SUNY Geneseo KnightScholar Geneseo Authors Milne Library Publishing 1-1-2015 A Friendly Introduction to Mathematical Logic Christopher C. Leary SUNY Geneseo Lars Kristiansen Follow this and additional works at: https://knightscholar.geneseo.edu/geneseo-authors This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License. Recommended Citation Leary, Christopher C. and Kristiansen, Lars, "A Friendly Introduction to Mathematical Logic" (2015). Geneseo Authors. 6. https://knightscholar.geneseo.edu/geneseo-authors/6 This Book is brought to you for free and open access by the Milne Library Publishing at KnightScholar. It has been accepted for inclusion in Geneseo Authors by an authorized administrator of KnightScholar. For more information, please contact [email protected]. oduc Intr tio y n dl to n ie r F A Mathematical Logic 2nd Edition Christopher C. Leary Lars Kristiansen A Friendly Introduction to Mathematical Logic A Friendly Introduction to Mathematical Logic 2nd Edition Second printing With corrections and some renumbered exercises Christopher C. Leary State University of New York College at Geneseo Lars Kristiansen The University of Oslo Milne Library, SUNY Geneseo, Geneseo, NY c 2015 Christopher C. Leary and Lars Kristiansen ISBN: 978-1-942341-07-9 Milne Library SUNY Geneseo One College Circle Geneseo, NY 14454 Lars Kristiansen has received financial support from the Norwegian Non-fiction Literature Fund Contents Preface ix 1 Structures and Languages 1 1.1 Na¨ıvely . .3 1.2 Languages . .5 1.3 Terms and Formulas . .9 1.4 Induction . 13 1.5 Sentences . 19 1.6 Structures . 22 1.7 Truth in a Structure . 27 1.8 Substitutions and Substitutability . 33 1.9 Logical Implication . 36 1.10 Summing Up, Looking Ahead . 38 2 Deductions 41 2.1 Na¨ıvely . 41 2.2 Deductions . 43 2.3 The Logical Axioms . 48 2.4 Rules of Inference . 50 2.5 Soundness . 54 2.6 Two Technical Lemmas . 58 2.7 Properties of Our Deductive System . 62 2.8 Nonlogical Axioms . 66 2.9 Summing Up, Looking Ahead . 71 3 Completeness and Compactness 73 3.1 Na¨ıvely . 73 3.2 Completeness . 74 3.3 Compactness . 87 3.4 Substructures and the L¨owenheim{Skolem Theorems . 94 3.5 Summing Up, Looking Ahead . 102 v vi CONTENTS 4 Incompleteness from Two Points of View 103 4.1 Introduction . 103 4.2 Complexity of Formulas . 105 4.3 The Roadmap to Incompleteness . 108 4.4 An Alternate Route . 109 4.5 How to Code a Sequence of Numbers . 109 4.6 An Old Friend . 113 4.7 Summing Up, Looking Ahead . 115 5 Syntactic Incompleteness|Groundwork 117 5.1 Introduction . 117 5.2 The Language, the Structure, and the Axioms of N ..... 118 5.3 Representable Sets and Functions . 119 5.4 Representable Functions and Computer Programs . 129 5.5 Coding|Na¨ıvely . 133 5.6 Coding Is Representable . 136 5.7 G¨odelNumbering . 139 5.8 G¨odelNumbers and N ..................... 142 5.9 Num and Sub Are Representable . 147 5.10 Definitions by Recursion Are Representable . 153 5.11 The Collection of Axioms Is Representable . 156 5.12 Coding Deductions . 158 5.13 Summing Up, Looking Ahead . 167 6 The Incompleteness Theorems 169 6.1 Introduction . 169 6.2 The Self-Reference Lemma . 170 6.3 The First Incompleteness Theorem . 174 6.4 Extensions and Refinements of Incompleteness . 182 6.5 Another Proof of Incompleteness . 185 6.6 Peano Arithmetic and the Second Incompleteness Theorem 187 6.7 Summing Up, Looking Ahead . 193 7 Computability Theory 195 7.1 The Origin of Computability Theory . 195 7.2 The Basics . 197 7.3 Primitive Recursion . 204 7.4 Computable Functions and Computable Indices . 215 7.5 The Proof of Kleene's Normal Form Theorem. 225 7.6 Semi-Computable and Computably Enumerable Sets . 235 7.7 Applications to First-Order Logic . 244 7.8 More on Undecidability . 254 CONTENTS vii 8 Summing Up, Looking Ahead 265 8.1 Once More, With Feeling . 266 8.2 The Language LBT and the Structure B........... 266 8.3 Nonstandard LBT -structures . 271 8.4 The Axioms of B ........................ 271 8.5 B extended with an induction scheme . 274 8.6 Incompleteness . 276 8.7 Off You Go . 