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Tables of Contents for Set Theory
Chapter/Section Title
Page #
Page Count
Part I. Basic Set Theory
Axioms of Set Theory
3
14
Axioms of Zermelo-Fraenkel
Why Axiomatic Set Theory
Language of Set Theory, Formulas
Classes
Extensionality
Pairing
Separation Schema
Union
Power Set
Infinity
Replacement Schema
Exercises
Historical Notes
Ordinal Numbers
17
10
Linear and Partial Ordering
Well-Ordering
Ordinal Numbers
Induction and Recursion
Ordinal Arithmetic
Well-Founded Relations
Exercises
Historical Notes
Cardinal Numbers
27
10
Cardinality
Alephs
The Canonical Well-Ordering of α x α
Cofinality
Exercises
Historical Notes
Real Numbers
37
10
The Cardinality of the Continuum
The Ordering of R
Suslin's Problem
The Topology of the Real Line
Borel Sets
Lebesgue Measure
The Baire Space
Polish Spaces
Exercises
Historical Notes
The Axiom of Choice and Cardinal Arithmetic
47
16
The Axiom of Choice
Using the Axiom of Choice in Mathematics
The Countable Axiom of Choice
Cardinal Arithmetic
Infinite Sums and Products
The Continuum Function
Cardinal Exponentiation
The Singular Cardinal Hypothesis
Exercises
Historical Notes
The Axiom of Regularity
63
10
The Cumulative Hierarchy of Sets
E-Induction
Well-Founded Relations
The Bernays-Godel Axiomatic Set Theory
Exercises
Historical Notes
Filters, Ultrafilters and Boolean Algebras
73
18
Filters and Ultrafilters
Ultrafilters on ω
κ-Complete Filters and Ideals
Boolean Algebras
Ideals and Filters on Boolean Algebras
Complete Boolean Algebras
Complete and Regular Subalgebras
Saturation
Distributivity of Complete Boolean Algebras
Exercises
Historical Notes
Stationary Sets
91
16
Closed Unbounded Sets
Mahlo Cardinals
Normal Filters
Silver's Theorem
A Hierarchy of Stationary Sets
The Closed Unbounded Filter on Pκ(λ)
Exercises
Historical Notes
Combinatorial Set Theory
107
18
Partition Properties
Weakly Compact Cardinals
Trees
Almost Disjoint Sets and Functions
The Tree Property and Weakly Compact Cardinals
Ramsey Cardinals
Exercises
Historical Notes
Measurable Cardinals
125
14
The Measure Problem
Measurable and Real-Valued Measurable Cardinals
Measurable Cardinals
Normal Measures
Strongly Compact and Supercompact Cardinals
Exercises
Historical Notes
Borel and Analytic Sets
139
16
Borel Sets
Analytic Sets
The Suslin Operation A
The Hierarchy of Projective Sets
Lebesgue Measure
The Property of Baire
Analytic Sets: Measure, Category, and the Perfect Set Property
Exercises
Historical Notes
Models of Set Theory
155
20
Review of Model Theory
Godel's Theorems
Direct Limits of Models
Reduced Products and Ultraproducts
Models of Set Theory and Relativization
Relative Consistency
Transitive Models and Δ0 Formulas
Consistency of the Axiom of Regularity
Inaccessibility of Inaccessible Cardinals
Reflection Principle
Exercises
Historical Notes
Part II. Advanced Set Theory
Constructible Sets
175
26
The Hierarchy of Constructible Sets
Godel Operations
Inner Models of ZF
The Levy Hierarchy
Absoluteness of Constructibility
Consistency of the Axiom of Choice
Consistency of the Generalized Continuum Hypothesis
Relative Constructibility
Ordinal-Definable Sets
More on Inner Models
Exercises
Historical Notes
Forcing
201
24
Forcing Conditions and Generic Sets
Separative Quotients and Complete Boolean Algebras
Boolean-Valued Models
The Boolean-Valued Model VB
The Forcing Relation
The Forcing Theorem and the Generic Model Theorem
Consistency Proofs
Independence of the Continuum Hypothesis
Independence of the Axiom of Choice
Exercises
Historical Notes
Applications of Forcing
225
42
Cohen Reals
Adding Subsets of Regular Cardinals
The k-Chain Condition
Distributivity
Product Forcing
Easton's Theorem
Forcing with a Class of Conditions
The Levy Collapse
Suslin Trees
Random Reals
Forcing with Perfect Trees
More on Generic Extensions
Symmetric Submodels of Generic Models
Exercises
Historical Notes
Iterated Forcing and Martin's Axiom
267
18
Two-Step Iteration
Iteration with Finite Support
Martin's Axiom
Independence of Suslin's Hypothesis
More Applications of Martin's Axiom
Iterated Forcing
Exercises
Historical Notes
Large Cardinals
285
26
Ultrapowers and Elementary Embeddings
Weak Compactness
Indescribability
