Abstract
In 1887, when he prepared his Zahlen for publication, Dedekind wrote down a proof of CBT. The proof, dated July 11, 1887, was found in Dedekind’s Nachlass and was first published in 1932, in Dedekind’s collected works (Dedekind 1930–32 p 447). The proof is for both the single-set and the two-set formulations. In his letter to Dedekind of November 5, 1882, Cantor only mentioned the theorem in its single-set formulation, from which the derivation of the two-set formulation is not natural. Thus the question arises how did Dedekind learn of the two-set formulation. It is possible that Cantor mentioned both formulations to Dedekind in their discussion of the theorem in September 1882, recollected in the above referenced letter to Dedekind.
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Notes
- 1.
A similar suggestion is made in Mańka-Wojciechowska 1984 p 195.
- 2.
Kekulé used ‘kette’ in his structure of benzene (1865).
- 3.
Dedekind writes 10 though he is aware of the distinction between a thing and the set that contains only it (see Sect. 8.4).
- 4.
- 5.
In our translation, from Dedekind 1930–2 vol 3 p 447–8. We use “set” for Dedekind’s “system”, “mapping” for “Abbildung”, “equivalence” or “1–1 mapping” for “ähnlich abbildung”; the translation in Zahlen is “transformation” and “similar transformation” or “similarity”, respectively. We use + for union instead of Dedekind’s W.
- 6.
ϕ is not explicitly introduced.
- 7.
This is the single-set formulation.
- 8.
Actually differentiating this case is not necessary because the proof that follows holds even if U is empty, in which case U0 is also empty, but in Zahlen (p 45) Dedekind avoided the empty set.
- 9.
Namely, U = S-T only Dedekind does not use the difference between sets operation.
- 10.
It seems that Dedekind made a mistake here in his use of his own terminology: U0 is the “chain of U” (Zahlen p 58 #4) while the image-chain refers to the chain generated by the image of U, i.e., U′0 (Zahlen p 59 #57 where “transform” is used for our “image”, Ferreirós 1993 p 355, Hessenberg 1906 p 689).
- 11.
Reference is to Zahlen.
- 12.
Namely, ψ(s) = s.
- 13.
This is a definition of a new mapping ψ.
- 14.
This is an editing remark Dedekind made for himself. He meant that the argumentation regarding the sets and their images should be made prior to the definition of ψ. Indeed Dedekind implemented this remark as we will note below.
- 15.
There is a typo here in the original and V0 is written instead of U0.
- 16.
This is the two-set formulation.
- 17.
In its single-set formulation which Ferreirós describes as a “crucial lemma in the proof of the Cantor-Bernstein theorem” rather than an alternative formulation of it.
- 18.
Here in Ewald there is a reference to §8 which seems to be a typo; it is not in Cantor 1932 p 449.
- 19.
We quote from Dedekind 1963. The translation does have some Germanized English.
- 20.
- 21.
- 22.
- 23.
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Hinkis, A. (2013). Dedekind’s Proof of CBT. 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_9
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