Conditional Proof. Proof by Counter Example. Negation of Quantified Predicates. PRELIMINARIES all of mathematics. . Multiple Quantifiers. Set Theory \A set is a Many that allows itself to be thought of as a One." De ning a set formally is a pretty delicate matter, for now, we will be happy to consider an intuitive de nition, namely: De nition 24. Set Theory and Logic Supplementary Materials Math 103: Contemporary Mathematics with Applications A. Calini, E. Jurisich, S. Shields c 2008. Predicate Logic and Quantifiers. (Georg Cantor) In the previous chapters, we have often encountered "sets", for example, prime numbers form a set, domains in predicate logic form sets as well. The axioms of set theory.....67 6. g Yet the density of squares goes down as we go up. A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. – Ian Stewart Does God play dice? 8 In the Two New Sciences book, his final masterpiece, Galileo observed that there is a one-to-one correspondence between natural numbers and squares: n !n2 f1,2,3,. IV. Unique Existence. Methods of Proof. For example, a deck of cards, every student enrolled in Math 103, the collection of all even integers, these are all examples of sets of things. g !f1,4,9,. Predicates. First-order logic.....12 3. The completeness theorem .....42 ELEMENTARY SET THEORY 5. 8. 1.1 Statements A proof in mathematics demonstrates the truth of certain statement . Indirect Proof. Sentential logic.....1 2. Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Universal and Existential Quantifiers. III. 2 CHAPTER 1. V. Naïve Set Theory. ii. Our main purpose here is to learn how to state mathematical results clearly and how to prove them. 1.1 Sets Mathematicians over the last two centuries have been used to the idea of considering a collection of objects/numbers as a single entity. . TABLE OF CONTENTS LOGIC 1. 2. Formal Proof. Informal Proof. The proof that p = t in Chapter 34 is based upon notes of Fremlin and a thesis of Roccasalvo. Proofs .....24 4. We start with the basic set theory. This chapter will be devoted to understanding set theory, relations, functions. Mathematical Induction. Also, their activity led to the view that logic + set theory can serve as a basis for 1. Abstraction is what makes mathematics work. . This era did not produce theorems in mathematical logic of any real depth, 1 but it did bring crucial progress of a conceptual nature, and the recognition that logic as used in mathematics obeys mathematical rules that can be made fully explicit. Notes on Set Theory and Logic August 29, 2013. These entities are … . It is therefore natural to begin with a brief discussion of statements. Chapter 1 Set Theory 1.1 Basic definitions and notation A set is a collection of objects. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. academic career, you may wish to study set theory and logic in greater detail. The consistency proofs in Chapter 35 are partly from Kunen and partly from the author.