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Set Theory (Springer Monographs in Mathematics)の画像
Ｔ．Ｊｅｃｈの集合論（第３版）
Set Theory (Springer Monographs in Mathematics) （3rd. Corr. 4th printing）

Jech, Thomas
Springer （2003/01 出版）

Hardcover:ハードカバー版
ISBN： 9783540440857
Academic Descriptors： KNO6230
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詳細

Academic Descriptors： KNO6230

Kinokuniya Annotation：
A standard reference in set theory. Set Theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. The present book covers each of these areas, giving the reader an understanding of the ideas involved. It can be used for introductory students and is broad and deep enough to bring the reader near the boundaries of current research. Students and researchers in the field will find the book invaluable both as a study material and as a desktop reference.
Review- Thomas Jech’s text has long been considered a classic survey of the state of the set theory …

KNV Description：
§16§Set Theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. The present book covers each of these areas, giving the reader an understanding of the ideas involved. It can be used for introductory students and is broad and deep enough to bring the reader near the boundaries of current research. Students and researchers in the field will find the book invaluable both as a study material and as a desktop reference. §04§I. Basic Set Theory.- Axioms of Set Theory.- Ordinal Numbers.- Cardinal Numbers.- Real Numbers.- The Axiom of Choice and Cardinal Arithmetic.- The Axiom of Regularity.- Filters, Ultrafilters and Boolean Algebras.- Stationary Sets.- Combinatorial Set Theory.- Measurable Cardinals.- Borel and Analytic Sets.- Models of Set Theory.- II. Advanced Set Theory.- Constructible Sets.- Forcing.- Applications of Forcing.- Iterated Forcing and Martin's Axiom.- Large Cardinals.- Large Cardinals and L.- Iterated Ultrapowers and LÄUÜ.- Very Large Cardinals.- Large Cardinals and Forcing.- Saturated Ideals.- The Nonstationary Ideal.- The Singular Cardinal Problem.- Descriptive Set Theory.- The Real Line.- III. Selected Topics.- Combinatorial Principles in L.- More Applications of Forcing.- More Combinatorial Set Theory.- Complete Boolean Algebras.- Proper Forcing.- More Descriptive Set Theory.- Determinacy.- Supercompact Cardinals and the Real Line.- Inner Models for Large Cadinals.- Forcing and La §04§rge Cardinals.- Martin's Maximum.- More on Stationary Sets.- Bibliography.- Notation.- Index.- Name Index.

