Book:Alan G. Hamilton/Numbers, Sets and Axioms

Subject Matter

 * Abstract Algebra

Contents

 * Preface


 * 1 Numbers
 * 1.1 Natural numbers and integers
 * 1.2 Rational numbers
 * 1.3 Real numbers
 * 1.4 Decimal notation


 * 2 The size of a set
 * 2.1 Finite and countable sets
 * 2.2 Uncountable sets
 * 2.3 Cardinal numbers


 * 3 Ordered sets
 * 3.1 Order relations and ordered sets
 * 3.2 Lattices and Boolean algebras


 * 4 Set theory
 * 4.1 What is a set?
 * 4.2 The Zermelo-Fraenkel axioms
 * 4.3 Mathematics in ZF
 * 4.4 Sets and classes
 * 4.5 Models of set theory


 * 5 The axiom of choice
 * 5.1 The axiom of choice and direct applications
 * 5.2 Zorn's lemma and the well-ordering theorem
 * 5.3 Other consequences of the axiom of choice


 * 6 Ordinal and cardinal numbers
 * 6.1 Well-ordered sets and ordinal numbers
 * 6.2 Transfinite recursion and ordinal arithmetic
 * 6.3 Cardinal numbers


 * Hints and solutions to selected exercises
 * References
 * Index of symbols
 * Subject index