Abstract
In 1924, 42 years after Cantor had first mentioned CBT, and after over 15 published proofs, Banach offered a new proof, with new gestalt and new metaphors.
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- 1.
Banach refers to CBT as the Schröder-Bernstein theorem or the Equivalence Theorem.
- 2.
Namely, ψ is also 1–1.
- 3.
Namely, a partitioning.
- 4.
Banach uses X for ∩ and 0 is tacitly used as his sign for the empty set.
- 5.
Mańka and Wojciechowska (1984 p 196) noticed that Banach’s original proof is based on an idea from the proof of J. Kőnig to CBT. This is an example of an identification of proof-processing relation between proofs that appears often in the literature.
- 6.
The C(a) are the parts of the strings defined in J. Kőnig 1906 which belong to A, for J. Kőnig’s strings included alternatively elements of A and B (using alternatively at each step either ϕ γ or ψ γ, where γ is either 1 or −1). Banach defines the C(a) not by J. Kőnig’s inductive procedure but by the impredicative metaphor applied in Zermelo’s 1908b CBT proof, which leveraged on Dedekind’s definition of the chain of a set. D. Kőnig (1926 p 130f, see Sect. 30.3) pointed out the indebtedness of Banach’s result (D. Kőnig mentions Banach’s theorem 2 but it seems that he refers to theorem 1) to J. Kőnig’s 1906 paper. D. Kőnig’s claim that Banach’s result is implicit in J. Kőnig’s paper is, however, not exact: not only is the use of the impredicative definition of C(a) missing from J. Kőnig, but the focus of Banach is different: he points out the correlated partitionings and this result leads him to structuralistic conclusions – a train of thoughts alien to J. Kőnig.
- 7.
It seems to us that such conviction can be gathered only using complete induction. If one wants to avoid this procedure then the presentation of C(a) as (S) can only be used as heuristic.
- 8.
Some of the C(a) may be finite and their (S) cyclic, so apparently, Banach, like Sierpiński, allows the ω* + ω sequences to contain repetitions.
- 9.
Proof of (2) without complete induction could be as follows: if C(a 1) and C(a 2) are not disjoint, let d be an element of both. Then C(d)⊆C(a 1) because C(d) is an intersection of all subsets of A that fulfill i–iii. But a 1 and d are connected by i–iii and this connection is symmetric, so a 1 must be in C(d) and therefore C(a 1)⊆C(d) so C(d) = C(a 1). Likewise C(a 2) = C(d), hence (2). It seems, however, that Banach was not interested in having his proof avoid the notion of number.
- 10.
Banach uses ⊂ for not necessarily proper subsets. Note that for all C(a), C(a)-{a} is a subset of ψ-1(B). Banach prefers to define A2 first and leave for A1 the C(a) for which the (S) sequence begins with a member of A-ψ -1(B). The partitioning here defined is not the only partitioning possible: all or part of the (S) which are finite or of type ω* + ω could be moved from A2 to A1.
- 11.
There is a typo in the original where in place of our ω* + ω, ω + ω* is printed.
- 12.
Banach’s proof could profit heuristically if the structure induced in B by the structure in A (the C(a)’s) would be explicitly described.
- 13.
Namely, A1 is composed of the frames A-ψ -1(B), ψ -1(ϕ(A)-ϕψ -1(B)), etc. Banach presents here A1 as a chain composed of frames (see Sect. 9.2).
- 14.
Apparently, the meaning here is that ϕ is onto.
- 15.
There seems to be a typo in the original: the sets A′ and B′ are denoted by A1 and B1, but clearly from theorem 1 the sets given in the conditions of the theorem (ϕ(A) and ψ -1(B)) are not the decomposition sets. When theorem 2 is then invoked for the proof of theorem 3, A1 and B1 of theorem 3 correspond to A′ and B′ of theorem 2.
- 16.
A≅B signifies congruence, namely, that there is a 1–1 mapping between A and B that preserves the distance between any two points of A. The notion of equivalence by finite decomposition was no doubt borrowed from the Euclidean geometry notion of scissors congruence.
References
Banach S, Tarski A. Sur la décomposition des ensembles de points en parties respectivement congruentes. Fund Math. 1924;6:244–77.
Kőnig D. Sur les correspondence multivoque des ensemble. Fund Math. 1926a;8:114–34.
Kőnig J. Sur le théorie des ensemble. Comptes Rendus Hebdomedaire des Séances de l’Academie des Science, Paris. 1906;143:110–2.
Mańka R, Wojciechowska A. On two Cantorian theorems. Annals of the Polish Mathematical Society, Series II: Mathematical News. 1984;25:191–8.
Zermelo E. Untersuchungen über die Grundlagen der Mengenlehre. I. Mathematische Annalen 1908b;65:261–81. English translation: van Heijenoort 1967. pp. 199–215.
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Hinkis, A. (2013). Banach’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_29
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