Constructible Sets with Applications

Name: Constructible Sets with Applications
Brand: Elsevier
Price: 215.0 EUR
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Lev D. Beklemishev(Autor*in)

Elsevier (Verlag)

Erschienen am 14. Mai 2014

281 Seiten

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978-0-08-095776-0 (ISBN)

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Front Cover
Constructible Sets With Applications
Copyright Page
CONTENTS
Chapter I. Axioms and auxiliary notions
1. Set theory ZF of Zermelo-Fraenkel
2. The meta-theory of classes
3. Definitions by transfinite induction. Ranks
4. Models, satisfaction
5. Derived semantical notions
the Skolem-Löwenheim theorem
6. The contraction lemma
Chapter II. General principles of construction
1. Sufficient conditions for a class to be a model
2. The reflection theorem
3. Predicatively closed classes
4. The fundamental operations
Chapter III. Constructible sets
1. Enumeration
2. Constructible sets
3. Properties of constructible sets
4. Constructibility of ordinals
5. Models containing with each element its mappings into ordinals
6. Examples of functions satisfying (B.0)-(B.6)
7. Sets constructible in a class
Chapter IV. Functors and their definability
1. Strongly definable functors
2. Properties of strongly definable functors and relations
3. Examples of strongly definable functors
4. Definitions by transfinite induction
5. Why is all that necessary?
Chapter V. Constructible sets as values of a functor
1. Uniformly definable functions
2. Examples of uniformly definable functions
3. Uniform definability of the function C?B(a)
4. A generalization
5. Further properties of constructible sets
Chapter VI. Cp(A)(a) as a model
1. The reflection theorem again
2. Satisfiability of the power set axiom and of the axiom of substitution
3. Existence of models
4. Minimal models
Chapter VII. Consistency of the axiom of choice and of the continuum hypothesis
1. Axiom of choice
2. Auxiliary functors
3. Formulation of the generalized continuum hypothesis
4. A sufficient condition for the validity of GCH
5. Construction of models in which the continuum hypothesis is valid
6. Definability of the contracting function
7. A refinement of the Skolem-Löwenheim theorem
8. Consistency of GCH
9. Axioms of constructibility. Final remarks
Chapter VIII. Reduction of models
1. A reflection lemma
2. The validity of the power set axiom
Chapter IX. Generic points and forcing
general theory
1. Auxiliary topological notions
2. Valuations
3. The forcing relation
4. A special valuation
5. Application of forcing to constructions of models
Chapter X. Polynomials
1. Polynomials
2. Reduction of Condition IV
3. Reduction of Condition V
Chapter XI. Explicit construction of polynomials for functions Bmin, Bo, BZ
1. The partial ordering &&o
2. Auxiliary polynomials
3. Expressing EBaß and JBaß as polynomials of EBI(a,ß) and JBI(a,ß)
4. Generalization to the cases B = BZ and B = B0
5. Final reduction of Conditions IVand V
6. Appendix: list of the polynomials fj, j = 12
Chapter XII. Examples of models and of independence proofs
1. Examples of topological spaces
2. Examples of mappings p a(p)
3. Proof of Condition VIII for sequences of the first and the second category
4. Examples of models
5. A theorem on generic points
6. Independence of the strong axiom of constructibility
Chapter XIII. The continuum hypothesis
1. Auxiliary notions concerning cardinals
2. The Souslin coefficient
3. The Souslin coefficient of product spaces
4. Relative cardinals, relative cofinality and relative Souslin coefficients
5. Determination of the relative Souslin coefficient
6. Absoluteness of cardinals and of the cofinality index
7. The function exp of a model
8. The independence of the continuum hypothesis
Chapter XIV. Independence of the axiom of choice
1. Action of homeomorphisms onto sets C?(a(p))
2. Homeomorphisms and forcing
3. Invariance properties
4. Independence of the axiom of choice
5. The ordering of P(P(?))
6. The existence of maximal ideals in P(?)
7. Cofinality of ?1
Chapter XV. Problems of definability
1. Definable relations between ordinals
2. Non-definability of the well-orderings of P(?)
3. Definable well-ordered subsets of P(?)
Appendix
Bibliography
List of important symbols
Author index
Subject index

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