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Models and Games Incomplete version for students of easllc2012 only. Models and Games JOUKO VÄÄNÄNEN University of Helsinki University of Amsterdam Incomplete version for students of easllc2012 only. cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521518123 © J. Väänänen 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library ISBN 978-0-521-51812-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Incomplete version for students of easllc2012 only. Contents Preface page xi 1 Introduction 1 2 Preliminaries and Notation 3 2.1 Finite Sequences 3 2.2 Equipollence 5 2.3 Countable sets 6 2.4 Ordinals 7 2.5 Cardinals 9 2.6 Axiom of Choice 10 2.7 Historical Remarks and References 11 Exercises 11 3 Games 14 3.1 Introduction 14 3.2 Two-Person Games of Perfect Information 14 3.3 The Mathematical Concept of Game 20 3.4 Game Positions 21 3.5 Infinite Games 24 3.6 Historical Remarks and References 28 Exercises 28 4 Graphs 35 4.1 Introduction 35 4.2 First-Order Language of Graphs 35 4.3 The Ehrenfeucht–Fra¨ısse´ Game on Graphs 38 4.4 Ehrenfeucht–Fra¨ısse´ Games and Elementary Equivalence 43 4.5 Historical Remarks and References 48 Exercises 49 Incomplete version for students of easllc2012 only. viii Contents 5 Models 53 5.1 Introduction 53 5.2 Basic Concepts 54 5.3 Substructures 62 5.4 Back-and-Forth Sets 63 5.5 The Ehrenfeucht–Fra¨ısse´ Game 65 5.6 Back-and-Forth Sequences 69 5.7 Historical Remarks and References 71 Exercises 71 6 First-Order Logic 79 6.1 Introduction 79 6.2 Basic Concepts 79 6.3 Characterizing Elementary Equivalence 81 6.4 The Lowenheim–Skolem¨ Theorem 85 6.5 The Semantic Game 93 6.6 The Model Existence Game 98 6.7 Applications 102 6.8 Interpolation 107 6.9 Uncountable Vocabularies 113 6.10 Ultraproducts 119 6.11 Historical Remarks and References 125 Exercises 126 7 Infinitary Logic 139 7.1 Introduction 139 7.2 Preliminary Examples 139 7.3 The Dynamic Ehrenfeucht–Fra¨ısse´ Game 144 7.4 Syntax and Semantics of Infinitary Logic 157 7.5 Historical Remarks and References 170 Exercises 171 8 Model Theory of Infinitary Logic 176 8.1 Introduction 176 8.2 Lowenheim–Skolem¨ Theorem for L ! 176 1 8.3 Model Theory of L!1! 179 8.4 Large Models 184 8.5 Model Theory of L+! 191 8.6 Game Logic 201 8.7 Historical Remarks and References 222 Exercises 223 Incomplete version for students of easllc2012 only. Contents ix 9 Stronger Infinitary Logics 228 9.1 Introduction 228 9.2 Infinite Quantifier Logic 228 9.3 The Transfinite Ehrenfeucht–Fra¨ısse´ Game 249 9.4 A Quasi-Order of Partially Ordered Sets 254 9.5 The Transfinite Dynamic Ehrenfeucht–Fra¨ısse´ Game 258 9.6 Topology of Uncountable Models 270 9.7 Historical Remarks and References 275 Exercises 278 10 Generalized Quantifiers 283 10.1 Introduction 283 10.2 Generalized Quantifiers 284 10.3 The Ehrenfeucht–Fra¨ısse´ Game of Q 296 10.4 First-Order Logic with a Generalized Quantifier 307 10.5 Ultraproducts and Generalized Quantifiers 312 10.6 Axioms for Generalized Quantifiers 314 10.7 The Cofinality Quantifier 333 10.8 Historical Remarks and References 342 Exercises 343 References 353 Index 362 Incomplete version for students of easllc2012 only. 1 Introduction A recurrent theme in this book is the concept of a game. There are essentially three kinds of games in logic. One is the Semantic Game, also called the Eval- uation Game, where the truth of a given sentence in a given model is at issue. Another is the Model Existence Game, where the consistency in the sense of having a model, or equivalently in the sense of impossibility to derive a con- tradiction, is at issue. Finally there is the Ehrenfeucht–Fra¨ısse´ Game, where separation of a model from another by finding a property that is true in one given model but false in another is the goal. The three games are closely linked to each other and one can even say they are essentially variants of just one basic game. This basic game arises from our understanding of the quantifiers. The purpose of this book is to make this