Abstract
There are two types of comparabilities in Cantor’s writings: comparability of sets, referred to also as the comparability of powers, and comparability of cardinal numbers. The latter applies only to consistent sets and the first to any sets. The distinction between the two types of comparabilities is, however, not clear because Cantor presented the terms ‘power’ and ‘cardinal number’ as synonyms. We will outline now how these notions feature in Cantor’s writings and how they related to CBT.
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Notes
- 1.
Declaring this synonymity was not necessary to the development of Cantor’s theory.
- 2.
In 1887 Mitteilungen Cantor introduced the notation of double over-line on a letter denoting a set, to signify its cardinal number, obtained by the operation of abstraction first introduced in that paper.
- 3.
Thus a cardinal number can be represented by any set of that cardinal number with regard to the order relation between cardinal numbers.
- 4.
It is still needed to warrant the smoothness of classes of the same cardinal or ordinal number even though those numbers are generated by abstraction – see Sect. 3.1.
- 5.
This is explicitly stated in Cantor’s letter to Mittag-Leffler quoted above.
- 6.
Our translation; Ewald’s is “C makes it clear that I was right when I stated [1895] the theorem”. The original is “Wir erkennen ferner aus C die Richtigkeit des [reference to 1895 Beiträge] ausgesprochenen Satzes”.
- 7.
Note that for finite M and any N and for finite N and any M (which is then necessarily finite), Corollary D can be proved directly using complete induction on the power of the set assumed finite. A proof of Corollary D that avoids comparability and uses instead Zorn’s lemma, which has the heuristic advantage of avoiding the notions of infinite numbers or well-ordering, was given in Abian 1963.
- 8.
In the language of powers, (1) of A* is: M and N have equal power; (2) is: M has greater power than N; (3) is: N has greater power than M.
- 9.
Cantor noted this point in passing, with regard to Theorem A, in the attachment for Schoenflies to his letter to Hilbert of June 28, 1899 (Meschkowski-Nilson 1991 p 403).
- 10.
In this notation \( {\text{E}} = \neg \Lambda \& \Delta \to \neg \Gamma \equiv \neg (\neg \Lambda \& \Delta )|\neg \Gamma \equiv (\Lambda |\neg \Delta )|\neg \Gamma \equiv (\neg \Delta |\neg \Gamma )|\Lambda \equiv \neg (\Delta \& \Gamma )|\Lambda \equiv \Delta \& \Gamma \to \Lambda = {\text{B}} \), so that indeed E and B are equivalent.
References
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Hinkis, A. (2013). Comparability in Cantor’s Writings. 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_5
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