Axiomatic Set Theory

Name: Axiomatic Set Theory
Brand: Elsevier
Price: 195.0 EUR
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R. B. Chuaqui(Autor*in)

Elsevier (Verlag)

Erschienen am 30. August 2011

385 Seiten

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978-0-08-087162-2 (ISBN)

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Inhalt

Front Cover
Axiomatic Set Theory: Impredicative Theories of Classes
Copyright Page
Table of Contents
PART 1: INTRODUCTION
Chapter 1.1 Informal background
Chapter 1.2 Axioms
1.2.1 General axioms
1.2.2 Limiting axioms
PART 2: GENERAL CLASS THEORY
Chapter 2.1 Introduction to General Class Theory
2.1.1 Axioms for G
2.1.2 Ordered pairs and Cartesian product
2.1.3 Generalized operations
(+) 2.1.4 G is a subtheory of B
Chapter 2.2 Relations
2.2.1 Operations with relations
2.2.2 Relations as superclasses
2.2.3 Types of relations
Chapter 2.3 Functions and Operations
2.3.1 Functional relations
2.3.2 Monotone operations
2.3.3 Additive and multiplicative operations
Chapter 2.4 Natural numbers
2.4.1 Fundamental properties
2.4.2 Arithmetic of natural numbers
Chapter 2.5 Generalized Recursive Definitions
2.5.1 The ancestry relation
2.5.2 Recursion over well-founded relations
2.5.3 Well-founded classes
Chapter 2.6 Well-orderings
2.6.1 Isomorphisms and simple orderings
2.6.2 Well - ordering relations
Chapter 2.7 Cardinality
2.7.1 Equipollency
2.7.2 Cardinal addition and multiplication
2.7.3 Finite and denumerable classes
2.7.4 Finite and denumerable orderings
PART 3: MORSE-KELLEY-TARSKI CLASS THEORY
Chapter 3.1 Introduction to Morse-Kelley-Tarski Class Theory
3.1.1 Axiomatic system
3.1.2 Elementary consequences
3.1.3 Relations and functions
(+) 3.1.4 MKT is a subtheory of B
Chapter 3.2 Structure of well-founded classes
3.2.1 Well -founded classes in MKT
3.2.2 Rank
3.2.3 Universal domains with a given rank
Chapter 3.3 Ordinal Numbers
3.3.1 General properties of ordinals
3.3.2 Ordinal functions
3.3.3 Iteration of functions
3.3.4 Ordinal addition
3.3.5 Ordinal multiplication
3.3.6 Ordinal exponentiation
Chapter 3.4 Cardinal Numbers
3.4.1 Definition of cardinal numbers
3.4.2 Cardinal exponentiation
3.4.3 Finite multiples of cardinals and countable sums
3.4.4 Bradford's result and its consequences
Chapter 3.5 Simple Orderings
3.5.1 Types of simple orderings
3.5.2 Scattered, dense, and continuous orderings
3.5.3 Well -Orderings
Chapter 3.6 Alephs
3.6.1 Arithmetic of alephs
3.6.2 Hartog's function
Chapter 3.7 The Local Axiom of Choice and the Generalized Continuum Hypothesis
3.7.1 Forms of the local axiom of choice
3.7.2 Maximal principles
3.7.3 Cardinal equivalences
3.7.4 The generalized continuum hypothesis implies AC
Chapter 3.8 Gauging Sizes of Sets and Classes
3.8.1 Initial ordinals
3.8.2 Degree of cofinality
3.8.3 The exponential scale
3.8.4 Inaccessible sets
3.8.5 Closed unbounded and stationary classes
PART 4: MORSE-KELLEY-TARSKI CLASS THEORY WITH CHOICE
Chapter 4.1 The Global Axiom of Choice
4.1.1 Axioms of choice
4.1.2 Immediate consequences
Chapter 4.2 Cardinality Theory
4.2.1 Initial ordinals as cardinals
4.2.2 Infinite sums and products
4.2.3 Degree of cofinality with choice
4.2.4 Cardinal exponentiation of initial ordinals
4.2.5 The exponential scale of initial ordinals
Chapter 4.3 Filters , Ideals, and Trees
4.3.1 Filters and ideals
4.3.2 The closed unbounded filter and the thin ideal
4.3.3 Trees
PART 5: BERNAYS CLASS THEORY
Chapter 5.1 Inaccessibility
5.1.1 Inaccessible sets
5.1.2 Inaccessible cardinals
5.1.3 Classification of inaccessible cardinals
Chapter 5.2 Weakly Compact and Larger Cardinals
5.2.1 Weakly compact cardinals
5.2.2 Larger cardinals
References
Index of Symbols

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