Sets: Naïve, Axiomatic and Applied

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.

Table of Contents


  • 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

Product details

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

About the Authors

D. Van Dalen

H. C. Doets

H. De Swart

About the Editor

I. N. Sneddon

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