by connectives in C . Notice that if x ∈ [0, 1],   n  _ i i 1 x = lim ∧¬ x→ . n 2 n n i=1 Hence, since the truncated sum a ⊕ b
= min(a + b, 1) = ¬x → y is in C , so are all the connectives 21 x + · · · + 21n x, for any positive integer n. Definition 1.1.13. Let L be a [0, 1]-valued logic. We say that L is closed under existential quantifiers if given any L-formula ϕ(x) there exists an L-formula ∃xϕ such that for every L-structure M one has (∃xϕ)M = supa∈M (ϕ[a]M ). Similarly, we say that L is closed under universal quantifiers if given any L-formula ϕ(x) there exists an L-formula ∀xϕ such that for every L-structure M one has (∀xϕ)M = inf a∈M (ϕ[a]M ).

II

Metric structures and logics for metric structures

Definition 1.1.14. A metric structure is a [0, 1]-valued structure M such that one of the predicates of M is a metric d on the universe of M, and all the functions and predicates of M are uniformly continuous with respect to d. Note that classical structures are metric structures; we regard them as being endowed with the discrete metric. The predicate for this metric is ¬(x = y). For this reason, we refer to classical structures as discrete structures. Definition 1.1.15. A logic for metric structures is a [0, 1]-valued logic L such that the structures of L are metric structures and L is closed under the basic connectives and the existential and universal quantifiers (see Definitions 1.1.8 and 1.1.13). Remark 1.1.16. To any logic L for metric structures there corresponds a f for discrete structures, i.e., for models of the sentence logic L ∀x∀y(d(x, y) = 0 ∨ d(x, y) = 1). f It follows trivially from the definition of logic for metric structures that L extends classical (discrete) first-order logic Lωω .

Metastable convergence and logical compactness

III

11

Examples of logics for metric structures

In Subsection V we shall turn our attention to compactness. Examples (2)–(5) below are examples of compact logics. 1. Two-valued logics. Certainly, any Lindström logic that is closed under the Boolean connectives and classical quantifiers can be seen as a two-valued logic and hence as a logic for metric structures. 2. Basic continuous logic. This logic, which we will temporarily denote as Lbasic , is defined in the following manner. The class of structures of Lbasic is the class of all metric structures. The class of sentences of Lbasic is defined as follows. For a vocabulary L, the concept of L-term is defined as in first-order logic. If t(x1 , . . . , xn ) is an L-term (where x1 , . . . , xn are the variables that occur in t), M is an L-structure, and a1 , . . . , an are elements of the universe of M, the interpretation tM [a1 , . . . , an ] is defined as in first-order logic as well. The atomic formulas of L are all the expressions of the form d(t1 , t2 ) or R(t1 , . . . , tn ), where R is an n-ary predicate symbol of L. If ϕ(x1 , . . . , xn ) is an atomic L-formula with variables x1 , . . . , xn and a1 , . . . , an are elements of an L-structure M, the interpretation ϕM [a1 , . . . , an ] is defined naturally by letting M R(t1 , . . . , tn )M [a1 , . . . , an ] = RM (tM 1 [a1 , . . . , an ], . . . , tn [a1 , . . . , an ])

and

M d(t1 , t2 )M [a1 , . . . , an ] = dM (tM 1 [a1 , . . . , an ], t2 [a1 , . . . , an ])

(where dM is the metric in M). The L-formulas of Lbasic are the syntactic expressions that result from closing the atomic formulas of L under the Łukasiewicz implication, the Pavelka constants, and the existential quantifier. A sentence of Lbasic is a formula without free variables, and the truth value of a L-sentence ϕ in an L-structure M is ϕM . We write M |=Lbasic ϕ[a1 , . . . , an ] if ϕ[a1 , . . . , an ]M = 1. Recall that in any [0, 1]-valued logic that is closed under the basic connectives, the expressions ¬ϕ, ϕ ∨ ψ, ϕ ∧ ψ, ϕ ≤ r, and ϕ ≥ r are written as abbreviations of ϕ → 0, (ϕ → ψ) → ψ, ¬(¬ϕ ∨ ¬ψ), ϕ → r, and r → ϕ, respectively. In Lbasic we also regard ∀xϕ as an abbreviation of ¬∃x¬ϕ. 3. Full continuous logic. This logic, temporarily denoted Lfull , is the same as Lbasic above with the difference that, instead of taking the closure under the Łukasiewicz-Pavelka connectives, one takes the closure under all continuous connectives (and the existential quantifier). Proposition 1.1.12 yields the following remark, which allows one to transfer model-theoretic results between Lbasic and Lfull : Remark 1.1.17. For every L-formula ϕ(¯ x) of Lfull and for every  > 0 there exists a formula ψ(¯ x) of Lbasic such that |ϕM [¯ a] − ψ M [¯ a]| ≤  for every complete L-structure M and every tuple a ¯ in the universe M of M with `(¯ a) = `(¯ x). It follows that if M, N are L-structures, then M ≡Lbasic N if and only if M ≡Lfull N, and M Lbasic N if and only if M Lfull N. Moreover, every structure is equivalent in Lbasic (and Lfull ) to its metric completion.

