Book:Thomas Jech/Set Theory/Third Edition

Part of the Springer Monographs in Mathematics series.

Subject Matter

 * Set Theory

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.


 * 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.


 * 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.


 * Bibliography
 * Notation
 * Name Index
 * Index