1st Edition

# Classic Set Theory For Guided Independent Study

296 Pages
by Chapman & Hall

296 Pages
by Chapman & Hall

296 Pages
by Routledge

Also available as eBook on:

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

### Biography

Goldrei, D.C.