# Book:Thomas Jech/Set Theory/Third Edition

## Thomas Jech: Set Theory (3rd Millennium Edition)

Published $2002$, Springer

ISBN 3-540-44085-2.

Part of the Springer Monographs in Mathematics series.

### Contents

Part I. Basic Set Theory
1. Axioms of Set Theory
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.
2. Ordinal Numbers
Linear and Partial Ordering.
Well-Ordering.
Ordinal Numbers.
Induction and Recursion.
Ordinal Arithmetic.
Well-Founded Relations.
Exercises.
Historical Notes.
3. Cardinal Numbers
Cardinality.
Alephs.
The Canonical Well-Ordering of $\alpha \times \alpha$.
Cofinality.
Exercises.
Historical Notes.
4. Real Numbers
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.
5. The Axiom of Choice and Cardinal Arithmetic
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.
6. The Axiom of Regularity
The Cumulative Hierarchy of Sets.
$\in$-Induction.
Well-Founded Relations.
The Bernays-Gödel Axiomatic Set Theory.
Exercises.
Historical Notes.
7. Filters, Ultrafilters and Boolean Algebras
Filters and Ultrafilters.
Ultrafilters on $\omega$.
$\kappa$-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.
8. Stationary Sets
Closed Unbounded Sets.
Mahlo Cardinals.
Normal Filters.
Silver's Theorem.
A Hierarchy of Stationary Sets.
The Closed Unbounded Filter on $P_\kappa(\lambda)$.
Exercises.
Historical Notes.
9. Combinatorial Set Theory
Partition Properties.
Weakly Compact Cardinals.
Trees.
Almost Disjoint Sets and Functions.
The Tree Property and Weakly Compact Cardinals.
Ramsey Cardinals.
Exercises.
Historical Notes.
10. Measurable Cardinals
The Measure Problem.
Measurable and Real-Valued Measurable Cardinals.
Measurable Cardinals.
Normal Measures.
Strongly Compact and Supercompact Cardinals.
Exercises.
Historical Notes.
11. Borel and Analytic Sets
Borel Sets.
Analytic Sets.
The Suslin Operation $\mathcal A$.
The Hierarchy of Projective Sets.
Lebesgue Measure.
The Property of Baire.
Analytic Sets: Measure, Category, and the Perfect Set Property.
Exercises.
Historical Notes.
12. Models of Set Theory
Review of Model Theory.
Gödel's Theorems.
Direct Limits of Models.
Reduced Products and Ultraproducts.
Models of Set Theory and Relativization.
Relative Consistency.
Transitive Models and $\Delta_0$ Formulas.
Consistency of the Axiom of Regularity.
Inaccessibility of Inaccessible Cardinals.
Reflection Principle.
Exercises.
Historical Notes.
13. Constructible Sets
The Hierarchy of Constructible Sets.
Gödel Operations.
Inner Models of ZF.
The Lévy 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.
14. Forcing
Forcing Conditions and Generic Sets.
Separative Quotients and Complete Boolean Algebras.
Boolean-Valued Models.
The Boolean-Valued Model $V^B$.
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.
15. Applications of Forcing
Cohen Reals.
The $\kappa$-Chain Condition.
Distributivity.
Product Forcing.
Easton's Theorem.
Forcing with a Class of Conditions.
The Lévy Collapse.
Suslin Trees.
Random Reals.
Forcing with Perfect Trees.
More on Generic Extensions.
Symmetric Submodels of Generic Models.
Exercises.
Historical Notes.
16. Iterated Forcing and Martin's Axiom
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.
17. Large Cardinals
Ultrapowers and Elementary Embeddings.
Weak Compactness.
Indescribability.
Partitions and Models.
Exercises.
Historical Notes.
18. Large Cardinals and $L$
Silver Indiscernibles.
Models with Indiscernibles.
Proof of Silver's Theorem and $0^\sharp$.
Elementary Embeddings of $L$.
Jensen's Covering Theorem.
Exercises.
Historical Notes.
19. Iterated Ultrapowers and $L[U]$
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 $L[\mathcal U]$.
Exercises.
Historical Notes.
20. Very Large Cardinals
Strongly Compact Cardinals.
