The Mathematics of Infinity: A Guide to Great Ideas

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.

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.

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