Partitions and Models
Exercises
Historical Notes
Large Cardinals and L
311
28
Silver Indiscernibles
Models with Indiscernibles
Proof of Silver's Theorem and 0#
Elementary Embeddings of L
Jensen's Covering Theorem
Exercises
Historical Notes
Iterated Ultrapowers and L[U]
339
26
The Model L[U]
Iterated Ultrapowers
Representation of Iterated Ultrapowers
Uniqueness of the Model L[D]
Indiscernibles for L[D]
General Iterations
The Mitchell Order
The Models L[U]
Exercises
Historical Notes
Very Large Cardinals
365
24
Strongly Compact Cardinals
Supercompact Cardinals
Beyond Supercompactness
Extenders and Strong Cardinals
Exercises
Historical Notes
Large Cardinals and Forcing
389
20
Mild Extensions
Kunen-Paris Forcing
Silver's Forcing
Prikry Forcing
Measurability of N1 in ZF
Exercises
Historical Notes
Saturated Ideals
409
32
Real-Valued Measurable Cardinals
Generic Ultrapowers
Precipitous Ideals
Saturated Ideals
Consistency Strength of Precipitousness
Exercises
Historical Notes
The Nonstationary Ideal
441
16
Some Combinatorial Principles
Stationary Sets in Generic Extensions
Precipitousness of the Nonstationary Ideal
Saturation of the Nonstationary Ideal
Reflection
Exercises
Historical Notes
The Singular Cardinal Problem
457
22
The Galvin-Hajnal Theorem
Ordinal Functions and Scales
The pcf Theory
The Structure of pcf
Transitive Generators and Localization
Shelah's Bound on 2Nω
Exercises
Historical Notes
Descriptive Set Theory
479
32
The Hierarchy of Projective Sets
Π11 Sets
Trees, Well-Founded Relations and k-Suslin Sets
Σ12 Sets
Projective Sets and Constructibility
Scales and Uniformization
Σ12 Well-Orderings and Σ12 Well-Founded Relations
Borel Codes
Exercises
Historical Notes
The Real Line
511
34
Random and Cohen reals
Solovay Sets of Reals
The Levy Collapse
Solovay's Theorem
Lebesgue Measurability of Σ12 Sets
Ramsey Sets of Reals and Mathias Forcing
Measure and Category
Exercises
Historical Notes
Part III. Selected Topics
Combinatorial Principles in L
545
12
The Fine Structure Theory
The Principle κ
The Jensen Hierarchy
Projecta, Standard Codes and Standard Parameters
Diamond Principles
Trees in L
Canonical Functions on ω1
Exercises
Historical Notes
More Applications of Forcing
557
16
A Nonconstructible Δ13 Real
Namba Forcing
A Cohen Real Adds a Suslin Tree
Consistency of Borel's Conjecture
κ +-Aronszajn Trees
Exercises
Historical Notes
More Combinatorial Set Theory
573
12
Ramsey Theory
Gaps in ωω
The Open Coloring Axiom
Almost Disjoint Subsets of ω1
Functions from ω1 into ω
Exercises
Historical Notes
Complete Boolean Algebras
585
16
Measure Algebras
Cohen Algebras
Suslin Algebras
Simple Algebras
Infinite Games on Boolean Algebras
Exercises
Historical Notes
Proper Forcing
601
14
Definition and Examples
Iteration of Proper Forcing
The Proper Forcing Axiom
Applications of PFA
Exercises
Historical Notes
More Descriptive Set Theory
615
12
Π11 Equivalence Relations
Σ11 Equivalence Relations
Constructible Reals and Perfect Sets
Projective Sets and Large Cardinals
Universally Baire sets
Exercises
Historical Notes
Determinacy
627
20
Determinacy and Choice
Some Consequences of AD
AD and Large Cardinals
Projective Determinacy
Consistency of AD
Exercises
Historical Notes
Supercompact Cardinals and the Real Line
647
12
Woodin Cardinals
Semiproper Forcing
The Model L(R)
Stationary Tower Forcing
Weakly Homogeneous Trees
Exercises
Historical Notes
Inner Models for Large Cardinals
659
10
The Core Model
The Covering Theorem for K
The Covering Theorem for L(U)
The Core Model for Sequences of Measures
Up to a Strong Cardinal
Inner Models for Woodin Cardinals
Exercises
Historical Notes
Forcing and Large Cardinals
669
12
Violating GCH at a Measurable Cardinal
The Singular Cardinal Problem
Violating SCH at Nω
Radin Forcing
Stationary Tower Forcing
Exercises
Historical Notes
Martin's Maximum
681
14
RCS iteration of semiproper forcing
Consistency of MM
Applications of MM
Reflection Principles
Forcing Axioms
Exercises
Historical Notes
More on Stationary Sets
695
12
The Nonstationary Ideal on N1
Saturation and Precipitousness
Reflection
Stationary Sets in Pκ (λ)
Mutually Stationary Sets
Weak Squares
Exercises
Historical Notes
Bibliography
707
26
Notation
733
10
Name Index
743
6
Index
749