This book offers an excursion through the developmental area of research mathematics. It presents some 40 papers, published between the 1870s and the 1970s, on proofs of the Cantor-Bernstein theorem and the related Bernstein division theorem. While the emphasis is placed on providing accurate proofs, similar to the originals, the discussion is broadened to include aspects that pertain to the methodology of the development of mathematics and to the philosophy of mathematics. Works of prominent mathematicians and logicians are reviewed, including Cantor, Dedekind, Schröder, Bernstein, Borel, Zermelo, Poincaré, Russell, Peano, the Königs, Hausdorff, Sierpinski, Tarski, Banach, Brouwer and several others mainly of the Polish and the Dutch schools. In its attempt to present a diachronic narrative of one mathematical topic, the book resembles Lakatos’ celebrated book Proofs and Refutations. Indeed, some of the observations made by Lakatos are corroborated herein. The analogy between the two books is clearly anything but superficial, as the present book also offers new theoretical insights into the methodology of the development of mathematics (proof-processing), with implications for the historiography of mathematics.
Contents: 1. Cantor's CBT Proof for Sets of the Power of (II) — 2. Generalizing Cantor's CBT Proof — 3. CBT in Cantor's 1878 Beitrag — 4. The Theory of Inconsistent Sets — 5. Comparability in Cantor's Writings — 6. The Scheme of Complete Disjunction — 7. Ruptures in the Cantor-Dedekind Correspondence — 8. The Inconsistency of Dedekind's Infinite Set — 9. Dedekind's Proof of CBT — 10. Schröder's Proof of CBT — 11. Bernstein, Borel and CBT — 12. Schoenflies' 1900 Proof of CBT — 13. Zermelo's 1901 Proof of CBT — 14. Bernstein's Division Theorem — 15. Russell's 1902 Proof of CBT — 16. The Role of CBT in Russell's Paradox — 17. Jourdain's 1904 Generalization of Grundlagen — 18. Harward 1905 on Jourdain 1904 — 19. Poincaré and CBT — 20. Peano's Proof of CBT — 21. J. Kőnig's Strings Gestalt — 22. From Kings to Graphs — 23. Jourdain's Improvements Round — 24. Zermelo's 1908 Proof of CBT — 25. Korselt's Proof of CBT — 26. Proofs of CBT in Principia Mathematica — 27. The Origin of Hausdorff Paradox in BDT — 28. Sierpiński's Proofs of BDT — 29. Banach's Proof of CBT — 30. Kuratowski's proof of BDT — 31. Early Fixed-Point CBT Proofs: Whitaker; Tarski-Knaster — 32. CBT and BDT for Order-Types — 33. Sikorski's Proof of CBT for Boolean Algebras — 34. Tarski's Proofs of BDT and the Inequality-BDT — 35. Tarski's Fixed-Point Theorem and CBT — 36. Reichbach's Proof of CBT — 37. Hellmann's Proof of CBT — 38. CBT and Intuitionism — 39. CBT in Category Theory
Author(s): Arie Hinkis
Series: Science Networks. Historical Studies, Vol. 45
Publisher: Birkhäuser (Springer Basel)
Year: 2013
Language: English
Pages: 455
Cover......Page 1
Title Page......Page 5
Introduction......Page 11
Contents......Page 19
List of Figures and Tables......Page 25
Part I: Cantor and Dedekind......Page 27
1.1 The Generation Principles......Page 29
1.2 Proof of CBT for Sets of the Power of (II)......Page 31
1.3 The Limitation Principle......Page 35
1.4 The Union Theorem......Page 37
1.5 The Principles of Arithmetic......Page 39
2 Generalizing Cantor's CBT Proof......Page 41
2.1 The Scale of Number-Classes......Page 42
2.2 The Induction Step......Page 44
2.3 The Declaration of Infinite Numbers......Page 49
3.1 Equivalence Classes......Page 53
3.2 The Order Relation Between Powers......Page 54
3.3 A Direct Allusion......Page 55
3.4 CBT for the Continuum......Page 56
3.5 Generalized Proofs......Page 57
3.6 The Different Powers of 1878 Beitrag......Page 59
3.6.3 The Set of Denumerable Numbers (II)......Page 60
3.6.4 Dating the Infinity Symbols......Page 61
3.6.5 The Berlin Circumstances......Page 63
4 The Theory of Inconsistent Sets......Page 65
4.1 Inconsistent Sets Contain an Image of Ω......Page 67
4.2 Views in the Literature......Page 69
