Classic Set Theory : For Guided Independent Study book cover
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Classic Set Theory
For Guided Independent Study





ISBN 9780412606106
Published July 1, 1996 by Chapman and Hall/CRC
296 Pages

 
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Book 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

    ...
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