Abstract
We review two proofs of BDT (2m = 2n → m = n; see Chap. 14) given by Sierpiński: the first from 1922 without use of the axiom of choice and the second from 1947 in the context of Lebesgue measure problem. We suggest that the proofs can be extended to prove the generalized BDT (ν m = ν n → m = n for any finite ν). We then review the proof of the inequality-BDT for ν = 2 (2m ≤ 2n → m ≤ n) from another paper of Sierpiński from 1947.
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Notes
- 1.
Sierpiński, like D. Kőnig (1916), refers to BDT as Bernstein’s theorem.
- 2.
D. Kőnig agreed with this view, see Sect. 22.2.
- 3.
The difference between S and T is not analogous to the difference between the single-set and the two-set formulations of CBT because we cannot glide from S to T, or vice-versa, as we can glide from the two-set formulation of CBT to its single-set alternative. But the fact that in both theorems there are two arrangements to the given sets is perhaps worth noting as a gestalt of the situation given by the conditions of the proof and the question arises whether similar alternative arrangements show up in other proofs that lead to apply the metaphors of CBT or BDT.
- 4.
Sierpiński introduces the convention, for any two functions ψ 1, ψ 2, that ψ 1(ψ 2(a)) is written as ψ 1 ψ 2(a) and ψ 1 ψ 1(a) as ψ 21 (a).
- 5.
Unlike Bernstein, Sierpiński does not point this out though he proves the 1–1 character of the χ’s by using their inverses.
- 6.
Compare Banach 1924 (see Sect. 29.1) notions (S) and C(a).
- 7.
We have somewhat shortened Sierpiński’s argument here.
- 8.
Already in Sierpiński’s 1914 paper with Mazurkiewitcz, such strings appear. In that paper a result akin to Hausdorff’s paradox was presented. The use of alternating mappings in Mazurkiewitcz-Sierpiński 1914 seems to be the reverse of its use in CBT: instead of the shrinkage evident in CBT proofs (e.g., by Schröder or Schoenflies) an expansion occurs. One begins with one point and generates more points through the application of interlaced mappings (with the set of these points having naturally two subsets congruent to it). This paper seems to us to provide an example of proof-processing and it may hide a whole research project.
- 9.
This results from Sierpiński’s choice of T and would not result for Bernstein’s S.
- 10.
Apparently, Sierpiński regards even strings composed of finite cycles to be of order-type ω* + ω.
- 11.
Note that our discussion here is about the metaphor of the proof.
- 12.
Strangely, Sierpiński seems to be working with a particular S(g) for each K but his construction by-passes this appearance. Also Sierpiński does not define K n for every n but only for such n that K n is not empty. The procedure works without change even when K is composed of only a finite number of different elements.
- 13.
Sierpiński uses the notation K ω following perhaps Schoenflies (see Sect. 12.1).
- 14.
D. Kőnig (Kőnig 1926 p 133 footnote 1) conceded that Sierpiński’s proof does not make use of AC. He added that this achievement seemed to him impossible when the 1923 paper was written (1914). He seems to be using hindsight here because it seems that in 1914 he was not aware of his use of AC. Unfortunately, D. Kőnig maintained there (the text to the referenced footnote) that only for ν = 2 Bernstein’s proof avoided AC, while also Bernstein’s general proof made no use of AC.
- 15.
It seems improbable that Cantor would have accepted an ambiguous definition of a set with his emphasis on ‘well-defined’ (see Sect. 3.1). Anyway, accepting ambiguous sets as effective seems to be a matter of convention.
- 16.
Tarski (1949a p 84) said of a similar family of mappings that it can be so represented because it is denumerable. A specific enumeration can be defined by complete induction from the enumeration of the set of all finite sequences of natural numbers as given in Troelstra or from Cantor’s enumeration of the rationals.
- 17.
Sierpiński notes (p 41) that he was not able to prove the equivalence of BDT and the Factoring Theorem without the axiom of choice. As the Factoring Theorem clearly implies BDT, it must be the opposite implication that requires AC.
- 18.
The bracketed statement was added by Sierpiński; see shortly.
- 19.
So it makes sense that proving the Factoring Theorem from BDT would require AC.
- 20.
The four cases appear in J. Kőnig’s proof of CBT (see Sect. 21.2).
References
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Hinkis, A. (2013). Sierpiński’s Proofs of BDT. 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_28
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