Faticoni The Mathematics of Infinity

A Guide to Great Ideas
2. Auflage 2012
ISBN: 978-1-118-24382-4
Verlag: John Wiley & Sons
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 358 Seiten, E-Book

Reihe: Wiley Series in Pure and Applied Mathematics

Praise for the First Edition
". . . an enchanting book for those people in computer scienceor mathematics who are fascinated by the concept ofinfinity."--Computing Reviews
". . . a very well written introduction to set theory . . . easyto read and well suited for self-study . . . highlyrecommended."--Choice
The concept of infinity has fascinated and confused mankind forcenturies with theories and ideas that cause even seasonedmathematicians to wonder. The Mathematics of Infinity: A Guide toGreat Ideas, Second Edition uniquely explores how we can manipulatethese ideas when our common sense rebels at the conclusions we aredrawing.
Continuing to draw from his extensive work on the subject, theauthor provides a user-friendly presentation that avoidsunnecessary, in-depth mathematical rigor. This Second Editionprovides important coverage of logic and sets, elements andpredicates, cardinals as ordinals, and mathematical physics.Classic arguments and illustrative examples are provided throughoutthe book and are accompanied by a gradual progression ofsophisticated notions designed to stun readers' intuitive view ofthe world.
With an accessible and balanced treatment of both concepts andtheory, the book focuses on the following topics:
* Logic, sets, and functions
* Prime numbers
* Counting infinite sets
* Well ordered sets
* Infinite cardinals
* Logic and meta-mathematics
* Inductions and numbers
Presenting an intriguing account of the notions of infinity, TheMathematics of Infinity: A Guide to Great Ideas, Second Edition isan insightful supplement for mathematics courses on set theory atthe undergraduate level. The book also serves as a fascinatingreference for mathematically inclined individuals who areinterested in learning about the world of counterintuitivemathematics.

Autoren/Hrsg.

Faticoni, Theodore G.

Inhaltsverzeichnis

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

Über Autor(innen)

THEODORE G. FATICONI, PhD, is a Professor in the Department of Mathematics at Fordham University. His professional experience includes forty research papers in peer-reviewed journals and forty lectures on his research to his colleagues.