Sets: Naïve, Axiomatic and Applied
A Basic Compendium with Exercises for Use in Set Theory for Non Logicians, Working and Teaching Mathematicians and Students
- 1st Edition - May 9, 2014
- Imprint: Pergamon
- Authors: D. Van Dalen, H. C. Doets, H. De Swart
- Editor: I. N. Sneddon
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 1 7 9 7 - 3
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 5 0 3 9 - 0
Sets: Naïve, Axiomatic and Applied is a basic compendium on naïve, axiomatic, and applied set theory and covers topics ranging from Boolean operations to union, intersection, and… Read more
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Request a sales quoteSets: Naïve, Axiomatic and Applied is a basic compendium on naïve, axiomatic, and applied set theory and covers topics ranging from Boolean operations to union, intersection, and relative complement as well as the reflection principle, measurable cardinals, and models of set theory. Applications of the axiom of choice are also discussed, along with infinite games and the axiom of determinateness. Comprised of three chapters, this volume begins with an overview of naïve set theory and some important sets and notations. The equality of sets, subsets, and ordered pairs are considered, together with equivalence relations and real numbers. The next chapter is devoted to axiomatic set theory and discusses the axiom of regularity, induction and recursion, and ordinal and cardinal numbers. In the final chapter, applications of set theory are reviewed, paying particular attention to filters, Boolean algebra, and inductive definitions together with trees and the Borel hierarchy. This book is intended for non-logicians, students, and working and teaching mathematicians.
Preface
Acknowledgements
Introduction
Chapter I. Naïve Set Theory
1. Some Important Sets and Notations
Natural Numbers
Integers
Rationals
Reals
Singleton
2. Equality of Sets
Extensionality Axiom
3. Subsets
4. The Naïve Comprehension Principle and the Empty Set
5. Union, Intersection and Relative Complement
Complement
De Morgan's Laws
6. Power Set
7. Unions and Intersections of Families
Greatest and smallest Set with Property Φ
Disjoint
Pairwise Disjoint
8. Ordered Pairs
Unordered Pairs
Ordered n-Tuples
9. Cartesian Product
10. Relations
Binary Relation
Converse
Composition
Identity Relation
Domain
Range
Reflexive
Symmetric
Transitive
11. Equivalence Relations
Equivalence Class
Representative
Partition
Quotient Set
Ẕ,
Q,
12. Real Numbers
Chains of Segments
Dedekind Cut
Cauchy Sequence
13. Functions (Mappings)
Injection
Surjection
Bijection
Identity Mapping
Characteristic Function
Equality of Functions
Composition
Inverse
Restriction
Structure
Isomorphism
Cartesian Product of a Family
14. Orderings
Lexicographic Ordering
Partial Ordering
Diagram
Minimal Element
Smallest Element
Lower Bound
Infimum
Total(Linear) Ordering
Well-Ordering
Immediate Successor
Principle of Transfinite Induction
Initial Segment
15. Equivalence (Cardinality)
≤1
Diagonal Method
<1
Denumerable
Countable
Cantor-Bernstein
Uncountable
16. Finite and Infinite
Axiom of Choice
Choice Function
Dedekind-Infinite
17. Denumerable Sets
Hilbert Hotel
Closure Under Union and Product
Ramsey's Theorem
18. UncountabZe Sets
The Continuum
P(Ṉ)
Continuum-Hypothesis
19. The Paradoxes
Russell's Paradox
The Set of All Sets
The Separation Principle
20. The Set Theory of Zermelo-Fraenkel (ZF)
Formal Language
Axioms
Natural Numbers
є-Minimal
21. Peano's Arithmetic
Axioms
Definition by Recursion
Chapter II. Axiomatic Set Theory
1. The Axiom of Regularity
Motivation
Consistency
2. Induction and Recursion
Well-Foundedness
Transitive Closure
Induction Principles
Definition by Recursion
Representation Theorem for Well-Founded Extensional Structures
3. Ordinal Numbers
Von Neumann's Ordinals
4. The Cumulative Hierarchy
Rank
Partial universes
5. Ordinal Arithmetic
Addition
Multiplication
Exponentiation
Cantor's Normal Form
6. Normal Operations
Cofinality
Regularity
Fixed Points
Derivative
7. The Reflection Principle
8. Initial Numbers
Hartogs' Function
Weakly Inaccessible
Well-Ordering of ORXOR and Consequences
9. The Axiom of Choice
Motivation
Equivalents
Two Proofs of the Well-Ordering Theorem
Zorn's Lemma
GCH
DC
CC
10. Cardinal Numbers
Motivation
Possible Definitions
Alephs
Addition
Multiplication
Exponentiation
Regularity
Cofinality
König's Inequality
11. Models
Purpose and Meaning
Conceptual Difficulties
Standard Models and Absoluteness
Natural Models
Role of the Axioms of Infinity and Substitution
Strongly Inaccessible
Reflections from a Strongly Inaccessible Ordinal
Non-Finite Axiomatizability
Hereditarily < K
Role of the Axioms of Sum-, and Powerset
Hereditarily Finite
12. Measurable cardinals
Higher Infinities via Reflection
Measurables Giving Rise to Reflexive Situations
Ultrapowers
Chapter III. Applications
1. Filters
Free Filter
Ultrafilter
2. Boolean Algebra
Axioms
Representation Theorems
Duality
3. Order Types
Back-and-Forth Method
Type of Q
Ordinals
Sum
Product
4. Inductive Definitions
Minimal Fixed Point
Length
Cantor-Bendixson
5. Applications of the Axiom of Choice
Basis of a Vectorspace
Compactness of Products (Tychonov)
Non-Measurable Sets (Vitali)
Algebraic Closure (Steinitz)
6. The Borel Hierarchy
Fα, Gα
Universal Sets
Hierarchy
Separation
Reduction
7. Trees
Ordinals
Continuity
Infinity Lemma
8. The Axiom of Determinateness (AD)
Games
Strategy
CC
AD → l AC
Lebesgue Measure on Ṟ
Appendix
Symbols
Literature
Index
Other Titles in the Series
- Edition: 1
- Published: May 9, 2014
- Imprint: Pergamon
- Language: English
- Paperback ISBN: 9781483117973
- eBook ISBN: 9781483150390
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