Classic Set Theory: For Guided Independent Study, 1st Edition (Paperback) book cover

Classic Set Theory

For Guided Independent Study, 1st Edition

By D.C. Goldrei

Chapman and Hall/CRC

296 pages

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Description

Designed for undergraduate students of set theory, Classic Set Theory presents a modern perspective of the classic work of Georg Cantor and Richard Dedekin and their immediate successors. This includes:

  • The definition of the real numbers in terms of rational numbers and ultimately in terms of natural numbers

  • Defining natural numbers in terms of sets

  • The potential paradoxes in set theory

  • The Zermelo-Fraenkel axioms for set theory

  • The axiom of choice

  • The arithmetic of ordered sets

  • Cantor's two sorts of transfinite number - cardinals and ordinals - and the arithmetic of these.

    The book is designed for students studying on their own, without access to lecturers and other reading, along the lines of the internationally renowned courses produced by the Open University. There are thus a large number of exercises within the main body of the text designed to help students engage with the subject, many of which have full teaching solutions. In addition, there are a number of exercises without answers so students studying under the guidance of a tutor may be assessed.

    Classic Set Theory gives students sufficient grounding in a rigorous approach to the revolutionary results of set theory as well as pleasure in being able to tackle significant problems that arise from the theory.

  • Table of Contents

    INTRODUCTION

    Outline of the book

    Assumed knowledge

    THE REAL NUMBERS

    Introduction

    Dedekind's construction

    Alternative constructions

    The rational numbers

    THE NATURAL NUMBERS

    Introduction

    The construction of the natural numbers

    Arithmetic

    Finite sets

    THE ZERMELO-FRAENKEL AXIOMS

    Introduction

    A formal language

    Axioms 1 to 3

    Axioms 4 to 6

    Axioms 7 to 9

    CARDINAL (Without the Axiom of Choice)

    Introduction

    Comparing Sizes

    Basic properties of ˜ and =

    Infinite sets without AC-countable sets

    Uncountable sets and cardinal arithmetic without AC

    ORDERED SETS

    Introduction

    Linearly ordered sets

    Order arithmetic

    Well-ordered sets

    ORDINAL NUMBERS

    Introduction

    Ordinal numbers

    Beginning ordinal arithmetic

    Ordinal arithmetic

    The Às

    SET THEORY WITH THE AXIOM OF CHOICE

    Introduction

    The well-ordering principle

    Cardinal arithmetic and the axiom of choice

    The continuum hypothesis

    BIBLIOGRAPHY

    INDEX

    Subject Categories

    BISAC Subject Codes/Headings:
    MAT000000
    MATHEMATICS / General
    MAT028000
    MATHEMATICS / Set Theory