Baker&Taylor Table of Contents：
Part I. Basic Set Theory
  Axioms of Set Theory                             3  (14)
    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
  Ordinal Numbers                                  17 (10)
    Linear and Partial Ordering
    Well-Ordering
    Ordinal Numbers
    Induction and Recursion
    Ordinal Arithmetic
    Well-Founded Relations
    Exercises
    Historical Notes
  Cardinal Numbers                                 27 (10)
    Cardinality
    Alephs
    The Canonical Well-Ordering of α x
    α
    Cofinality
    Exercises
    Historical Notes
  Real Numbers                                     37 (10)
    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
  The Axiom of Choice and Cardinal Arithmetic      47 (16)
    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
  The Axiom of Regularity                          63 (10)
    The Cumulative Hierarchy of Sets
    E-Induction
    Well-Founded Relations
    The Bernays-Godel Axiomatic Set Theory
    Exercises
    Historical Notes
  Filters, Ultrafilters and Boolean Algebras       73 (18)
    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
  Stationary Sets                                  91 (16)
    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
  Combinatorial Set Theory                         107(18)
    Partition Properties
    Weakly Compact Cardinals
    Trees
    Almost Disjoint Sets and Functions
    The Tree Property and Weakly Compact
    Cardinals
    Ramsey Cardinals
    Exercises
    Historical Notes
  Measurable Cardinals                             125(14)
    The Measure Problem
    Measurable and Real-Valued Measurable
    Cardinals
    Measurable Cardinals
    Normal Measures
    Strongly Compact and Supercompact Cardinals
    Exercises
    Historical Notes
  Borel and Analytic Sets                          139(16)
    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
  Models of Set Theory                             155(20)
    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 Δ0 Formulas
    Consistency of the Axiom of Regularity
    Inaccessibility of Inaccessible Cardinals
    Reflection Principle
    Exercises
    Historical Notes
Part II. Advanced Set Theory
  Constructible Sets                               175(26)
    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
  Forcing                                          201(24)
    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
  Applications of Forcing                          225(42)
    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
    Historical Notes
  Iterated Forcing and Martin's Axiom              267(18)
    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
  Large Cardinals                                  285(26)
    Ultrapowers and Elementary Embeddings
    Weak Compactness
    Indescribability
    Partitions and Models
    Exercises
    Historical Notes
  Large Cardinals and L                            311(28)
    Silver Indiscernibles
    Models with Indiscernibles
    Proof of Silver's Theorem and 0#
    Elementary Embeddings of L
    Jensen's Covering Theorem
    Exercises
    Historical Notes
  Iterated Ultrapowers and L[U]                    339(26)
    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
  Very Large Cardinals                             365(24)
    Strongly Compact Cardinals
    Supercompact Cardinals
    Beyond Supercompactness
    Extenders and Strong Cardinals
    Exercises
    Historical Notes
  Large Cardinals and Forcing                      389(20)
    Mild Extensions
    Kunen-Paris Forcing
    Silver's Forcing
    Prikry Forcing
    Measurability of N1 in ZF
    Exercises
    Historical Notes
  Saturated Ideals                                 409(32)
    Real-Valued Measurable Cardinals
    Generic Ultrapowers
    Precipitous Ideals
    Saturated Ideals
    Consistency Strength of Precipitousness
    Exercises
    Historical Notes
  The Nonstationary Ideal                          441(16)
    Some Combinatorial Principles
    Stationary Sets in Generic Extensions
    Precipitousness of the Nonstationary Ideal
    Saturation of the Nonstationary Ideal
    Reflection
    Exercises
    Historical Notes
  The Singular Cardinal Problem                    457(22)
    The Galvin-Hajnal Theorem
    Ordinal Functions and Scales
    The pcf Theory
    The Structure of pcf
    Transitive Generators and Localization
    Shelah's Bound on 2Nω
    Exercises
    Historical Notes
  Descriptive Set Theory                           479(32)
    The Hierarchy of Projective Sets
    Π11 Sets
    Trees, Well-Founded Relations and k-Suslin
    Sets
    Σ12 Sets
    Projective Sets and Constructibility
    Scales and Uniformization
    Σ12 Well-Orderings and Σ12
    Well-Founded Relations
    Borel Codes
    Exercises
    Historical Notes
  The Real Line                                    511(34)
    Random and Cohen reals
    Solovay Sets of Reals
    The Levy 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
  Combinatorial Principles in L                    545(12)
    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
  More Applications of Forcing                     557(16)
    A Nonconstructible Δ13 Real
    Namba Forcing
    A Cohen Real Adds a Suslin Tree
    Consistency of Borel's Conjecture
    κ +-Aronszajn Trees
    Exercises
    Historical Notes
  More Combinatorial Set Theory                    573(12)
    Ramsey Theory
    Gaps in ωω
    The Open Coloring Axiom
    Almost Disjoint Subsets of ω1
    Functions from ω1 into ω
    Exercises
    Historical Notes
  Complete Boolean Algebras                        585(16)
    Measure Algebras
    Cohen Algebras
    Suslin Algebras
    Simple Algebras
    Infinite Games on Boolean Algebras
    Exercises
    Historical Notes
  Proper Forcing                                   601(14)
    Definition and Examples
    Iteration of Proper Forcing
    The Proper Forcing Axiom
    Applications of PFA
    Exercises
    Historical Notes
  More Descriptive Set Theory                      615(12)
    Π11 Equivalence Relations
    Σ11 Equivalence Relations
    Constructible Reals and Perfect Sets
    Projective Sets and Large Cardinals
    Universally Baire sets
    Exercises
    Historical Notes
  Determinacy                                      627(20)
    Determinacy and Choice
    Some Consequences of AD
    AD and Large Cardinals
    Projective Determinacy
    Consistency of AD
    Exercises
    Historical Notes
  Supercompact Cardinals and the Real Line         647(12)
    Woodin Cardinals
    Semiproper Forcing
    The Model L(R)
    Stationary Tower Forcing
    Weakly Homogeneous Trees
    Exercises
    Historical Notes
  Inner Models for Large Cardinals                 659(10)
    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
  Forcing and Large Cardinals                      669(12)
    Violating GCH at a Measurable Cardinal
    The Singular Cardinal Problem
    Violating SCH at Nω
    Radin Forcing
    Stationary Tower Forcing
    Exercises
    Historical Notes
  Martin's Maximum                                 681(14)
    RCS iteration of semiproper forcing
    Consistency of MM
    Applications of MM
    Reflection Principles
    Forcing Axioms
    Exercises
    Historical Notes
  More on Stationary Sets                          695(12)
    The Nonstationary Ideal on N1
    Saturation and Precipitousness
    Reflection
    Stationary Sets in Pκ (λ)
    Mutually Stationary Sets
    Weak Squares
    Exercises
    Historical Notes
Bibliography                                       707(26)
Notation                                           733(10)
Name Index                                         743(6)
Index                                              749