Supercompact Cardinals.
Beyond Supercompactness.
Extenders and Strong Cardinals.
Exercises.
Historical Notes.
21. Large Cardinals and Forcing
Mild Extensions.
Kunen-Paris Forcing.
Silver's Forcing.
Prikry Forcing.
Measurability of $\aleph_1$ in ZF.
Exercises.
Historical Notes.
22. Saturated Ideals
Real-Valued Measurable Cardinals.
Generic Ultrapowers.
Precipitous Ideals.
Saturated Ideals.
Consistency Strength of Precipitousness.
Exercises.
Historical Notes.
23. The Nonstationary Ideal
Some Combinatorial Principles.
Stationary Sets in Generic Extensions.
Precipitousness of the Nonstationary Ideal.
Saturation of the Nonstationary Ideal.
Reflection.
Exercises.
Historical Notes.
24. The Singular Cardinal Problem
The Galvin-Hajnal Theorem.
Ordinal Functions and Scales.
The pcf Theory.
The Structure of pcf.
Transitive Generators and Localization.
Shelah's Bound on $2^{\aleph_\omega}$.
Exercises.
Historical Notes.
25. Descriptive Set Theory
The Hierarchy of Projective Sets.
$\Pi_1^1$ Sets.
Trees, Well-Founded Relations and $\kappa$-Suslin Sets.
$\Sigma_2^1$ Sets.
Projective Sets and Constructibility.
Scales and Uniformization.
$\Sigma_2^1$ Well-Orderings and $\Sigma_2^1$ Well-Founded Relations.
Borel Codes.
Exercises.
Historical Notes.
26. The Real Line
Random and Cohen reals.
Solovay Sets of Reals.
The Lévy Collapse.
Solovay's Theorem.
Lebesgue Measurability of $\Sigma_2^1$ Sets.
Ramsey Sets of Reals and Mathias Forcing.
Measure and Category.
Exercises.
Historical Notes.
Part III. Selected Topics
27. Combinatorial Principles in $L$
The Fine Structure Theory.
The Principle $\square_\kappa$.
The Jensen Hierarchy.
Projecta, Standard Codes and Standard Parameters.
Diamond Principles.
Trees in $L$.
Canonical Functions on $\omega_1$.
Exercises.
Historical Notes.
28. More Applications of Forcing
A Nonconstructible $\Delta_3^1$ Real.
Namba Forcing.
A Cohen Real Adds a Suslin Tree.
Consistency of Borel's Conjecture.
$\kappa^+$-Aronszajn Trees.
Exercises.
Historical Notes.
29. More Combinatorial Set Theory
Ramsey Theory.
Gaps in $\omega^\omega$.
The Open Coloring Axiom.
Almost Disjoint Subsets of $\omega_1$.
Functions from $\omega_1$ into $\omega$.
Exercises.
Historical Notes.
30. Complete Boolean Algebras
Measure Algebras.
Cohen Algebras.
Suslin Algebras.
Simple Algebras.
Infinite Games on Boolean Algebras.
Exercises.
Historical Notes.
31. Proper Forcing
Definition and Examples.
Iteration of Proper Forcing.
The Proper Forcing Axiom.
Applications of PFA.
Exercises.
Historical Notes.
32. More Descriptive Set Theory
$Pi_1^1$ Equivalence Relations.
$\Sigma_1^1$ Equivalence Relations.
Constructible Reals and Perfect Sets.
Projective Sets and Large Cardinals.
Universally Baire sets.
Exercises.
Historical Notes.
33. Determinacy
Determinacy and Choice.
Projective Determinacy.
Exercises.
Historical Notes.
34. Supercompact Cardinals and the Real Line
Woodin Cardinals.
Semiproper Forcing.
The Model $L(R)$.
Stationary Tower Forcing.
Weakly Homogeneous Trees.
Exercises.
Historical Notes.
35. Inner Models for Larger Cardinals
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.
36. Forcing and Large Cardinals
Violating GCH at a Measurable Cardinal.
The Singular Cardinal Problem.
Violating SCH at $\aleph_\omega$.
Stationary Tower Forcing.
Exercises.
Historical Notes.
37. Martin's Maximum
RCS iteration of semiproper forcing.
Consistency of MM.
Applications of MM.
Reflection Principles.
Forcing Axioms.
Exercises.
Historical Notes.
38. More on Stationary Sets
The Nonstationary Ideal on $\aleph_1$.
Saturation and Precipitousness.
Reflection.
Stationary Sets in $P_\kappa(\lambda)$.
Mutually Stationary Sets.
Weak Squares.
Exercises.
Historical Notes.
Bibliography
Notation
Name Index
Index