Abstract
Cantor’s first paper on general set theory was his 1878 Beitrag. More precisely, general set theory is dealt with in the first two pages of the paper; the rest of the paper, which consists of 13 pages (no division is explicit in the text) deals with the equivalence of the unit segment and n-dimensional space.
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Notes
- 1.
Our translation uses common rather than literal terms (e.g., ‘set’ not ‘manifold’).
- 2.
Hallett translated ‘middle’. Brower later implicitly attacked this definition (see Sect. 38.1).
- 3.
Cantor is not saying here that the continuum is equivalent to the second number-class as Parpart (1976 p 51) implies.
- 4.
P. Tannery (1884) noted that to establish the continuum hypothesis it is required to prove that the continuum does not contain a subset of greater power. Cantor must have been aware to this point.
- 5.
- 6.
Cf. Kuratowski-Mostowski 1968 p 188.
- 7.
It is possible that an earlier draft of 1878 Beitrag was more specific regarding CBT and what we see as hints, are traces of the older version.
- 8.
For the familiar mapping of the line onto the unit segment see Fraenkel 1966 p 49.
- 9.
In Dedekind’s letter of July 2, 1877 (Cavailles 1962 p 215, Ewald 1996 vol 2 p 863), Dedekind notes that Cantor’s mapping in the first proof is continuous!
- 10.
A direct proof that the set of all real functions has power greater than the power of R is easy to obtain. Assume that there is a mapping from R to the set of all real functions and let f x be the function corresponding to x. Let A be the set of numbers such that f x (x) = 0. Let g be the real function that assigns to every member of A the value 1 and to every other number y the value f y (y) + 1. Then there is no x such that g = f x .
- 11.
This proof leads to the proof of the Limitation Theorem for (II) which leads to the Limitation Principle and thus to regard Ω as a legitimate infinite number through which the generation of more number-classes could proceed. But Cantor may have lacked at the time a general proof of the Union Theorem.
- 12.
Cantor did not use the term.
- 13.
The second paper in the series Ueber unendliche, lineare Punkmannichfaltigkeiten, Cantor 1932 p 145.
- 14.
The third paper of the same series, Cantor 1932 p 149.
- 15.
Of the original publication in Mathematische Annalen, vol 17, not in Cantor 1932.
- 16.
The footnote appears apparently only in the booklet format of Grundlagen. It is in Ewald but neither in Cantor 1932 nor in the original Mathematische Annalen.
- 17.
Cantor slipped here into using ‘power’ as an entity, outside its defined context in 1878 Beitrag of ‘equal’ or ‘smaller/greater’ power. Cf. Fraenkel 1966 p 61.
- 18.
We note in passing our belief that Cantor had obtained both his infinity symbols and the set named after him (and thus, perhaps, the first example of non-denumerable set), by proof-processing, in 1869–1870, Dirichlet’s Condition (Ferreirós 1999 p 150) that the of exception points must be nowhere dense.
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Hinkis, A. (2013). CBT in Cantor’s 1878 Beitrag . 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_3
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