strategic aspect of logic perfectly transpar- ent and to show that it underlies not only first-order logic but infinitary logic and logic with generalized quantifiers alike. We call the close link between the three games the Strategic Balance of Logic (Figure 1.1). This balance is perfectly commutative, in the sense that winning strategies can be transferred from one game to another. This mere fact is testimony to the close connection between logic and games, or, thinking semantically, between games and models. This connection arises from the na- ture of quantifiers. Introducing infinite disjunctions and conjunctions does not upset the balance, barring some set-theoretic issues that may surface. In the last chapter of this book we consider generalized quantifiers and show that the Strategic Balance of Logic persists even in the presence of generalized quanti- fiers. The purpose of this book is to present the Strategic Balance of Logic in all its glory. Incomplete version for students of easllc2012 only. 2 Introduction Figure 1.1 The Strategic Balance of Logic. Incomplete version for students of easllc2012 only. 2 Preliminaries and Notation We use some elementary set theory in this book, mainly basic properties of countable and uncountable sets. We will occasionally use the concept of count- able ordinal when we index some uncountable sets. There are many excellent books on elementary set theory. (See Section 2.7.) We give below a simplified account of some basic concepts, the barest outline necessary for this book. We denote the set 0, 1, 2,... of all natural numbers by N, the set of ra- { } tional numbers by Q, and the set of all real numbers by R. The power-set operation is written (A)= B : B A . P { ✓ } We use A B to denote the set-theoretical difference of the sets A and B. \ 1 If f is a function, f 00X is the set f(x):x X and f − (X) is the set { 2 } x dom(f):f(x) X . Composition of two functions f and g is denoted { 2 2 } g f and defined by (g f)(x)=g(f(x)). We often write fa for f(a). ◦ ◦ The notation id is used for the identity function A A which maps every A ! element of A to itself, i.e. id (a)=a for a A. A 2 2.1 Finite Sequences The concept of a finite (ordered) sequence s =(a0,...,an 1) − of elements of a given set A plays an important role in this book. Examples of finite sequences of elements of N are (8, 3, 9, 67, 200, 0) (8, 8, 8) Incomplete version for students of easllc2012 only. 4 Preliminaries and Notation (24). We can identify the sequence s =(a0,...,an 1) with the function − s0 : 0,...,n 1 A, { − } ! where s0(i)=ai. The main property of finite sequences is: (a0,...,an 1)=(b0,...,bm 1) if − − and only if n = m and ai = bi for all i<n. The number n is called the length of the sequence s =(a0,...,an 1) and is denoted len(s). A special case is − the case len(s)=0. Then s is called the empty sequence. There is exactly one empty sequence and it is denoted by . ; The Cartesian product of two sets A and B is written A B = (a, b):a A, b B . ⇥ { 2 2 } More generally A0 ... An 1 = (a0,...,an 1):ai Ai for all i<n . ⇥ ⇥ − { − 2 } An = A ... A (n times). ⇥ ⇥ According to this definition, A1 = A. The former consists of sequences of 6 length 1 of elements of A. Note that A0 = . {;} Finite Sets A set A is finite if it is of the form a0,...,an 1 for some natural number n. { − } This means that the set A has at most n elements. If A has exactly n elements we write A = n and call A the cardinality of A. A set which is not finite is | | | | infinite. Finite sets form a so-called ideal, which means that: 1. is finite. ; 2. If A and B are finite, then so is A B. [ 3. If A is finite and B A, then also B is finite. ✓ Further useful properties of finite sets are: 1. If A and B are finite, then so is A B. ⇥ 2. If A is finite, then so is (A). P Incomplete version for students of easllc2012 only. 2.2 Equipollence 5 The Axiom of Choice says that for every set A of non-empty sets there is a function f such that f(a) a for all a A.
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