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By Andrew O Lindstrum

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The set of positive integers is such a set and therefore every set that contains an image of it, as M0 in the single-set formulation, satisfies this requirement. Note that the process of removing members from a set is a numbering process. The method used in this proof of the denumerable CBT is important because it was proof-processed by Cantor, leveraged by transfinite induction, and applied in many instances, including in the proof of the Fundamental Theorem to be given shortly. We call it the enumeration-by method of proof (Abz€ ahlbar durch, Cantor 1932 p 169 footnote 1, 193, Ewald 1996 vol 2 p 885 footnote 1, 905).

At other times we will simply use juxtaposition. 9 It is anachronistic to use the term ‘ordered-pair’ in this context; not the notion itself. Cantor used ‘indexed dummy’ for such purpose in letters to Dedekind of 1873. 10 We note in passing that the familiar method was probably due to an idea of Dedekind and that Cantor’s original method was different (see Cantor’s letter to Dedekind of December 2, 1973; Cavailles 1962 p 188f, Meschkowski-Nilson 1991 p 33f (in handwriting), Ewald 1996 vol 2, p 844f).

In Grundlagen (}2) Cantor introduced well-ordered set as a set that has the following properties: (i) it has a first member, (ii) every element except the last has a sequent, (iii) every succession (not cofinal with the set) has a sequent [in the set]. But he did not link it to the properties of Lemma 2. He did make this link in 1897 Beitr€ age }12. Cantor only proved that Lemma 2(i) entails 2(ii) which is easy. Of 2(i) he only said that it follows from the definitions of the second number-class (II).

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