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.

  • $\begingroup$ There has not been any "reliable reference of a mathematician (other than Cantor or Hausdorff) before 1930 applying the method of well-ordering reported by Hausdorff." So the bounty cannot be awarded. $\endgroup$
    – Franz Kurz
    Mar 25, 2018 at 18:28
  • $\begingroup$ How can I award this bounty? $\endgroup$
    – Franz Kurz
    Apr 2, 2018 at 16:20

1 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.

  • $\begingroup$ However I could not find any mathematician of that time who in written form has accepted it. By "that time", do you mean the 1880s when Cantor's letter was written or around 1932 when Zermelo's excerpt was published? Also, when Zermelo says "well-known", he might be referring to students or to mathematicians not involved in set theory (who might talk about these issues in discussions with others but don't actually publish something with this argument). He might also be thinking of discussions in upper level and graduate level textbooks, which is where I would look for this being said. $\endgroup$ Mar 18, 2018 at 16:04
  • $\begingroup$ See my bounty: before 1930. I am so interested in that matter because it is, according to my judgement, so obviously in contradiction with the successor axiom of finite ordinals that I was taken aback when I read Hausdorff's paragraph for the first time. $\endgroup$
    – Franz Kurz
    Mar 18, 2018 at 16:09
  • $\begingroup$ Zermelo might also simply be talking about an attempt that everyone knew was logically deficient. Indeed, the wording in the quote you gave does not really suggest that people some people actually accepted this --- Zermelo merely states that this way of trying to prove the result was well known, perhaps simply because it is a natural approach to use. $\endgroup$ Mar 18, 2018 at 16:10
  • $\begingroup$ See Hausdorff's statement of 1914 in my original question: It was usual to argue as follows. How else should a set be well-ordered before Zermelo? $\endgroup$
    – Franz Kurz
    Mar 18, 2018 at 16:24
  • $\begingroup$ I looked in several old books I have and the issue just doesn't arise. I suspect that, aside from a very few mathematicians who specifically investigated general set notions, these issues were not something they ever concerned themselves with, and trying to analyze with a logical-microscope what two or three mathematicians said somewhere is probably not going to lead to anything historically informative. Keep in mind also that they didn't have the ability to almost endlessly modify manuscripts like we do --- it was a big deal to have to retype a manuscript when changes affected the paging. $\endgroup$ Mar 18, 2018 at 16:59

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