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Full Description
This book provides an account of those parts of contemporary set theory of direct relevance to other areas of pure mathematics. The intended reader is either an advanced-level mathematics undergraduate, a beginning graduate student in mathematics, or an accomplished mathematician who desires or needs some familiarity with modern set theory. The book is written in a fairly easy-going style, with minimal formalism. In Chapter 1, the basic principles of set theory are developed in a 'naive' manner. Here the notions of 'set', 'union', 'intersection', 'power set', 'rela tion', 'function', etc., are defined and discussed. One assumption in writing Chapter 1 has been that, whereas the reader may have met all of these 1 concepts before and be familiar with their usage, she may not have con sidered the various notions as forming part of the continuous development of a pure subject (namely, set theory). Consequently, the presentation is at the same time rigorous and fast.
Contents
1 Naive Set Theory.- 1.1 What is a Set?.- 1.2 Operations on Sets.- 1.3 Notation for Sets.- 1.4 Sets of Sets.- 1.5 Relations.- 1.6 Functions.- 1.7 Well-Or der ings and Ordinals.- 1.8 Problems.- 2 The Zermelo—Fraenkel Axioms.- 2.1 The Language of Set Theory.- 2.2 The Cumulative Hierarchy of Sets.- 2.3 The Zermelo—Fraenkel Axioms.- 2.4 Classes.- 2.5 Set Theory as an Axiomatic Theory.- 2.6 The Recursion Principle.- 2.7 The Axiom of Choice.- 2.8 Problems.- 3 Ordinal and Cardinal Numbers.- 3.1 Ordinal Numbers.- 3.2 Addition of Ordinals.- 3.3 Multiplication of Ordinals.- 3.4 Sequences of Ordinals.- 3.5 Ordinal Exponentiation.- 3.6 Cardinality, Cardinal Numbers.- 3.7 Arithmetic of Cardinal Numbers.- 3.8 Regular and Singular Cardinals.- 3.9 Cardinal Exponentiation.- 3.10 Inaccessible Cardinals.- 3.11 Problems.- 4 Topics in Pure Set Theory.- 4.1 The Borel Hierarchy.- 4.2 Closed Unbounded Sets.- 4.3 Stationary Sets and Regressive Functions.- 4.4 Trees.- 4.5 Extensions of Lebesgue Measure.- 4.6 A Result About the GCH.- 5 The Axiom of Constructibility.- 5.1 Constructible Sets.- 5.2 The Constructible Hierarchy.- 5.3 The Axiom of Constructibility.- 5.4 The Consistency of V = L.- 5.5 Use of the Axiom of Constructibility.- 6 Independence Proofs in Set Theory.- 6.1 Some Undecidable Statements.- 6.2 The Idea of a Boolean-Valued Universe.- 6.3 The Boolean-Valued Universe.- 6.4 VB and V.- 6.5 Boolean-Valued Sets and Independence Proofs.- 6.6 The Nonprovability of the CH.- 7 Non-Well-Founded Set Theory.- 7.1 Set-Membership Diagrams.- 7.2 The Anti-Foundation Axiom.- 7.3 The Solution Lemma.- 7.4 Inductive Definitions Under AFA.- 7.5 Graphs and Systems.- 7.6 Proof of the Solution Lemma.- 7.7 Co-Inductive Definitions.- 7.8 A Model of ZF- +AFA.- Glossary of Symbols.
