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【Set Theory】【Thomas Jech】
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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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