Thomas Jech. Set Theory. The Third Millennium Edition, revised and expanded. 4y Springer
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1 Thomas Jech Set Theory The Third Millennium Edition, revised and expanded 4y Springer
2 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. 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 a x a. Cofinality. Exercises. 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. 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. 6. The Axiom of Regularity 63 The Cumulative Hierarchy of Sets. S-Induction. Well-Founded Relations. The Bernays-Godel Axiomatic Set Theory. Exercises. 7. Filters, Ultrafilters and Boolean Algebras 73 Filters and Ultrafdters. Ultrafilters on LJ. K-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.
3 8. Stationary Sets 91 Closed Unbounded Sets. Mahlo Cardinals. Normal Filters. Silver's Theorem. A Hierarchy of Stationary Sets. The Closed Unbounded Filter on P K (\). Exercises. 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. 10. Measurable Cardinals 125 The Measure Problem. Measurable and Real-Valued Measurable Cardinals. Measurable Cardinals. Normal Measures. Strongly Compact and Supercompact Cardinals. Exercises. 11. Borel and Analytic Sets 139 Borel Sets. Analytic Sets. The Suslin Operation A. The Hierarchy of Projective Sets. Lebesgue Measure. The Property of Baire. Analytic Sets: Measure, Category, and the Perfect Set Property. Exercises. 12. Models of Set Theory 155 Review of Model Theory. Godel's Theorems. Direct Limits of Models. Reduced Products and Ultraproducts. Models of Set Theory and Relativization. Relative Consistency. Transitive Models and Ao Formulas. Consistency of the Axiom of Regularity. Inaccessibility of Inaccessible Cardinals. Reflection Principle. Exercises. Part II. Advanced Set Theory 13. Constructible Sets 175 The Hierarchy of Constructible Sets. Godel Operations. Inner Models of ZF. The Levy 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. 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. 15. Applications of Forcing 225 Cohen Reals. Adding Subsets of Regular Cardinals. The K-Chain Condition. Distributivity. Product Forcing. Easton's Theorem. Forcing with a Class of Conditions. The Levy Collapse. Suslin Trees. Random Reals. Forcing with Perfect Trees. More on Generic Extensions. Symmetric Submodels of Generic Models. Exercises.
4 XI 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. 17. Large Cardinals 285 Ultrapowers and Elementary Embeddings. Weak Compactness. Indescribability. Partitions and Models. Exercises. 18. Large Cardinals and L 311 Silver Indiscernibles. Models with Indiscernibles. Proof of Silver's Theorem and 0 3. Elementary Embeddings of L. Jensen's Covering Theorem. Exercises. 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. 20. Very Large Cardinals 365 Strongly Compact Cardinals. Supercompact Cardinals. Beyond Supercompactness. Extenders and Strong Cardinals. Exercises. 21. Large Cardinals and Forcing 389 Mild Extensions. Kunen-Paris Forcing. Silver's Forcing. Prikry Forcing. Measurability of Ki in ZF. Exercises. 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. 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 on2 s "'. Exercises. 25. Descriptive Set Theory 479 The Hierarchy of Projective Sets, n} Sets. Trees, Well-Founded Relations and K-Suslin Sets. 2 Sets. Projective Sets and Constructibility. Scales and Uniformization. 2 Well-Orderings and Ti\ Well-Founded Relations. Borel Codes. Exercises.
5 XII Table of Contents 26. The Real Line 511 Random and Cohen reals. Solovay Sets of Reals. The Levy Collapse. Solovay's Theorem. Lebesgue Measurability of S2 Sets. Ramsey Sets of Reals and Mathias Forcing. Measure and Category. Exercises. Part III. Selected Topics 27. Combinatorial Principles in I 545 The Fine Structure Theory. The Principle D K. The Jensen Hierarchy. Projecta, Standard Codes and Standard Parameters. Diamond Principles. Trees in L. Canonical Functions on ui\. Exercises. 28. More Applications of Forcing 557 A Nonconstructible A3 Real. Namba Forcing. A Cohen Real Adds a Suslin Tree. Consistency of Borel's Conjecture. K + -Aronszajn Trees. Exercises. Historical Notes. 29. More Combinatorial Set Theory 573 Ramsey Theory. Gaps in ui". The Open Coloring Axiom. Almost Disjoint Subsets of wi. Functions from LJ\ into UJ. Exercises. 30. Complete Boolean Algebras 585 Measure Algebras. Cohen Algebras. Suslin Algebras. Simple Algebras. Infinite Games on Boolean Algebras. Exercises. 31. Proper Forcing 601 Definition and Examples. Iteration of Proper Forcing. The Proper Forcing Axiom. Applications of PFA. Exercises. 32. More Descriptive Set Theory 615 II} Equivalence Relations. S} Equivalence Relations. Constructible Reals and Perfect Sets. Projective Sets and Large Cardinals. Universally Baire sets. Exercises. 33. Determinacy 627 Determinacy and Choice. Some Consequences of AD. AD and Large Cardinals. Projective Determinacy. Consistency of AD. Exercises. 34. Supercompact Cardinals and the Real Line 647 Woodin Cardinals. Semiproper Forcing. The Model L(R). Stationary Tower Forcing. Weakly Homogeneous Trees. Exercises. 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.
6 XIII 36. Forcing and Large Cardinals 669 Violating GCH at a Measurable Cardinal. The Singular Cardinal Problem. Violating SCH at N^. Radin Forcing. Stationary Tower Forcing. Exercises. 37. Martin's Maximum 681 RCS iteration of semiproper forcing. Consistency of MM. Applications of MM. Reflection Principles. Forcing Axioms. Exercises. 38. More on Stationary Sets 695 The Nonstationary Ideal on Ni. Saturation and Precipitousness. Reflection. Stationary Sets in P K (A). Mutually Stationary Sets. Weak Squares. Exercises. Bibliography 707 Notation 733 Name Index 743 Index 749
Springer Monographs in Mathematics. Set Theory. The Third Millennium Edition, revised and expanded. Bearbeitet von Thomas Jech
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