The mathematics of infinity : a guide to great ideas / Theodore G. Faticoni.

Format:

Book

Author/Creator:

Faticoni, Theodore G. (Theodore Gerard), 1954-

Series:

Pure and applied mathematics (John Wiley & Sons : Unnumbered)

Pure and applied mathematics

Language:

English

Subjects (All):

Cardinal numbers.

Set theory.

Infinite.

Physical Description:

xii, 287 pages : illustrations ; 25 cm.

Place of Publication:

Hoboken, N.J. : Wiley-Interscience, [2006]

Summary:

The concept of infinity has fascinated and confused mankind for centuries with concepts and ideas that cause even seasoned mathematicians to wonder. For instance, the idea that a set is infinite if it is not a finite set is an elementary concept that jolts our common sense and imagination. The Mathematics of Infinity: A Guide to Great Ideas uniquely explores how we can manipulate these ideas when our common sense rebels at the conclusions we are drawing.

Writing with clear knowledge and affection for the subject, the author introduces and explores infinite sets, infinite cardinals, and ordinals, thus challenging the readers' intuitive beliefs about infinity. Requiring little mathematical training and a healthy curiosity, the book presents a user-friendly approach to ideas involving the infinite. Readers will discover the main ideas of infinite cardinals and ordinal numbers without experiencing in-depth mathematical rigor. Classic arguments and illustrative examples are provided throughout the book and are accompanied by a gradual progression of sophisticated notions designed to stun your intuitive view of the world.

Recommended as recreational reading for the mathematically inquisitive or as supplemental reading for curious college students, The Mathematics of Infinity: A Guide to Great Ideas gently leads readers into the world of counterintuitive mathematics.

Contents:

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 Hibert'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 145

5 Well Ordered Sets 163

5.1 Successors of Elements 163

5.2 The Arithmetic of Ordinals 173

5.3 Cardinals as Ordinals 184

5.4 Magnitude versus Cardinality 197

6 Inductions and Numbers 205

6.1 Mathematical Induction 205

6.2 Transfinite Induction 222

6.3 Mathematical Recursion 231

6.4 Number Theory 237

6.5 The Fundamental Theorem of Arithmetic 240

6.6 Perfect Numbers 242

7 Prime Numbers 247

7.1 Prime Number Generators 247

7.2 The Prime Number Theorem 251

7.3 Products of Geometric Series 254

7.4 The Riemann Zeta Function 261

7.5 Real Numbers 265

8 Logic and Meta-Mathematics 271

8.1 The Collection of All Sets 271

8.2 Other Than True or False 274.

Notes:

Includes bibliographical references (page 283) and index.

ISBN:

0471794325

OCLC:

65065422

Publisher Number:

9780471794325