Sets, Logic, Computation An Open Logic Text Winter 2017 Sets, Logic, Computation The Open Logic Project Instigator Richard Zach, University of Calgary Editorial Board Aldo Antonelli,y University of California, Davis Andrew Arana, Université Paris I Panthénon–Sorbonne Jeremy Avigad, Carnegie Mellon University Walter Dean, University of Warwick Gillian Russell, University of North Carolina Nicole Wyatt, University of Calgary Audrey Yap, University of Victoria Contributors Samara Burns, University of Calgary Dana Hägg, University of Calgary Sets, Logic, Computation An Open Logic Text Remixed by Richard Zach Winter 2017 The Open Logic Project would like to acknowledge the generous support of the Faculty of Arts and the Taylor Institute of Teaching and Learning of the University of Calgary. This resource was funded by the Alberta Open Educational Re- sources (ABOER) Initiative, which is made possible through an investment from the Alberta government. Illustrations by Matthew Leadbeater, used under a Creative Com- mons Attribution-NonCommercial 4.0 International License. Typeset in Baskervald X and Universalis ADF Standard by LATEX. Sets, Logic, Computation by Richard Zach is licensed under a Creative Commons Attribution 4.0 Interna- tional License. It is based on The Open Logic Text by the Open Logic Project, used under a Creative Commons At- tribution 4.0 International License. Contents Preface xi I Sets, Relations, Functions1 1 Sets2 1.1 Basics......................... 2 1.2 Some Important Sets................ 4 1.3 Subsets........................ 5 1.4 Unions and Intersections.............. 6 1.5 Proofs about Sets .................. 8 1.6 Pairs, Tuples, Cartesian Products ......... 10 1.7 Russell’s Paradox .................. 11 Summary.......................... 12 Problems .......................... 12 2 Relations 14 2.1 Relations as Sets................... 14 2.2 Special Properties of Relations........... 16 2.3 Orders ........................ 18 2.4 Graphs........................ 21 2.5 Operations on Relations .............. 22 Summary.......................... 23 Problems .......................... 24 v CONTENTS vi 3 Functions 26 3.1 Basics......................... 26 3.2 Kinds of Functions ................. 28 3.3 Inverses of Functions................ 29 3.4 Composition of Functions ............. 30 3.5 Isomorphism..................... 31 3.6 Partial Functions................... 31 3.7 Functions and Relations .............. 32 Summary.......................... 33 Problems .......................... 34 4 The Size of Sets 35 4.1 Introduction..................... 35 4.2 Countable Sets.................... 35 4.3 Uncountable Sets .................. 41 4.4 Reduction ...................... 45 4.5 Equinumerous Sets ................. 46 4.6 Comparing Sizes of Sets .............. 48 Summary.......................... 50 Problems .......................... 51 II First-order Logic 55 5 Syntax and Semantics 56 5.1 Introduction..................... 56 5.2 First-Order Languages ............... 58 5.3 Terms and Formulas ................ 60 5.4 Unique Readability ................. 63 5.5 Main operator of a Formula ............ 67 5.6 Subformulas..................... 68 5.7 Free Variables and Sentences............ 70 5.8 Substitution ..................... 71 5.9 Structures for First-order Languages . 73 5.10 Covered Structures for First-order Languages . 75 5.11 Satisfaction of a Formula in a Structure . 76 CONTENTS vii 5.12 Extensionality.................... 82 5.13 Semantic Notions.................. 83 Summary.......................... 85 Problems .......................... 87 6 Theories and Their Models 89 6.1 Introduction..................... 89 6.2 Expressing Properties of Structures . 92 6.3 Examples of First-Order Theories......... 93 6.4 Expressing Relations in a Structure . 96 6.5 The Theory of Sets................. 98 6.6 Expressing the Size of Structures . 101 Summary..........................103 Problems ..........................103 7 Natural Deduction 105 7.1 Introduction.....................105 7.2 Rules and Derivations . 107 7.3 Examples of Derivations . 111 7.4 Proof-Theoretic Notions . 120 7.5 Properties of Derivability . 121 7.6 Soundness......................125 7.7 Derivations with Identity predicate . 128 7.8 Soundness of Identity predicate Rules . 130 Summary..........................130 Problems ..........................131 8 The Completeness Theorem 133 8.1 Introduction.....................133 8.2 Outline of the Proof . 134 8.3 Maximally Consistent Sets of Sentences . 136 8.4 Henkin Expansion..................138 8.5 Lindenbaum’s Lemma . 140 8.6 Construction of a Model . 141 8.7 Identity........................143 8.8 The Completeness Theorem . 147 CONTENTS viii 8.9 The Compactness Theorem . 