Notes on Mathematical Logic David W. Kueker University of Maryland, College Park E-mail address: [email protected] URL: http://www-users.math.umd.edu/~dwk/ Contents Chapter 0. Introduction: What Is Logic? 1 Part 1. Elementary Logic 5 Chapter 1. Sentential Logic 7 0. Introduction 7 1. Sentences of Sentential Logic 8 2. Truth Assignments 11 3. Logical Consequence 13 4. Compactness 17 5. Formal Deductions 19 6. Exercises 20 20 Chapter 2. First-Order Logic 23 0. Introduction 23 1. Formulas of First Order Logic 24 2. Structures for First Order Logic 28 3. Logical Consequence and Validity 33 4. Formal Deductions 37 5. Theories and Their Models 42 6. Exercises 46 46 Chapter 3. The Completeness Theorem 49 0. Introduction 49 1. Henkin Sets and Their Models 49 2. Constructing Henkin Sets 52 3. Consequences of the Completeness Theorem 54 4. Completeness Categoricity, Quantifier Elimination 57 5. Exercises 58 58 Part 2. Model Theory 59 Chapter 4. Some Methods in Model Theory 61 0. Introduction 61 1. Realizing and Omitting Types 61 2. Elementary Extensions and Chains 66 3. The Back-and-Forth Method 69 i ii CONTENTS 4. Exercises 71 71 Chapter 5. Countable Models of Complete Theories 73 0. Introduction 73 1. Prime Models 73 2. Universal and Saturated Models 75 3. Theories with Just Finitely Many Countable Models 77 4. Exercises 79 79 Chapter 6. Further Topics in Model Theory 81 0. Introduction 81 1. Interpolation and Definability 81 2. Saturated Models 84 3. Skolem Functions and Indescernables 87 4. Some Applications 91 5. Exercises 95 95 Appendix A. Appendix A: Set Theory 97 1. Cardinals and Counting 97 2. Ordinals and Induction 100 Appendix B. Appendix B: Notes on Validities and Logical Consequence 103 1. Some Useful Validities of Sentential Logic 103 2. Some Facts About Logical Consequence 104 Appendix C. Appendix C: Gothic Alphabet 105 Bibliography 107 Index 109 CHAPTER 0 Introduction: What Is Logic? Mathematical logic is the study of mathematical reasoning. We do this by developing an abstract model of the process of reasoning in mathematics. We then study this model and determine some of its properties. Mathematical reasoning is deductive; that is, it consists of drawing (correct) inferences from given or already established facts. Thus the basic concept is that of a statement being a logical consequence of some collection of statements. In ordinary mathematical English the use of \therefore" customarily means that the statement following it is a logical consequence of what comes before. Every integer is either even or odd; 7 is not even; therefore 7 is odd. In our model of mathematical reasoning we will need to precisely define logical consequence. To motivate our definition let us examine the everyday notion. When we say that a statement σ is a logical consequence of (\follows from") some other statements θ1; : : : ; θn, we mean, at the very least, that σ is true provided θ1; : : : ; θn are all true. Unfortunately, this does not capture the essence of logical consequence. For example, consider the following: Some integers are odd; some integers are prime; therefore some integers are both odd and prime. Here the hypotheses are both true and the conclusion is true, but the reasoning is not correct. The problem is that for the reasoning to be logically correct it cannot depend on properties of odd or prime integers other than what is explicitly stated. Thus the reasoning would remain correct if odd, prime, and integer were changed to something else. But in the above example if we replaced prime by even we would have true hypotheses but a false conclusion. This shows that the reasoning is false, even in the original version in which the conclusion was true. The key observation here is that in deciding whether a specific piece of rea- soning is or is not correct we must consider alMathematical logic is the study of mathematical reasoning. We do this by developing an abstract model of the process of reasoning in mathematics. We then study this model and determine some of its properties. Mathematical reasoning is deductive; that is, it consists of drawing (correct) inferences from given or already established facts. Thus the basic concept is that of a statement being a logical consequence of some collection of statements. In ordinary mathematical English the use of \therefore" customarily means that the statement following it is a logical consequence of what l ways of interpreting the undefined concepts|integer, odd, and prime in the above example. This is conceptually easier 1 2 0. INTRODUCTION: WHAT IS LOGIC? in a formal language in which the basic concepts