4.3 The Origin of the Inconsistent Sets Theory......Page 70
4.4 The End of the Limitation Principle......Page 72
5 Comparability in Cantor's Writings......Page 75
5.1 The Definition of Order Between Cardinal Numbers......Page 77
5.2 The Comparability Theorem for Cardinal Numbers......Page 78
5.4 The Comparability of Sets......Page 79
6.1 The Scheme and Schoenflies......Page 83
6.2 The Scheme and Dedekind......Page 85
6.3 The Scheme and Borel......Page 86
6.4 Schröder's Scheme......Page 88
6.5 The Origin of the Scheme of Complete Disjunction......Page 90
7.1 Views in the Literature......Page 93
7.2 The Aufgabe Complex......Page 94
7.3 The 1874 Rupture......Page 95
7.4 The 1899 Rupture......Page 98
7.5 By the Way......Page 101
8.1 Dedekind's Infinite Set......Page 103
8.2 Bernstein's Recollections......Page 105
8.3 Cantor's Criticism......Page 108
8.4 Dedekind's Concerns......Page 110
9 Dedekind's Proof of CBT......Page 113
9.1 Summary of the Theory of Chains......Page 116
9.2.1 The First Proof......Page 118
9.2.2 The Second Proof......Page 119
9.2.4 Comparing the Proofs......Page 120
9.3 The Origin of Dedekind's Proof......Page 122
9.4 Descriptors for Dedekind's Proof......Page 124
9.5 Comparison to Cantor's Proof......Page 126
Part II: The Early Proofs......Page 129
10 Schröder's Proof of CBT......Page 131
10.1 Schröder's Proof......Page 132
10.2 Criticism of Schröder's Proof......Page 137
10.3 Comparison with Cantor and Dedekind......Page 142
11 Bernstein, Borel and CBT......Page 143
11.1 Borel's Proof......Page 145
11.2 Bernstein's Original Proof......Page 147
11.3 Comparison with Earlier Proofs......Page 149
12.1 Cantor's 1899 Proof......Page 151
12.2 Schoenflies' Proof......Page 152
12.3 Comparisons......Page 153
13 Zermelo's 1901 Proof of CBT......Page 155
13.1 The Proof......Page 156
13.2 The Reemergence Argument......Page 160
13.3 Convex-Concave......Page 161
14 Bernstein's Division Theorem......Page 165
14.1 The Proof's Plan......Page 166
14.2 The Proof......Page 168
14.3 Generalizations of the Theorem......Page 174
14.4 The Inequality-BDT......Page 178
Part III: Under the Logicist Sky......Page 179
15 Russell's 1902 Proof of CBT......Page 181
15.1 The Core Arguments......Page 182
15.2 The Definition of ℵ₀......Page 186
16 The Role of CBT in Russell's Paradox......Page 191
16.1 Russell's Proof of Cantor's Theorem......Page 192
16.2 Derivation of Russell's Paradox......Page 193
16.4 Corroborating Lakatos......Page 195
17 Jourdain's 1904 Generalization of Grundlagen......Page 197
17.1 The Ordinals and the Alephs......Page 198
17.2 The Power of the Continuum......Page 200
17.3 Inconsistent Aggregates......Page 201
17.4 The Corollaries......Page 204
17.5 Jourdain's Rendering of Zermelo's 1901 CBT Proof......Page 205
17.6 The Sum and Union Theorems......Page 207
17.7 Comparison with Cantor's Theory......Page 210
18 Harward 1905 on Jourdain 1904......Page 211
18.1 Proof of CBT......Page 212
18.2 Harward's Unlimited Classes and Other Basic Notions......Page 213
18.4 Constructing the Number-Classes......Page 215
18.5 The Union Theorem......Page 216
19 Poincaré and CBT......Page 221
19.1 The First Proof by Complete Induction......Page 223
19.2 The Second Proof by Complete Induction......Page 225
19.3 The Russell-Like Argument......Page 227
19.4 On Impredicativity and Poincaré's Influence on Russell......Page 229
19.5 Criticism of Zermelo's Proof......Page 230
19.6 Criticism of Cantor's Proof......Page 232
19.7 CBT from the Well-Ordering Theorem......Page 233
20 Peano's Proof of CBT......Page 235
20.1 Peano's Inductive Proof......Page 236
20.2 Addressing Poincaré's Challenge......Page 239
20.3 A Model for Arithmetic......Page 240
21 J. Kőnig's Strings Gestalt......Page 243
21.1 J. Kőnig's Ideology......Page 244
21.2 J. Kőnig's CBT Proof......Page 246
21.3 More Comments on the Proof......Page 247
21.5 Comparison with Earlier Proofs......Page 249
22.1 D. Kőnig's Proof that