278 Appendix: Just Enough Set Theory to Be Dangerous 279 Solutions to Selected Exercises 283 Bibliography 359 Preface Preface to the First Edition This book covers the central topics of first-order mathematical logic in a way that can reasonably be completed in a single semester. From the core ideas of languages, structures, and deductions we move on to prove the Soundness and Completeness Theorems, the Compactness Theorem, and G¨odel'sFirst and Second Incompleteness Theorems. There is an introduction to some topics in model theory along the way, but I have tried to keep the text tightly focused. One choice that I have made in my presentation has been to start right in on the predicate logic, without discussing propositional logic first. I present the material in this way as I believe that it frees up time later in the course to be spent on more abstract and difficult topics. It has been my experience in teaching from preliminary versions of this book that students have responded well to this choice. Students have seen truth tables before, and what is lost in not seeing a discussion of the completeness of the propositional logic is more than compensated for in the extra time for G¨odel'sTheorem. I believe that most of the topics I cover really deserve to be in a first course in mathematical logic. Some will question my inclusion of the L¨owenheim{Skolem Theorems, and I freely admit that they are included mostly because I think they are so neat. If time presses you, that sec- tion might be omitted. You may also want to soft-pedal some of the more technical results in Chapter 5. The list of topics that I have slighted or omitted from the book is de- pressingly large. I do not say enough about recursion theory or model theory. I say nothing about linear logic or modal logic or second-order logic. All of these topics are interesting and important, but I believe that they are best left to other courses. One semester is, I believe, enough time to cover the material outlined in this book relatively thoroughly and at a reasonable pace for the student. Thanks for choosing my book. I would love to hear how it works for you. ix x Preface To the Student Welcome! I am really thrilled that you are interested in mathematical logic and that we will be looking at it together! I hope that my book will serve you well and will help to introduce you to an area of mathematics that I have found fascinating and rewarding. Mathematical logic is absolutely central to mathematics, philosophy, and advanced computer science. The concepts that we discuss in this book|models and structures, completeness and incompleteness|are used by mathematicians in every branch of the subject. Furthermore, logic pro- vides a link between mathematics and philosophy, and between mathe- matics and theoretical computer science. It is a subject with increasing applications and of great intrinsic interest. One of the tasks that I set for myself as I wrote this book was to be mindful of the audience, so let me tell you the audience that I am trying to reach with this book: third- or fourth-year undergraduate students, most likely mathematics students. The student I have in mind may not have taken very many upper-division mathematics courses. He or she may have had a course in linear algebra, or perhaps a course in discrete mathematics. Neither of these courses is a prerequisite for understanding the material in this book, but some familiarity with proving things will be required. In fact, you don't need to know very much mathematics at all to follow this text. So if you are a philosopher or a computer scientist, you should not find any of the core arguments beyond your grasp. You do, however, have to work abstractly on occasion. But that is hard for all of us. My suggestion is that when you are lost in a sea of abstraction, write down three examples and see if they can tell you what is going on. At several points in the text there are asides that are indented and start with the word Chaff. I hope you will find these comments helpful. They are designed to restate difficult points or emphasize important things that may get lost along the way. Sometimes they are there just to break up the exposition. But these asides really are chaff, in the sense that if they were blown away in the wind, the mathematics that is left would be correct and secure. But do look at them|they are supposed to make your life easier. Just like every other math text, there are exercises and problems for you to work out. Please try to at least think about the problems. Mathematics is a contact sport, and until you are writing things down and trying to use and apply the material you have been studying, you don't really know the subject. I have tried to include problems of different levels of difficulty, so some will be almost trivial and others will give you a chance to show off. This is an elementary textbook, but elementary does not mean easy. It was not easy when we learned to add, or read, or write. You will find the going tough at times as we work our way through some very difficult and technical results.