Classic Set Theory: For Guided Independent Study

ISBN-13: 9780412606106 / Angielski / Miękka / 1996 / 296 str.

Goldrei

Opis

This volume is designed for undergraduate students of set theory. It covers the classic work of Georg Cantor and Richard Dedekind at the end of the 19th century, and their immediate successors, viewed from a modern perspective. The book 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. There are a number of exercises within the main body of the text, designed to help students engage with the subject, many of these with full teaching solutions. In addition there are a number of exercises without answers so that students studying under the guidance of a tutor may be assessed.

Spis treści

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

O autorze

Goldrei, D.C.