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[资源] 【Set Theory】【Thomas Jech】

Table of Contents
Part I. Basic Set Theory
1. Axioms of Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Linear and Partial Ordering. Well-Ordering. Ordinal Numbers. Induction and
Recursion. Ordinal Arithmetic. Well-Founded Relations. Exercises. Historical
Notes.
3. Cardinal Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Cardinality. Alephs. The Canonical Well-Ordering of α × α. Cofinality. Exercises.
Historical Notes.
4. Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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 . . . . . . . . . . . . . 47
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
The Cumulative Hierarchy of Sets. ∈-Induction. Well-Founded Relations. The
Bernays-G¨odel Axiomatic Set Theory. Exercises. Historical Notes.
7. Filters, Ultrafilters and Boolean Algebras . . . . . . . . . . . . . . . . . . 73
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.
8. Stationary Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
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.
9. Combinatorial Set Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Borel Sets. Analytic Sets. The Suslin Operation A. TheHierarchyofProjective
Sets. Lebesgue Measure. The Property of Baire. Analytic Sets: Measure,
Category, and the Perfect Set Property. Exercises. Historical Notes.
12. Models of Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Review of Model Theory. G¨odel’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
13. Constructible Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
The Hierarchy of Constructible Sets. G¨odel Operations. Inner Models of ZF.
The L′evy 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Cohen Reals. Adding Subsets of Regular Cardinals. The κ-Chain Condition.
Distributivity. Product Forcing. Easton’s Theorem. Forcing with a Class of
Conditions. The L′evy Collapse. Suslin Trees. Random Reals. Forcing with
Perfect Trees. More on Generic Extensions. Symmetric Submodels of Generic
16. Iterated Forcing and Martin’s Axiom . . . . . . . . . . . . . . . . . . . . . 267
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Ultrapowers and Elementary Embeddings. Weak Compactness. Indescribability.
Partitions and Models. Exercises. Historical Notes.
18. Large Cardinals and L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Silver Indiscernibles. Models with Indiscernibles. Proof of Silver’s Theorem
and 0. Elementary Embeddings of L. Jensen’s Covering Theorem. Exercises.
Historical Notes.
19. Iterated Ultrapowers and L[U] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
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.
20. Very Large Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Strongly Compact Cardinals. Supercompact Cardinals. Beyond Supercompactness.
Extenders and Strong Cardinals. Exercises. Historical Notes.
21. Large Cardinals and Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
Mild Extensions. Kunen-Paris Forcing. Silver’s Forcing. Prikry Forcing. Measurability
of ℵ1 in ZF. Exercises. Historical Notes.
22. Saturated Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
Real-Valued Measurable Cardinals. Generic Ultrapowers. Precipitous Ideals.
Saturated Ideals. Consistency Strength of Precipitousness. Exercises. Historical
Notes.
23. The Nonstationary Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
The Galvin-Hajnal Theorem. Ordinal Functions and Scales. The pcf Theory.
The Structure of pcf. Transitive Generators and Localization. Shelah’s Bound
on 2
ℵω . Exercises. Historical Notes.
25. Descriptive Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
The Hierarchy of Projective Sets. Π11
Sets. Trees, Well-Founded Relations
and κ-Suslin Sets. Σ12
Sets. Projective Sets and Constructibility. Scales and
Uniformization. Σ12
Well-Orderings and Σ12
Well-Founded Relations. Borel
26. The Real Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
Random and Cohen reals. Solovay Sets of Reals. The L′evy 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
27. Combinatorial Principles in L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
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.
28. More Applications of Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
A Nonconstructible Δ13
Real. Namba Forcing. A Cohen Real Adds a Suslin
Tree. Consistency of Borel’s Conjecture. κ+-Aronszajn Trees. Exercises. Historical
Notes.
29. More Combinatorial Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 573
Ramsey Theory. Gaps in ωω. The Open Coloring Axiom. Almost Disjoint
Subsets of ω1. Functions from ω1 into ω. Exercises. Historical Notes.
30. Complete Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
Measure Algebras. Cohen Algebras. Suslin Algebras. Simple Algebras. Infinite
Games on Boolean Algebras. Exercises. Historical Notes.
31. Proper Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
Definition and Examples. Iteration of Proper Forcing. The Proper Forcing
Axiom. Applications of PFA. Exercises. Historical Notes.
32. More Descriptive Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
Π11
Equivalence Relations. Σ11
Equivalence Relations. Constructible Reals
and Perfect Sets. Projective Sets and Large Cardinals. Universally Baire sets.
Exercises. Historical Notes.
33. Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627
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 . . . . . . . . . . . . . . . 647
Woodin Cardinals. Semiproper Forcing. The Model L(R). Stationary Tower
Forcing. Weakly Homogeneous Trees. Exercises. Historical Notes.
35. Inner Models for Large Cardinals . . . . . . . . . . . . . . . . . . . . . . . . . 659
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
Violating GCH at a Measurable Cardinal. The Singular Cardinal Problem.
Violating SCH at ℵω. Radin Forcing. Stationary Tower Forcing. Exercises.
Historical Notes.
37. Martin’s Maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681
RCS iteration of semiproper forcing. Consistency of MM. Applications of MM.
Reflection Principles. Forcing Axioms. Exercises. Historical Notes.
38. More on Stationary Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
The Nonstationary Ideal on ℵ1. Saturation and Precipitousness. Reflection.
Stationary Sets in Pκ(λ). Mutually Stationary Sets. Weak Squares. Exercises.
Historical Notes.
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733
Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 749@tigou@hyly

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