are represented by symbols (like P , Q) without any standard or intuitive meanings to mislead one. Thus the fundamental building blocks of our model are the following: (1) a formal language L, (2) sentences of L: σ; θ; : : :, (3) interpretations for L: A; B;:::, (4) a relation j= between interpretations for L and sentences of L, with A j= σ read as \σ is true in the interpretation A," or \A is a model of σ." Using these we can define logical consequence as follows: Definition -1.1. Let Γ = fθ1; : : : ; θng where θ1; : : : ; θn are sentences of L, and let σ be a sentence of L. Then σ is a logical consequence of Γ if and only if for every interpretation A of L, A j= σ provided A j= θi for all i = 1; : : : ; n. Our notation for logical consequence is Γ j= σ. In particular note that Γ 6j= σ, that is, σ is not a logical consequence of Γ, if and only if there is some interpretation A of L such that A j= θi for all θi 2 Γ but A 6j= σ, A is not a model of σ. As a special limiting case note that ; j= σ, which we will write simply as j= σ, means that A j= σ for every interpretation A of L. Such a sentence σ is said to be logically true (or valid). How would one actually show that Γ j= σ for specific Γ and σ? There will be infinitely many different interpretations for L so it is not feasible to check each one in turn, and for that matter it may not be possible to decide whether a par- ticular sentence is or is not true on a particular structure. Here is where another fundamental building block comes in, namely the formal analogue of mathematical proofs. A proof of σ from a set Γ of hypotheses is a finite sequence of statements σ0; : : : ; σk where σ is σk and each statement in the sequence is justified by some explicitly stated rule which guarantees that it is a logical consequence of Γ and the preceding statements. The point of requiring use only of rules which are explicitly stated and given in advance is that one should be able to check whether or not a given sequence σ0; : : : ; σk is a proof of σ from Γ. The notation Γ ` σ will mean that there is a formal proof (also called a deduc- tion or derivation) of σ from Γ. Of course this notion only becomes precise when we actually give the rules allowed. Provided the rules are correctly chosen, we will have the implication if Γ ` σ then Γ j= σ. Obviously we want to know that our rules are adequate to derive all logical consequences. That is the content of the following fundamental result: Theorem -1.1 (Completeness Theorem (K. G¨odel)). For sentences of a first- order language L, we have Γ ` σ if and only if Γ j= σ. First-order languages are the most widely studied in modern mathematical logic, largely to obtain the benefit of the Completeness Theorem and its applica- tions. In these notes we will study first-order languages almost exclusively. Part ?? is devoted to the detailed construction of our \model of reasoning" for first-order languages. It culminates in the proof of the Completeness Theorem and derivation of some of its consequences. 0. INTRODUCTION: WHAT IS LOGIC? 3 Part ?? is an introduction to Model Theory. If Γ is a set of sentences of L, then Mod(Γ), the class of all models of Γ, is the class of all interpretations of L which make all sentences in Γ true. Model Theory discusses the properties such classes of interpretations have. One important result of model theory for first-order languages is the Compactness Theorem, which states that if Mod(Γ) = ; then there must be some finite Γ0 ⊆ Γ with Mod(Γ0) = ;. Part ?? discusses the famous incompleteness and undecidability results of G'odel, Church, Tarski, et al. The fundamental problem here (the decision problem) is whether there is an effective procedure to decide whether or not a sentence is logi- cally true. The Completeness Theorem does not automatically yield such a method. Part ?? discusses topics from the abstract theory of computable functions (Re- cursion Theory). Part 1 Elementary Logic CHAPTER 1 Sentential Logic 0. Introduction Our goal, as explained in Chapter 0, is to define a class of formal languages whose sentences include formalizations of the sttements commonly used in math- ematics and whose interpretatins include the usual mathematical structures. The details of this become quite intricate, which obscures the \big picture." We there- fore first consider a much simpler situation and carry out our program in this simpler context. The outline remains the same, and we will use some of the same ideas and techniques{especially the interplay of definition by recursion and proof by induction{when we come to first-order languages.