 # 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 - January 1, 1978

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• Authors: D. Van Dalen, H. C. Doets, H. De Swart
• eBook ISBN: 9781483150390

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## Description

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.

• 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

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

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

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

Lebesgue Measure on Ṟ

Appendix

Symbols

Literature

Index

Other Titles in the Series

## Product details

• No. of pages: 360
• Language: English
• Published: January 1, 1978
• Imprint: Pergamon
• eBook ISBN: 9781483150390