147 8.10 The Löwenheim-Skolem Theorem . 150 Summary..........................152 Problems ..........................153 9 Beyond First-order Logic 154 9.1 Overview.......................154 9.2 Many-Sorted Logic . 155 9.3 Second-Order logic . 157 9.4 Higher-Order logic . 162 9.5 Intuitionistic Logic . 165 9.6 Modal Logics ....................171 9.7 Other Logics.....................173 III Turing Machines 177 10 Turing Machine Computations 178 10.1 Introduction.....................178 10.2 Representing Turing Machines . 181 10.3 Turing Machines...................186 10.4 Configurations and Computations . 187 10.5 Unary Representation of Numbers . 189 10.6 Halting States....................190 10.7 Combining Turing Machines . 191 10.8 Variants of Turing Machines . 193 10.9 The Church-Turing Thesis . 195 Summary..........................196 Problems ..........................197 11 Undecidability 199 11.1 Introduction.....................199 11.2 Enumerating Turing Machines . 201 11.3 The Halting Problem . 203 11.4 The Decision Problem . 205 11.5 Representing Turing Machines . 206 CONTENTS ix 11.6 Verifying the Representation . 209 11.7 The Decision Problem is Unsolvable . 214 Summary..........................215 Problems ..........................216 A Induction 219 A.1 Introduction.....................219 A.2 Induction on N . 220 A.3 Strong Induction ..................223 A.4 Inductive Definitions . 224 A.5 Structural Induction . 227 B Biographies 229 B.1 Georg Cantor ....................229 B.2 Alonzo Church ...................230 B.3 Gerhard Gentzen ..................231 B.4 Kurt Gödel......................233 B.5 Emmy Noether ...................235 B.6 Bertrand Russell...................236 B.7 Alfred Tarski.....................238 B.8 Alan Turing .....................239 B.9 Ernst Zermelo....................241 Glossary 245 Photo Credits 251 Bibliography 253 About the Open Logic Project 258 Preface This book is an introduction to meta-logic, aimed especially at students of computer science and philosophy. “Meta-logic” is so- called because it is the discipline that studies logic itself. Logic proper is concerned with canons of valid inference, and its sym- bolic or formal version presents these canons using formal lan- guages, such as those of propositional and predicate, a.k.a., first- order logic. Meta-logic investigates the properties of these lan- guage, and of the canons of correct inference that use them. It studies topics such as how to give precise meaning to the ex- pressions of these formal languages, how to justify the canons of valid inference, what the properties of various proof systems are, including their computational properties. These questions are important and interesting in their own right, because the lan- guages and proof systems investigated are applied in many dier- ent areas—in mathematics, philosophy, computer science, and linguistics, especially—but they also serve as examples of how to study formal systems in general. The logical languages we study here are not the only ones people are interested in. For instance, linguists and philosophers are interested in languages that are much more complicated than those of propositional and first-order logic, and computer scientists are interested in other kinds of languages altogether, such as programming languages. And the methods we discuss here—how to give semantics for for- mal languages, how to prove results about formal languages, how xi PREFACE xii to investigate the properties of formal languages—are applicable in those cases as well. Like any discipline, meta-logic both has a set of results or facts, and a store of methods and techniques, and this text cov- ers both. Some students won’t need to know some of the results we discuss outside of this course, but they will need and use the methods we use to establish them. The Löwenheim-Skolem the- orem, say, does not often make an appearance in computer sci- ence, but the methods we use to prove it do. On the other hand, many of the results we discuss do have relevance for certain de- bates, say, in the philosophy of science and in metaphysics. Phi- losophy students may not need to be able to prove these results outside this course, but they do need to understand what the results are—and you really only understand these results if you have thought through the definitions and proofs needed to es- tablish them. These are, in part, the reasons for why the results and the methods covered in this text are recommended study—in some cases even required—for students of computer science and philosophy. The material is divided into three parts. Part 1 concerns it- self with the theory of sets. Logic and meta-logic is historically connected very closely to what’s called the “foundations of math- ematics.” Mathematical