# Book:Thomas Jech/Set Theory/Third Edition

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## Thomas Jech:

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

Published $2002$, **Springer**

- ISBN 3-540-44085-2.

Part of the **Springer Monographs in Mathematics** series.

### Subject Matter

### 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.

- 1. Axioms of Set Theory

- Part II. Advanced Set Theory
- 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.
- Adding Subsets of Regular Cardinals.
- 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.

- 13. Constructible Sets

- 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.
- Some Consequences of AD.
- AD and Large Cardinals.
- Projective Determinacy.
- Consistency of AD.
- 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 $\aleph_\omega$.
- Radin Forcing.
- 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.

- 27. Combinatorial Principles in $L$

- Bibliography
- Notation
- Name Index
- Index