The Scheme of Complete Disjunction
In the late 1890s a new gestalt for the presentation of the comparability of sets emerged. Following Fraenkel (1966 p 72f) we call it the scheme of complete disjunction. It was published first by Borel in his 1898 book, in the appendix where he brought Be
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The Scheme of Complete Disjunction
In the late 1890s a new gestalt for the presentation of the comparability of sets emerged. Following Fraenkel (1966 p 72f) we call it the scheme of complete disjunction.1 It was published first by Borel in his 1898 book, in the appendix where he brought Bernstein’s proof of CBT. It then appeared in two letters of Cantor from 1899, to Schoenflies and to Dedekind, and in Schoenflies’ report of 1900. The scheme is noteworthy because it brought logical analysis, of the propositional calculus kind we used in Sect. 5.4, to what seems to be a pure set theoretic context. We will argue that it was Cantor who developed the scheme, following an analysis of Schr€ oder (1896).
6.1
The Scheme and Schoenflies
In the attachment to Schoenflies, of June 28, 1899, (Meschkowski-Nilson 1991 p 401) Cantor presented the scheme as follows: Let M and N be any two sets, then one of the following four cases always occurs: I. No subset M1 of M is ~ N, however, there is a subset N1 of N (or even more such) such that M N1 . II. No subset N1 of N is ~ M, however, there is a subset M1 of M (or even more such) such that M N1 . III. There exists both a subset M1 of M such that N M1, as well as a subset N1 of N, such that M N1 . IV. There exists neither a subset M1 of M, such that N M1, nor a subset N1 of N such that M N1 .
Cantor then linked the four cases of the scheme to the three cases of the Comparability Theorem for cardinal numbers: cases (I), (II) correspond to the
1 Fraenkel may have picked the name from Schr€ oder’s (1898 p 346) “complete development” (vollst€ andigen “Entwickelung”).
A. Hinkis, Proofs of the Cantor-Bernstein Theorem, Science Networks. Historical Studies 45, DOI 10.1007/978-3-0348-0224-6_6, # Springer Basel 2013
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6 The Scheme of Complete Disjunction
inequality cases, and cases (III), (IV) both to the equality case. Cantor mentioned the proofs of Schr€ oder and Bernstein, stressing that Bernstein’s proof is independent of logical calculus, and he brought Bernstein’s proof (see Sect. 12.1). Following that proof Cantor added that so far a direct proof of case (IV) was not found and that he reckons that such proof does not exist. He then briefly outlined the proof of the Comparability Theorem, which leads indirectly to the conclusion that case (IV) is not a separate case. On the importance of the scheme in 1899 we have the following testimony of Schoenflies, in his reminiscences of Cantor from 1922 (Schoenflies 1922 pp 101–102): I am reminded especially of a question regarding the general construction [of set theory]; the comparability of sets, i.e., the possible relationships between the powers of two sets M and N.2 He [Cantor] based it on the four logically possible cases by which the subsets of M can stand to N and subsets of N can stand to M. He chose them so that first they exclude each other logically and second that one of them is necessarily realized. Today this method may seem to us self-evident or completely trivial; at that time it was in no way so; it
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