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

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

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  • 製本 Hardcover:ハードカバー版/ページ数 772 p.
  • 商品コード 9783540440857

基本説明

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 …

Full Description


This monograph covers the recent major advances in various areas of set theory. From the reviews:"One of the classical textbooks and reference books in set theory....The present 'Third Millennium' edition...is a whole new book. In three parts the author offers us what in his view every young set theorist should learn and master....This well-written book promises to influence the next generation of set theorists, much as its predecessor has done." --MATHEMATICAL REVIEWS

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