Jindrich Zapletal is Professor of Mathematics at University of Florida, specializing in mathematical logic and set theory. He received his Ph.D. in 1995 from the Pennsylvania State University, and held postdoctoral positions at MSRI Berkeley, Cal Tech, and Dartmouth College, before joining the University of Florida in 2000.
Contents
3. Zermelo–Fraenkel Set Theory
3.1 Historical Context
3.2 The Language of the Theory
3.5 Axiom Schema of Comprehension
3.7 Axiom Schema of Replacement
4. Natural Numbers and Countable Sets
4.1 Von Neumann’s Natural Numbers
4.3 Inductive and Recursive Definability
4.5 Countable and Uncountable Sets
5. Ordinal Numbers and the Transfinite
5.2 Transfinite Induction and Recursion
5.4 Ordinals and Well-Orderings
6. Cardinality and the Axiom of Choice
6.1 Equivalent Versions of the Axiom of Choice
6.2 Applications of the Axiom of Choice
7.1 Integers and Rational Numbers
7.4 Countable and Uncountable Sets of Reals
8.1 The Hereditarily Finite Sets
9.2 Countably Infinite Patterns
Chapter 1
Introduction
Set theory and mathematical logic compose the foundation of pure mathematics. Using the axioms of set theory, we can construct our universe of discourse, beginning with the natural numbers, moving on with sets and functions over the natural numbers, integers, rationals and real numbers, and eventually developing the transfinite ordinal and cardinal numbers. Mathematical logic provides the language of higher mathematics which allows one to frame the definitions, lemmas, theorems, and conjectures which form the everyday work of mathematicians. The axioms and rules of deduction set up the system in which we can prove our conjectures, thus turning them into theorems.
Chapter 2 begins with elementary naive set theory, including the algebra of sets under union, intersection, and complement and their connection with elementary logic. This chapter introduces the notions of relations, functions,