Abstract
Following his statement of CBT, in its single-set formulation, for sets of the power of (II), as a corollary to the Fundamental Theorem, Cantor said (Cantor 1932 p 201, Ewald 1996 vol 2 p 912 [12]):
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- 1.
Hallett (1984 §2.2) attempted to obtain such generalization but concluded (p 74) that it was not possible.
- 2.
Alone to state a view similar to ours, but with actually no details to support his thesis, is Tait (2005 p 164).
- 3.
The convention to denote by ω γ the initial number of the (γ + 2)th number-class (Jourdain 1904b p 295 footnote †) will not do for number-classes from the (ω + 1)th on.
- 4.
The confused notation originated with Cantor in the following quotation.
- 5.
Cantor did not use the terms ‘regular number’ and ‘singular number’, introduced by Hausdorff (1914a p 130).
- 6.
Cantor is using here the term ‘transfinite number’ which gained popularity after Jourdain used “transfinite numbers” in the title of his translation (Cantor 1915) of Cantor 1895/7, instead of Cantor’s Transfiniten Mengenlehre. But Cantor preferred in his 1897 the term ‘ordinal number’ and in Grundlagen he preferred unendlichen realen Zahlen, which Ewald translated into ‘infinite integers’ (Ewald 1996 vol 2 pp 883 [7], 908 [7]). We prefer to use ‘infinite numbers’ to stress that on the one hand we are attached to the Grundlagen presentation and on the other hand to the ‘number’ attribute of 1897.
- 7.
- 8.
Hessenberg says that his proof follows the proof of Bernstein (namely – Borel, see Sect. 11.2) with some changes. Actually his proof is similar to Peano’s first (inductive) proof (see Sect. 20.1), published in March 1906. Another case of simultaneity of proofs.
- 9.
U1 is the empty set. We assume by convention that when we use U k , we have κ > 1.
- 10.
Compare to Zermelo’s three principles of complete induction in theorems I, III, V, of his 1909 paper (p 192).
- 11.
That the part cannot be greater than the whole is provided by CBT.
- 12.
The definition of the sum of numbers from U γ will be discussed below.
- 13.
Here and in the rest of this paragraph, the + sign does not signify the sum operation but just the stage in which the number is generated by the first generation principle.
- 14.
Strangely, Zermelo did not refer in his comment also to the proof in Grundlagen.
- 15.
Cantor met Dedekind twice during September 1882; first in Harzburg, their favorite vacation retreat, and then in Eisenach, in a gathering of mathematicians.
- 16.
Manifold (Mannigfaltig) was Cantor’s term for set (Menge) before Grundlagen.
- 17.
The crucial step was probably a scheme for the proof of the Union Theorem.
- 18.
It does not make much sense to assume that the entire construction of Grundlagen was developed in that month, as some writers have suggested, because of the wealth of the ideas and technical details in Grundlagen. Cf. Ferreiro’s 1995 p 41, Meschkowski-Nilson 1991 p 90 (3).
- 19.
Reference is here made to Cantor’s 1880 paper, 1882 paper and 1883 paper, parts 2, 3, 4, in the series Ueber unendliche, lineare Punkmannichfaltigkeiten, Cantor 1932 pp 145, 149, 157, respectively.
- 20.
In 1883 Grundlagen Cantor denoted the result of the following definition by βα, but he reversed the notation in 1895 Beiträge. We use the latter convention that prevailed.
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Hinkis, A. (2013). Generalizing Cantor’s CBT Proof. In: Proofs of the Cantor-Bernstein Theorem. Science Networks. Historical Studies, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0224-6_2
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