  # Classic Set Theory

## For Guided Independent Study, 1st Edition

Chapman and Hall/CRC

296 pages

Paperback: 9780412606106
pub: 1996-07-01
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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.

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