.

0 (hodnocen0 x )

BK

3rd millennium ed., revised and expanded

Berlin : Springer, c2003

xiii, 769 s. ; 25 cm

ISBN 3-540-44085-2 (váz.)

Springer monographs in mathematics, ISSN 1439-7382

Popsáno dle 4. dotisku vydaného v roce 2006

Obsahuje bibliografii na s. [707]-732, bibliografické odkazy a rejstříky

000251004

Part 1. 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 a x a. 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-Godel Axiomatic Set Theory. Exercises. Historical Notes. // 7. Filters, Ultrafilters and Boolean Algebras ...73 // Filters and Ultrafilters. Ultrafilters on w. 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. 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 Pk(Ă€). // 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. 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 ...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. Historical Notes. // 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. // Historical Notes. // 14. Forcing ...201 // Forcing Conditions and Generic Sets. Separative Quotients and Complete // Boolean Algebras. Boolean-Valued Models. The Boolean-Valued Model VB. // 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 K-Chain Condition. // Distributivity. Product Forcing. Easton’s Theorem. Forcing with a Class of // Conditions. The Levy Collapse. Suslin "Frees. Random Reals. Forcing with // Perfect Trees. More on Generic Extensions. Symmetric Submodels of Generic // Models. Exercises. Historical Notes. // Table of Contents // 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. 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 LU] ...339 // The Model LU]. Iterated Ultrapowers. Representation of Iterated Ultrapowers. Uniqueness of the Model L[D]. Indiscernibles for L[D]. General Iterations. // The Mitchell Order. The Models LU]. 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 R1 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 // Fhe Galvin-Hajnal Theorem. Ordinal Functions and Scales. The pcf Theory, // rhe Structure of pcf. Transitive Generators and Localization. Shelah’s Bound on 28?. Exercises. Historical Notes. // 25. Descriptive Set Theory ...479 // rhe Hierarchy of Projective Sets. nJ Sets. Trees, Well-Founded Relations // and K-Suslin Sets. 22 Sets. Projective Sets and Constructibility. Scales and // Uniformization. 22 Well-Orderings and 22 Well-Founded Relations. Borel // Codes. Exercises. Historical Notes. // 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 22 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 •k. The Jensen Hierarchy. Projecta, // Standard Codes and Standard Parameters. Diamond Principles. Trees in L. // Canonical Functions on w1. Exercises. Historical Notes. // 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 w". The Open Coloring Axiom. Almost Disjoint // Subsets of U1. Functions from 01 into w. 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 // 11} Equivalence Relations. 21 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 LU]. The Core Model for Sequences of Measures. Up to a Strong Cardinal. // Inner Models for Woodin Cardinals. Exercises. Historical Notes. // Table of Contents XIII // 36. Forcing and Large Cardinals ...669 // Violating GCH at a Measurable Cardinal. The Singular Cardinal Problem. // Violating SCH at Ru. 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 R1. Saturation and Precipitousness. Reflection. // Stationary Sets in Pk(Ă€). Mutually Stationary Sets. Weak Squares. Exercises. // Historical Notes. // Bibliography ...707 // Notation ...733 // Name Index ...743 // Index ...749