Table of contents : Cover Title Page Copyright Preface Contents Chapter 0 Introduction 0.1 Why Read This Book? 0.2 Prerequisites 0.3 How to Read This Book Chapter I Background Material I.1 The Formalist Philosophy of Mathematics I.1.1 On doing things twice I.2 Formal Logic I.3 The Axioms of Set Theory I.4 Extensionality, Comprehension, Pairing, Union I.5 Infinity, Replacement, and Power Set I.6 Discrete Mathematics; Relations and Functions I.7 Ordinals I I.8 Ordinals II I.9 Induction and Recursion and Foundation I.10 Cardinalities I.11 Uncountable Cardinalities I.12 The Axiom of Choice (AC) I.13 Cardinal Arithmetic I.14 The Identity of Mathematical Objects I.15 Model Theory I.16 Models of Set Theory I.17 Recursion Theory Chapter II Easy Consistency Proofs II.1 Informal Remarks on Consistency Proofs II.2 The Consistency of Foundation II.3 The Last Word on Foundation II.4 More on Absoluteness II.5 Reflection Theorems II.6 The Constructible Sets II.7 The Forcing Idea II.8 Ordinal Definable Sets II.9 The Independence of Foundation II.10 Set Theory with Classes Chapter III Infinitary Combinatorics III.1 Some Small Cardinals III.2 The Countable Chain Condition III.3 Martin's Axiom III.4 Equivalents of MA III.5 Trees III.6 Club Filters III.7 ◊ and ◊+ III.8 Elementary Submodels Chapter IV Forcing IV.1 The Forcing Idea IV.2 Generic Extensions IV.3 Computing Cardinal Exponentiation IV.4 Embeddings of Posets IV.5 The Metamathematics of Forcing IV.5.1 Countable Transitive Models IV.5.2 Forcing over the Universe IV.6 Current Forcing Notation IV.7 Further Results and Posets IV.8 Independence of the Axiom of Choice Chapter V Iterated Forcing V.1 Products V.2 Applications of Products V.3 General Two-Step Iteration V.4 Finite Support Iteration V.5 Other Iterations V.6 Independence from Martin's Axiom V.7 Proper Forcing V.7.1 Some Applications V.7.2 A Combinatorial Equivalent V.7.3 Another Application V.7.4 Concluding Remarks Bibliography Indices Index of Symbols General Index Back Cover