The Mathematics of Infinity: A Guide to Great Ideas (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)

A balanced and clearly explained treatment of infinity in mathematics. 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. With a thoughtful and balanced treatment of both concepts and theory, The Mathematics of Infinity focuses on the following topics: Sets and Functions Images and Preimages of Functions Hilbert's Infinite Hotel Cardinals and Ordinals The Arithmetic of Cardinals and Ordinals the Continuum Hypothesis Elementary Number Theory The Riemann Hypothesis The Logic of Paradoxes 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. The Mathematics of Infinity A Guide to Great Ideas......Page 6 Contents......Page 10 Preface......Page 12 1 Elementary Set Theory......Page 18 1.1 Sets......Page 19 1.2 Cartesian Products......Page 34 1.3 Power Sets......Page 37 1.4 Something From Nothing......Page 39 1.5 Indexed Families of Sets......Page 44 2 Functions......Page 54 2.1 Functional Preliminaries......Page 55 2.2 Images and Preimages......Page 69 2.3 One-to-one and Onto Functions......Page 78 2.4 Bijections......Page 82 2.5 Inverse Functions......Page 85 3.1 Finite Sets......Page 92 3.2 Hilbert’s Infinite Hotel......Page 99 3.3 Equivalent Sets and Cardinality......Page 115 4 Infinite Cardinals......Page 120 4.1 Countable Sets......Page 121 4.2 Uncountable Sets......Page 134 4.3 Two Infinities......Page 143 4.4 Power Sets......Page 149 4.5 The Arithmetic of Cardinals......Page 162 5.1 Successors of Elements......Page 180 5.2 The Arithmetic of Ordinals......Page 190 5.3 Cardinals as Ordinals......Page 201 5.4 Magnitude versus Cardinality......Page 214 6.1 Mathematical Induction......Page 222 6.2 Transfinite Induction......Page 239 6.3 Mathematical Recursion......Page 248 6.4 Number Theory......Page 254 6.5 The Fundamental Theorem of Arithmetic......Page 257 6.6 Perfect Numbers......Page 259 7.1 Prime Number Generators......Page 264 7.2 The Prime Number Theorem......Page 268 7.3 Products of Geometric Series......Page 271 7.4 The Riemann Zeta Function......Page 278 7.5 Real Numbers......Page 282 8.1 The Collection of All Sets......Page 288 8.2 Other Than True or False......Page 291 Bibliography......Page 300 Index......Page 301 Facticoni starts at the basic concept of sets and functions, discusses infinite cardinalities, and reaches the topics of transfinite induction and cardinal arithmetics. He also discusses some famous paradoxes, and there are excursions to number theory, namely, to prime numbers and to the Riemann zeta function The Mathematics of Infinity addresses infinite cardinals and is appropriate for readers at any level. Inviting the reader to imagine constructing an infinite chain infinities, which are called cardinals, the author successfully prepares and motivates readers for topics covered within the book The ideal writing style ascribed to by mathematicians is that in writing mathematics, less is more. Theodore G. Faticoni. Includes Bibliographical References (p. 283) And Index.