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The Mathematics of Infinity: A Guide to Great Ideas, 2nd Edition

Book Description

Praise for the First Edition

". . . an enchanting book for those people in computer science or mathematics who are fascinated by the concept of infinity."—Computing Reviews

". . . a very well written introduction to set theory . . . easy to read and well suited for self-study . . . highly recommended."—Choice

The concept of infinity has fascinated and confused mankind for centuries with theories and ideas that cause even seasoned mathematicians to wonder. The Mathematics of Infinity: A Guide to Great Ideas, Second Edition uniquely explores how we can manipulate these ideas when our common sense rebels at the conclusions we are drawing.

Continuing to draw from his extensive work on the subject, the author provides a user-friendly presentation that avoids unnecessary, in-depth mathematical rigor. This Second Edition provides important coverage of logic and sets, elements and predicates, cardinals as ordinals, and mathematical physics. Classic arguments and illustrative examples are provided throughout the book and are accompanied by a gradual progression of sophisticated notions designed to stun readers' intuitive view of the world.

With an accessible and balanced treatment of both concepts and theory, 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, The Mathematics of Infinity: A Guide to Great Ideas, Second Edition is an insightful supplement for mathematics courses on set theory at the undergraduate level. The book also serves as a fascinating reference for mathematically inclined individuals who are interested in learning about the world of counterintuitive mathematics.

Table of Contents

  1. Cover Page
  2. Title Page
  3. Copyright
  4. Dedication
  5. Contents
  6. Preface for the Second Edition
  7. Chapter 1: Logic
    1. 1.1 Axiomatic Method
    2. 1.2 Tabular Logic
    3. 1.3 Tautology
    4. 1.4 Logical Strategies
    5. 1.5 Implications From Implications
    6. 1.6 Universal Quantifiers
    7. 1.7 Fun With Language and Logic
  8. Chapter 2: Sets
    1. 2.1 Elements and Predicates
    2. 2.2 Equality
    3. 2.3 Cartesian Products
    4. 2.4 Power Sets
    5. 2.5 Something From Nothing
    6. 2.6 Indexed Families of Sets
  9. Chapter 3: Functions
    1. 3.1 Functional Preliminaries
    2. 3.2 Images and Preimages
    3. 3.3 One-to-One and Onto Functions
    4. 3.4 Bijections
    5. 3.5 Inverse Functions
  10. Chapter 4: Counting Infinite Sets
    1. 4.1 Finite Sets
    2. 4.2 Hilbert's Infinite Hotel
    3. 4.3 Equivalent Sets and Cardinality
  11. Chapter 5: Infinite Cardinals
    1. 5.1 Countable Sets
    2. 5.2 Uncountable Sets
    3. 5.3 Two Infinities
    4. 5.4 Power Sets
    5. 5.5 The Arithmetic of Cardinals
  12. Chapter 6: Weil-Ordered Sets
    1. 6.1 Successors of Elements
    2. 6.2 Constructing Weil-Ordered Sets
    3. 6.3 Cardinals as Ordinals
    4. 6.4 Magnitude versus Cardinality
  13. Chapter 7: Inductions and Numbers
    1. 7.1 Mathematical Induction
    2. 7.2 Sums of Powers of Integers
    3. 7.3 Transfinite Induction
    4. 7.4 Mathematical Recursion
    5. 7.5 Number Theory
    6. 7.6 The Fundamental Theorem of Arithmetic
    7. 7.7 Perfect Numbers
  14. Chapter 8: Prime Numbers
    1. 8.1 Prime Number Generators
    2. 8.2 The Prime Number Theorem
    3. 8.3 Products of Geometric Series
    4. 8.4 The Riemann Zeta Function
    5. 8.5 Real Numbers
  15. Chapter 9: Logic and Meta-Mathematics
    1. 9.1 The Collection of All Sets
    2. 9.2 Other Than True or False
    3. 9.3 No Theory of Everything
  16. Bibliography
  17. Index