Set Theory · A First Course by Cunningham, Daniel W. -- Read -- Imperial Library of Trantor

Index

Cover Half title Series Title Copyright Table of 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