Endorsers of the method of well-ordering reported by Hausdorff?

Question

In the course of preparing a lecture for graduate students about errors of early set theory I have come across this nice example reported by Hausdorff which violates Peano's successor axiom but nevertheless seems to have been accepted for more than 40 years:

It was usual to argue as follows: From the set $A$ to be well-ordered take by arbitrary choice an element and denote it as $a_0$, then from the set $A - \{a_0\}$ an element $a_1$, then an element from the set $A - \{a_0, a_1\}$ and so on. If the set $\{a_0, a_1, a_2, …\}$ is not yet the complete set $A$, we can choose from $A - \{a_0, a_1, a_2, …\}$ an element $a_{\omega}$, then an element $a_{\omega+1}$, and so on. This procedure must come to an end, because beyond the set $W$ of ordinal numbers which are mapped on elements of $A$, there are greater numbers; these obviously cannot be mapped on elements of $A$.

This is reported as late as in 1914 by Felix Hausdorff, obviously without reservations because he remarks:

“We cannot share most of the doubts which have been raised against this method.”

Hausdorff only deplores the undesired impression of a temporal process but confirms that the element $a_{\omega}$ is fully determined in the sense of transfinite induction and claims that every single action of choosing an element as well as their order has to be understood as timeless. [F. Hausdorff: “Grundzüge der Mengenlehre”, Veit, Leipzig (1914); reprinted: Chelsea Publishing Company, New York (1965) p. 133f]

I am interested to know who of our great colleagues has originally endorsed this method.

Remark: This question had been asked in MathOverflow where it disappeared without an answer, receiving some startling comments though. History of Science and Math seems better suited anyway.

Answer 1

Cantor claimed that it is always possible to transform a well-defined set into the form of a well-ordered set.

In a letter to Mittag-Leffler of Dec. 17, 1882 Cantor explains this well-ordering :

1) There is a first element $E_1$.

2) For every element $E'$, except the last one, if a last one is existing, there exists an element $E''$ following next upon $E'$. (*)

3) For every finite or infinite part $(E')$ of $(E)$ there is an element $E$* which is following next upon all $E'$ - except that an element following next upon all $E'$ is not existing.

Cantor applies this method of stepwise well-ordering also in his proof of theorem A that every transfinite set $T$ has a subset of cardinality $\aleph_0$:

After having removed according to some rule a finite number of elements $t_1, t_2, ...t_{\nu-1}$ from the set $T$ then there is always the possibility to remove another element $t_{\nu}$. The set $\{t_{\nu}\}$, where $\nu$ is an arbitrary finite cardinal number, is a partial set of ${T}$ having cardinal number $\aleph_0$ because $\{t_{\nu}\}$ ~ $\{\nu\}$. [E. Zermelo: "Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 293]

Zermelo comments:

The "proof" of theorem A, which is purely intuitive and logically unsatisfactory is reminiscent of the well-known primitive attempt to obtain a well-ordering of a given set by successively taking away arbitrary elements. [loc. cit. p. 352]

Therefore this "primitive attempt" must have been rather popular. However I could not find any mathematician of that time who in written form has accepted it.

(*) Note the phrase "following next upon all" which Cantor often used, for instance denoting $\omega$ as following next upon all finite ordinals. It is in clear contradiction with the linear character of any well-ordering of natural numbers.