# Set Theory 9781848900509

645 124 4MB

English Pages 412 Year 2013

Cover
Title Page
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