Set Theory: A First Course

Description

Set theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises. Contents Preface The Greek Alphabet 1 Introduction 1.1 Elementary Set Theory 1.2 Logical Notation 1.3 Predicates and Quantifiers 1.4 A Formal Language for Set Theory 1.5 The Zermelo–Fraenkel Axioms 2 Basic Set-Building Axioms and Operations 2.1 The First Six Axioms 2.1.1 The Extensionality Axiom 2.1.2 The Empty Set Axiom 2.1.3 The Subset Axiom 2.1.4 The Pairing Axiom 2.1.5 The Union Axiom 2.1.6 The Power Set Axiom 2.2 Operations on Sets 2.2.1 De Morgan’s Laws for Sets 2.2.2 Distributive Laws for Sets 3 Relations and Functions 3.1 Ordered Pairs in Set Theory 3.2 Relations 3.2.1 Operations on Relations 3.2.2 Reflexive, Symmetric, and Transitive Relations 3.2.3 Equivalence Relations and Partitions 3.3 Functions 3.3.1 Operations on Functions 3.3.2 One-to-One Functions 3.3.3 Indexed Sets 3.3.4 The Axiom of Choice 3.4 Order Relations 3.5 Congruence and Preorder 4 The Natural Numbers 4.1 Inductive Sets 4.2 The Recursion Theorem on ω 4.2.1 The Peano Postulates 4.3 Arithmetic on ω 4.4 Order on ω 5 On the Size of Sets 5.1 Finite Sets 5.2 Countable Sets 5.3 Uncountable Sets 5.4 Cardinality 6 Transfinite Recursion 6.1 Well-Ordering 6.2 Transfinite Recursion Theorem 6.2.1 Using a Set Function 6.2.2 Using a Class Function 7 The Axiom of Choice (Revisited) 7.1 Zorn’s Lemma 7.1.1 Two Applications of Zorn’s Lemma 7.2 Filters and Ultrafilters 7.2.1 Ideals 7.3 Well-Ordering Theorem 8 Ordinals 8.1 Ordinal Numbers 8.2 Ordinal Recursion and Class Functions 8.3 Ordinal Arithmetic 8.4 The Cumulative Hierarchy 9 Cardinals 9.1 Cardinal Numbers 9.2 Cardinal Arithmetic 9.3 Closed Unbounded and Stationary Sets Notes References Index of Special Symbols Index