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4. The continuous logic framework of Ben Yaacov and Usvyatsov. This logic is the restriction of Lfull to the class of complete metric structures; it was introduced by Ben Yaacov and Usvyatsov [BYU10] as a reformulation of Henson’s logic for metric spaces, based on the concept of continuous model theory developed by Chang and Keisler [Cha61, CK66]. 5. Łukasiewicz-Pavelka logic. The formulas of Łukasiewicz-Pavelka logic are like those of basic continuous logic, with the following difference: in place of the distinguished metric d, one uses the similarity relation x ≈ y. However, there is a precise correspondence between the two relations, namely, d(x, y) is 1 − (x ≈ y) (in other words, the two relations are weak negations of each other) — see Section 5.6 of [Háj98], especially Example 5.6.3-(1). Also, in Łukasiewicz-Pavelka logic, for each n-ary operation symbol f , one has the axiom (x1 ≈ y1 ∧ · · · ∧ xn ≈ yn ) → (f (x1 . . . , xn ) ≈ f (y1 , . . . yn )), and similarly, for each n-ary predicate symbol R, one has the axiom (x1 ≈ y1 ∧ · · · ∧ xn ≈ yn ) → (R(x1 . . . , xn ) ↔ R(y1 , . . . yn )), where “ϕ ↔ ψ” abbreviates “(ϕ → ψ) ∧ (ψ → ϕ)” ([Háj98], Definition 5.6.5). Thus, Łukasiewicz-Pavelka logic is the restriction of basic continuous logic to the class of 1-Lipschitz structures, i.e., structures whose operations and predicates are 1-Lipschitz. Historically, Pavelka extended Łukasiewicz propositional logic by adding the rational constants, and proved a form of approximate completeness for the resulting logic. See [Pav79a, Pav79b, Pav79c] (see also Section 5.4 of [Háj98].) This is known as Pavelka-style completeness. Łukasiewicz-Pavelka logic is also referred to in the literature as rational Pavelka logic, or Pavelka many-valued logic. Novák proved Pavelka-style completeness for predicate ŁukasiewiczPavelka logic, which he calls “first-order fuzzy logic”, first using ultrafilters [Nov89, Nov90], and later using a Henkin-type construction [Nov95]. Another proof of Pavelka-style completeness for predicate Łukasiewicz-Pavelka logic was given by Hajek ([Háj97] and [Háj98, Section 5.4]). 6. Infinitary [0, 1]-valued logics. Different [0, 1]-valued logics with infinitary formulas have been been studied by Ben Yaacov-Iovino [BYI09], Eagle [Eag14, Eag17], Grinstead [Gri13], Sequeira [Seq13], and Caicedo [Cai17]. See [Cai17] and [Eag17] for comparisons among these. Convention 1.1.18. Throughout the rest of the chapter, the symbol Lcont will denote any of the logics in Examples (1)–(5) above.

IV

Relativizations

The fact that a given predicate of a [0, 1]-structure (including the metric of a metric structure) takes on values in {0, 1}, can be expressed using only the connectives ∨ and ¬:

Metastable convergence and logical compactness

13

Definition 1.1.19. Let L be a [0, 1]-valued logic that is closed under the basic connectives and let M be an L-structure of L . Let P a predicate symbol of L or the symbol denoting the metric. We define Discrete(P ) to be the Lformula ∀¯ x(P (¯ x) ∨ ¬P (¯ x)), and call P M discrete if M |= Discrete P ). Let L be a vocabulary and let P (x) be a monadic predicate not in L. If M is an (L ∪ {P })-structure with universe M such that P M is discrete, and a valid L-structure of L is obtained by restricting the universe of M to {a ∈ M : M |=L P [a]}, then we denote this structure by M  {x : P (x)} or M  P . Note that if M is complete, the continuity of P ensures that M  P , when defined, is complete. Definition 1.1.20. A [0, 1]-valued logic L permits relativization to discrete predicates if for every vocabulary L, every L-sentence ϕ, and every monadic predicate symbol P not in L there exists an (L∪{P })-sentence, denoted ϕP or ϕ{x:P (x)} and called the relativization of ϕ to P , such that the following holds: If M is an (L ∪ {P })-structure with universe M such that P M is discrete, then · M  {x : P (x)} is an (L ∪ {P })-structure of L . · For all c ∈ M ,

(ϕP )M [c] = ϕMP [c].

