This title offers 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; and, 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.
Rezensionen / Stimmen
"...an enchanting book for those people in computer science or mathematics who are fascinated by the concept of infinity." (Computing Reviews.com, May 25, 2007) "...a very well written introduction to set theory...easy to read and well suited for self-study...highly recommended." (CHOICE, February 2007)
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ISBN-13
978-0-470-04914-3 (9780470049143)
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Schweitzer Klassifikation
Theodore G. Faticoni, PhD, is Professor in the Dept. of Mathematics at Fordham University. He received his PhD in Mathematics from the Univ. of Connecticut in 1981. His professional experience includes 30 research papers in peer reviewed journals and 30 lectures on his research to his colleagues.
Preface. 1. Elementary Set Theory. 1.1 Sets. 1.2 Cartesian Products. 1.3 Power Sets. 1.4 Something From Nothing. 1.5 Indexed Families of Sets. 2. Functions. 2.1 Functional Preliminaries. 2.2 Images and Preimages. 2.3 One-to-one and Onto . 2.4 Bijections. 2.5 Inverse Functions. 3. Counting Infinite Sets. 3.1 Finite Sets. 3.2 Hilbert's Infinite Hotel. 3.3 Equivalent Sets and Cardinality. 4. Infinite Cardinals. 4.1 Countable Sets. 4.2 Uncountable Sets. 4.3 Two Infinities. 4.4 Power Sets. 4.5 The Arithmetic of Cardinals. 5. Well Ordered Sets. 5.1 Successors of Elements. 5.2 The Arithmetic of Ordinals. 5.4 Magnitude versus Cardinality. 6. Inductions and Numbers. 6.1 Mathematical Induction. 6.2 Transfinite Induction. 6.3 Mathematical Recursion . 6.4 Number Theory. 6.5 The Fundamental Theorem of Arithmetic. 6.6 Perfect Numbers. 7. Prime Numbers. 7.1 Prime Number Generators. 7.2 The Prime Number Theorem. 7.3 Products of Geometric Series. 7.4 The Riernann Zeta Function. 7.5 Real Numbers. 8. Logic and Meta-Mathematics. 8.1 The Collection of All Sets. 8.2 Other Than True or False. Bibliography. Index.
