The mathematics of infinity : a guide to great ideas - Faticoni, Theodore G. (Theodore Gerard), 1954- (Hoboken, N.J. : John Wiley & Sons, 2012)
"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. Infinity, we are told, is as large as things get. This is not entirely true. This book does not refer to infinities, but rather to cardinals. This is to emphasize the point that what you thought you knew about infinity is probably incorrect or imprecise. Since the reader is assumed to be educated in mathematics, but not necessarily mathematically trained, an attempt has been made to convince the reader of the truth of a matter without resorting to the type of rigor found in professional journals. Therefore, the author has accompanied the proofs with illustrative examples. The examples are often a part of a larger proof. Important facts are included and their proofs have been excluded if the author has determined that the proof is beyond the scope of the discussion. For example, it is assumed and not proven within the book that a collection of cardinals is larger than any set or mathematical object. The topics covered within the book cannot be found within any other one book on infinity, and the work succeeds in being the only book on infinite cardinals for the high school educated person. Topical coverage includes: logic and sets; functions; counting infinite sets; infinite cardinals; well ordered sets; inductions and numbers; prime numbers; and logic and meta-mathematics. "--
Machine generated contents note: 1. Logic 1 1.1 Axiomatic Method 2 1.2 Tabular Logic 3 1.3 Tautology 9 1.4 Logical Strategies 15 1.5 Implications From Implications 17 1.6 Universal Quantifiers 20 1.7 Fun With Language and Logic 22 2. Sets 29 2.1 Elements and Predicates 30 2.2 Cartesian Products 45 2.3 Power Sets 48 2.4 Something From Nothing 50 2.5 Indexed Families of Sets 56 3. Functions 65 3.1 Functional Preliminaries 66 3.2 Images and Preimages 81 3.3 One-to-One and Onto Functions 90 3.4 Bijections 95 3.5 Inverse Functions 97 4. Counting Infinite Sets 105 4.1 Finite Sets 105 4.2 Hilbert's Infinite Hotel 113 4.3 Equivalent Sets and Cardinality 128 5. Infinite Cardinals 135 5.1 Countable Sets 136 5.2 Uncountable Sets 149 5.3 Two Infinites 159 5.4 Power Sets 166 5.5 The Arithmetic of Cardinals 180 6. Well Ordered Sets 199 6.1 Successors of Elements 199 6.2 The Arithmetic of Ordinals 210 6.3 Cardinals as Ordinals 222 6.4 Magnitude versus Cardinality 234 7. Inductions and Numbers 243 7.1 Mathematical Induction 243 7.2 Sums of Powers of Integers 260 7.3 Transfinite Induction 264 7.4 Mathematical Recursion 274 7.5 Number Theory 279 7.6 The Fundamental Theorem of Arithmetic 283 7.7 Perfect Numbers 285 8. Prime Numbers 289 8.1 Prime Number Generators 289 8.2 The Prime Number Theorem 292 8.3 Products of Geometric Series 296 8.4 The Riemann Zeta Function 302 8.5 Real Numbers 307 9. Logic and Meta-Mathematics 313 9.1 The Collection of All Sets 313 9.2 Other Than True or False 317 9.3 Logical Implications of A Theory of Everything 326 Bibliography 283 Index 284 .
Includes bibliographical references and index.
Keywords: Cardinal numbers; Set theory; Infinite; MATHEMATICS / Infinity