Description
Book SynopsisDesigned for undergraduate students of set theory, this book presents a modern perspective of the classic work of Georg Cantor and Richard Dedekin and their immediate successors. It aims to give students a grounding to the results of set theory as well as to tackle significant problems that arise from the theory.
Table of ContentsINTRODUCTION
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