As an example, if ϕ is an formula of Lcont , the relativization of ϕ to P can be defined by the following recursive rule: · If ϕ is atomic, then ϕP is ϕ. · If ϕ is of the form C(ψ1 , . . . , ψn ), where C is a connective, then ϕP is C(ψ1P , . . . , ψnP ). · If ϕ is of the form ∃yψ, then ϕP is ∃y(P (y) ∧ ψ P ). · If ϕ is of the form ∀yψ, then ϕP is ∀y(¬P (y) ∨ ψ P ). One may verify that all the basic examples of [0, 1]-valued logics discussed in Subsection III satisfy the following stronger property: Definition 1.1.21. A [0, 1]-valued logic L permits relativization to definable families of predicates if for every vocabulary L, every L-sentence ϕ, every binary predicate symbol R not in L, and any variable y, there is an (L ∪ {R})formula ψ(y), denoted ϕ{x:R(x,y)} or ϕR(·,y) , such that the following holds: Whenever M is an (L ∪ {R})-structure with universe M such that for every b ∈ M, · Either M |= R[a, b] or M |= ¬R[a, b] for every a ∈ M (i.e., the collection {RM (·, b) : b ∈ M } consists of discrete predicates), and

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Beyond First Order Model Theory, Volume 2 · M  {x : R(x, b)} (also denoted M  R(·, b)) is defined as an L-structure of L ,

one has,

(ϕR(·,b) )M = ϕMR(·,b) R(·,y)

Such formula ϕ

for all b ∈ M.

is a relativization of ϕ by R(x, y) with parameter y.

Definition 1.1.22. We will say that a logic for metric structures is regular if it permits relativization to definable families of predicates. All the logics mentioned in Subsection III and in Examples 1.1.28 of the next section are regular.

V

[κ, λ]-compactness and (κ, λ)-compactness

Recall that if (X, d) and (Y, ρ) are pseudometric spaces and F : X → Y is uniformly continuous, a modulus of uniform continuity for F is a function ∆ : (0, ∞) → [0, ∞) such that, for all x, y ∈ B and  > 0, d(x, y) < ∆()

⇒

ρ(F (x), F (y)) ≤ .

Definition 1.1.23. If L is a logic for metric structures and T is an L -theory, the class of models of T will be denoted ModL (T ). An L -elementary class is a class of the form ModL (ϕ), where ϕ is a sentence. Definition 1.1.24. Let L be a vocabulary and let C be a class of L-structures. We will say that C is a uniform class if for every function symbol f of L there exists ∆f : (0, ∞) → [0, ∞) such that for every structure M of C , the function ∆ is a modulus of uniform continuity for f M . The collection (∆f )f ∈L is called a modulus of uniform continuity for C . If L is a logic for metric structures and T is an L -theory, we will say that T is uniform if ModL (T ) is a uniform class. We will say that a family T of L-theories is uniform if there exists a common modulus of uniform continuity for ModL (T ) for all T ∈ T. Remark 1.1.25. The definition of uniform class given above applies only to [0, 1]-valued structures. For unbounded or non-uniformly bounded structures, a more general definition imposing local bounds in addition to local moduli of continuity is needed [DnI17]. Definition 1.1.26. Let L be a logic for metric structures and let κ, λ be infinite cardinals with λ ≤ κ ≤ ∞. 1. We will say that L is [κ, λ]-compact if the following holds: Whenever L is a vocabulary and T is a uniformly continuous family of L-theories S S of L of cardinality at most κ, the union T is satisfiable if T0 is satisfiable for every subfamily T0 ⊆ T of cardinality strictly less than λ. We will say that L is compact if and only if L is [∞, ω]-compact, i.e., [κ, ω]-compact for every κ.

