Problems and Theorems in Classical Set Theory

✍ Scribed by Péter Komjáth, Vilmos Totik

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 527
• Series: Problem Books in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.

✦ Table of Contents

Cover......Page 1
Series: Problem Books in Mathematics......Page 2
Problems and Theorems in Classical Set Theory......Page 4
Copyright......Page 5
Contents......Page 8
Preface......Page 12
Part I: Problems......Page 14
1. Operations on sets......Page 16
2. Countability......Page 22
3. Equivalence......Page 26
4. Continuum......Page 28
5. Sets of reals and real functions......Page 32
6. Ordered sets......Page 36
7. Order types......Page 46
8. Ordinals......Page 50
9. Ordinal arithmetic......Page 56
10. Cardinals......Page 64
11. Partially ordered sets......Page 68
12. Transfinite enumeration......Page 72
13. Euclidean spaces......Page 76
14. Zorn’s lemma......Page 78
15. Hamel bases......Page 80
16. The continuum hypothesis......Page 84
17. Ultrafilters on ω......Page 88
18. Families of sets......Page 92
19. The Banach–Tarski paradox......Page 94
20. Stationary sets in ω_1......Page 98
21. Stationary sets in larger cardinals......Page 102
22. Canonical functions......Page 106
23. Infinite graphs......Page 108
24. Partition relations......Page 114
25. Δ-systems......Page 120
26. Set mappings......Page 122
27. Trees......Page 124
28. The measure problem......Page 130
29. Stationary sets in [λ]^{<κ}......Page 136
30. The axiom of choice......Page 140
31. Well-founded sets and the axiom of foundation......Page 142
Part II: Solutions......Page 146
1. Operations on sets......Page 148
2. Countability......Page 160
3. Equivalence......Page 172
4. Continuum......Page 176
5. Sets of reals and real functions......Page 186
6. Ordered sets......Page 198
7. Order types......Page 226
8. Ordinals......Page 236
9. Ordinal arithmetic......Page 250
10. Cardinals......Page 278
11. Partially ordered sets......Page 288
12. Transfinite enumeration......Page 298
13. Euclidean spaces......Page 312
14. Zorn’s lemma......Page 322
15. Hamel bases......Page 330
16. The continuum hypothesis......Page 340
17. Ultrafilters on ω......Page 354
18. Families of sets......Page 364
19. The Banach–Tarski paradox......Page 372
20. Stationary sets in ω_1......Page 382
21. Stationary sets in larger cardinals......Page 390
22. Canonical functions......Page 398
23. Infinite graphs......Page 402
24. Partition relations......Page 418
25. Δ-systems......Page 434
26. Set mappings......Page 440
27. Trees......Page 446
28. The measure problem......Page 466
29. Stationary sets in [λ]^{<κ}......Page 476
30. The axiom of choice......Page 484
31. Well-founded sets and the axiom of foundation......Page 494
Part III: Appendix......Page 504
1. Glossary of Concepts......Page 506
2. Glossary of Symbols......Page 520
3. Index......Page 522