Abstract
Zermelo published two proofs of CBT. The first in 1901, in his first paper on set theory, we review in detail in this chapter. The second in 1908, in the paper where Zermelo first presented his axiomatic set theory, we review in Chap. 23.
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Notes
- 1.
For the proof of this theorem Zermelo proof-processed certain elements from Schoenflies’ 1901 rendering of Cantor’s 1878 Beitrag; see below.
- 2.
Sometimes we use ‘cardinal’ for ‘cardinal number’; Zermelo sometimes uses ‘number’ for ‘cardinal number’. Like Cantor in his 1895 Beiträge, Zermelo is using small Old English letters for cardinal numbers.
- 3.
Zermelo denotes ℵ0 by a.
- 4.
The two columns typographic arrangement leads us to read that the third (forth) expression implies the fifth (sixth).
- 5.
Zermelo remarked (p 38 towards the end of the paper) that Theorem I is an extension of CBT, which cannot be proved from it alone. He may have meant that to prove \( ({\text{M}}, {{\text{P}}_{{1}}},{ }{{\text{P}}_{{2}}}, \ldots )\sim {\text{M}} \) by CBT it is necessary to prove that (M, P1, P2, …) is equivalent to a subset of M. Theorem I requires less, that each \( ({\text{M}},{ }{{\text{P}}_{\lambda }})\sim {\text{M}} \), and so it extends CBT.
- 6.
Like Schoenflies, Zermelo uses Cantor’s notation (M, N) for the union of M, N.
- 7.
The axiom of choice is necessary to provide this sequence of mappings from assumption (1). Moore (1982 p 90) further notes that already for the definition of infinite sum of cardinals the axiom of choice is necessary. He refers probably to the choice of the P λ .
- 8.
Zermelo has here ‘,’ which is clearly a typo.
- 9.
With Zermelo the rightmost mapping is applied first, unlike Bernstein 1905 (see the next chapter).
- 10.
The use of ‘;’ in this notation seems to be a variant of Zermelo.
- 11.
If all the P λ are disjoint from the start, we can skip the construction of the P′ λ and begin the proof at (6). The construction can be simplified (eliminating the need to define the \( {\text{P}}{{^\prime}_{\lambda }} \)), if we apply m = m + p2 to M1 directly, and so on. Note that the P λ need not be different.
- 12.
Zermelo assumes here properties of a and the distributivity of cardinal multiplication over finite sum, besides the commutativity and associativity of cardinal addition.
- 13.
Zermelo uses here distributivity over denumerable sum.
- 14.
The extended form is unnecessarily emphasized; it results by theorem II applied on the conclusion of theorem I.
- 15.
Zermelo references Cantor 1878 Beitrag with regard to the mentioned property of infinite sets, pointing out that Dedekind used it as a definition of infinitude. He does not stress that Dedekind had devised his definition independently of Cantor (see Sect. 8.1).
- 16.
Zermelo was yet unaware that the axiom of choice is necessary for this assertion, which is equivalent to the reflexivity of an infinite set mentioned earlier.
- 17.
- 18.
- 19.
- 20.
Of the English department at Northwest Missouri State University.
- 21.
Literally ‘hight’ means ‘named’. Professor Mayer tends to think there were words Emily Dickinson simply misspelled, so that we should read ‘height’. But then, we can read “a Hight” as making the dying a kind of standard, a measuring rod.
- 22.
According to Professor Mayer, the dedication does not appear in Dickinson’s books and the identity of Alice Dickinson is unknown. Also the last line is considered an editorial addition. Check the web for ‘ringer of change’.
- 23.
Check out Winnicott 1958.
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Hinkis, A. (2013). Zermelo’s 1901 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_13
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