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2. We will say that L is (κ, λ)-compact if the following holds: Whenever L is a vocabulary and T is a uniform L-theory of L of cardinality at most κ, we have that T is satisfiable if every subtheory of T of cardinality strictly less than λ is satisfiable. Remark 1.1.27. Clearly, [κ, λ]-compactness is stronger than (κ, λ)compactness. However, the two properties become equivalent if, in the definition of [κ, λ]-compactness, we consider only theories of cardinality at most κ. Also, both are equivalent when κ = ∞. Examples 1.1.28. Let Lωω be first-order logic. Given a quantifier Q, we denote by Lωω (Q) the extension of Lωω by the quantifier Q. 1. If κ is an infinite cardinal and ∃≥κ is the quantifier that says “there exist κ-many”, then Lωω (∃≥κ ) is [ω, ω]-compact for κ = (2ω )+ , and in general [δ, δ]-compact for any κ of the form (λδ )+ [Fuh65]. In particular, firstorder logic extended with the quantifier “there exist at most continuum many” is [ω, ω]-compact. The logic Lωω (∃≥ℵ1 ) (i.e., first-order extended with the quantifier “there exist uncountably many”) is known for its good behavior [Kei70, Kau85]. This logic is (ω, ω)-compact [Vau64] but not [ω, ω]-compact [Cai99]. It is consistent to assume that Lωω (∃≥κ ) is (ω, ω)-compact for all κ > ω. See [SV] for a detailed account. 2. Stationary logic is the extension of first-order logic with the second-order quantifier that says “for almost all countable sets” (more precisely, for a close unbounded family of subsets of the universe). This logic is (ω, ω)compact. Stationary logic was introduced by Shelah [She71, She72], investigated in detail by Barwise-Kaufmann-Makkai [BKM78] and further by Mekler-Shelah [MS85, MS86]). cof 3. Shelah’s cofinality quantifier Qcof ω such that Qω x, y ϕ(x, y) says “ϕ(x, y) defines a linear order of cofinality ω” gives a fully compact proper extension of first-order logic [She71]. This logic is a sublogic of stationary logic [She85, Lemma 4.4].

4. Other compact extensions of first-order by second order quantifiers have been studied by Shelah [She75, She] and Mekler-Shelah [MS93]. 5. The infinitary logic Lκκ is (κ, κ)-compact if and only if κ is weakly compact, and it is (∞, κ)-compact if and only if κ is strongly compact. Definition 1.1.29. Let X be a topological space. If κ, λ are infinite cardinals with λ ≤ κ, a topological space is said to be [κ, λ]-compact if whenever F is a family of at most κ closed sets such that the intersection of any subfamily of T F of cardinality less than λ has nonempty intersection we must have F 6= ∅. Note that a topological space X is compact if and only if X is [∞, ω]-compact.

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Beyond First Order Model Theory, Volume 2

The concept of [κ, λ]-compactness was introduced in topology by Alexandroff and Urysohn [AU29]. Smirnov took up its systematic study and considered variations of the definition [Smi50, Smi51]; he was followed by Gál [Gál57, Gál58] and Noble [Nob71]. For a comprehensive introduction (including detailed comparisons among the variants introduced by Smirnov, Gaal, and Noble), see Vaughan’s paper [Vau75]. Remarks 1.1.30. If L is a vocabulary, the class of L-structures of L can be regarded as a topological space naturally by letting the classes of the form ModL (T ), where T is an L-theory, be the closed classes of the topology (in other words, elementary L -classes are the basic closed sets). Clearly, L is [κ, λ]-compact if and only if the class of L-structures of L is [κ, λ]-compact for every vocabulary L. In particular, L is [ω, ω]-compact if and only if, for every vocabulary L, the class of L-structures of L is countably compact. It is easy to verify that if λ < κ, then [κ, λ]-compactness is equivalent to [δ, δ]-compactness for all regular δ such that λ ≤ δ ≤ κ (see [Vau75]). For general compact extensions of Lωω from a topological viewpoint see [Cai99, Cai93]. Remark 1.1.31. The nomenclature for square-bracket compactness is not unified in logic and topology. The term “countable compactness” corresponds to [ω, ω]-compactness in topology and to (ω, ω)-compactness in logic. Also, [λ, κ]-compactness in topology, corresponds to [κ, λ]-compactness in logic. For the rest of this chapter, we will adhere to the usage within logic.

1.2

[κ, κ]-compactness and cofinality

Recall that a logic for metric structures is regular if it permits relativization to definable families of predicates (see Definitions 1.1.21 and 1.1.22). The following theorem is a version for metric structures of a theorem of Makowsky and Shelah [MS79]. Theorem 1.2.1. Let L be a logic for metric structures and let κ be a regular cardinal. Then (1) below implies (2). If L is regular, (2) implies (1). 1. L is [κ, κ]-compact. 2. If L is a vocabulary containing a monadic predicate symbol P , a binary predicate symbol
, and a family (α : α < κ) of constant symbols, then every satisfiable uniform theory of L extending the theory Tκ consisting of the sentences · Discrete(P ), Discrete(
), plus

Metastable convergence and logical compactness

17

· expressing that
is a linear order on the truth set of P , and · P (α), α
β, for α < β < κ, necessarily has a model M such that (αM : α < κ) is not cofinal in (P M ,
M ). Proof. (1) ⇒ (2): Let L be [κ, κ]-compact and let T be a satisfiable uniform theory extending Tκ . For δ < κ and a new constant c, the theory Tδ = T ∪ {P (c)} ∪ {α < c : α < δ} has a model (e.g., the expansion of a model M of Tκ obtained upon interpreting c by δ). By the hypothesis of S [κ, κ]-compactness, δ