The Mathematics of Infinity: A Guide to Great Ideas

307 Pages·2006·7.52 MB·English

by  Theodore G. Faticoni

The Mathematics of Infinity
A Guide to Great Ideas
Theodore G. Faticoni
Fordham University
Department of Mathematics
Bronx, NY
A JOHN WILEY & SONS, INC., PUBLICATION
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The Mathematics of Infinity
PURE AND APPLIED MATHEMATICS
A Wiley-Interscience Series of Texts, Monographs, and Tracts
Consulting Editor: David A. Cox
Founded by RICHARD COURANT
Editors Emeriti: MYRON B. ALLEN 111, DAVID A. COX, PETER HILTON,
HARRY HOCHSTADT, PETER LAX, JOHN TOLAND
A complete list of the titles in this series appears at the end of this volume.
The Mathematics of Infinity
A Guide to Great Ideas
Theodore G. Faticoni
Fordham University
Department of Mathematics
Bronx, NY
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright 0 2006 by John Wiley & Sons, lnc. All rights reserved
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Library of Congress Cataloging-in-Publication Data is available.
Faticoni, Theodore G.
The Mathematics of Infinity: A Guide to Great Ideas
ISBN-1 3 978-0-47 1-79432-5
ISBN-I0 0-471-79432-5
Printed in the United States of America.
I 0 9 8 7 6 5 4 3 2 1
To Professor Elliot Wolk who taught me Set Theory.
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Contents
Preface ix
1 Elementary Set Theory 1
1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Cartesian Products . . . . . . . . . . . . . . . . . . . 17
1.3 Power Sets . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Something From Nothing . . . . . . . . . . . . . . . . 22
1.5 Indexed Families of Sets . . . . . . . . . . . . . . . . 27
2 Functions 37
2.1 Functional Preliminaries . . . . . . . . . . . . . . . . 38
2.2 Images and Preimages . . . . . . . . . . . . . . . . . 52
2.3 One-to-one and Onto Functions . . . . . . . . . . . . 61
2.4 Bijections . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5 Inverse Functions . . . . . . . . . . . . . . . . . . . . 68
3 Counting Infinite Sets 75
3.1 Finite Sets . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 Hilbert’s Infinite Hotel . . . . . . . . . . . . . . . . . 82
3.3 Equivalent Sets and Cardinality . . . . . . . . . . . . 98
4 Infinite Cardinals 103
4.1 Countable Sets . . . . . . . . . . . . . . . . . . . . . 104
4.2 Uncountable Sets . . . . . . . . . . . . . . . . . . . . 117
4.3 Two Infinities . . . . . . . . . . . . . . . . . . . . . . 126
4.4 Power Sets . . . . . . . . . . . . . . . . . . . . . . . . 132
4.5 The Arithmetic of Cardinals . . . . . . . . . . . . . .1 45
vii