Sets: Naïve, Axiomatic and Applied

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 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.

Inhalt

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 F

   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

    ,

    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

    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( )

    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

    -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

    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

   Fa, Ga

   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

Appendix

Symbols

Literature

Index

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