I just assumed that he did since we owe the proof that $\mathbb{R}$ is uncountable to him, but a search of my favorite source for answering such questions -- https://mathshistory.st-andrews.ac.uk/Miller/mathword/ -- doesn't even mention uncountable.

  • $\begingroup$ I suspect, without documentable back-up, that this notion had indeed not settled out of the whole set-theory tumult until probably Fraenkel's time. Our hindsight on these ideas is very misleading, as far as I can tell from my (amateurish, barely German-math-literate) reading of those old sources. $\endgroup$ Apr 12, 2022 at 2:06
  • $\begingroup$ @njuffa - wow! Thank you seems very lame but it's all I got. I'll just get out of the way. $\endgroup$
    – TonyK
    Apr 12, 2022 at 2:59

1 Answer 1


Georg Cantor published a 47-page booklet, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre," Leipzig: Teubner 1883. The title translates to "Foundations of a general set theory" and various English translations show that he refers to countable and uncountable sets in it. I am unable to find a scan of the German original online. Mannigfaltigkeit (literally translated: manifold) was Cantor's term for a set, the modern German term for which is Menge.

The material that comprises this monograph appears to also have been published (in somewhat different form) in various individual publications, one of which is

G. Cantor, "Ueber unendliche, lineare Punktmannichfaltigkeiten", Mathematische Annalen, Vol. 21, 1883, pp. 52-58 (scan online). As far as I understand, Cantor's concept of a point set is a subset of the real line. Note that the title uses a spelling variant of Mannigfaltigkeit suggested by Northern German pronunciation. On page 54 of this publication we find:

Da nämlich alle Bestandtheile der rechten Seite abzählbar sind und die Anzahl dieser Bestandtheile eine abzählbar unendliche ist, so folgt daraus die Abzählbarkeit von $P'$ und nach Th. II diejenige von $P$. [...]

Theorem V. Jede Punktmenge $P$ zweiter Gattung, für welche $P^{(\alpha)}$ abzählbar, ist selbst abzählbar. [...]

Ist $P$ eine nicht abzählbare Punktmenge, so ist auch $P^{(\alpha)}$ nicht abzählbar, sowohl wenn $\alpha$ eine endliche ganze Zahl, wie auch wenn es eines der Unendlichkeitssymbole ist.

My translation: Since all components of the right-hand side are countable and the number of these components is countably infinite, it follows the countability of $P'$ and according to theorem II that of $P$. [...] Theorem V. Every point set $P$ of the second class, for which $P^{(\alpha)}$ is countable, is itself countable. [...] If $P$ is an uncountable point set, then $P^{(\alpha)}$ is also uncountable, both when $\alpha$ is a finite integer and when it is one of the infinity symbols.

  • $\begingroup$ a literal translation of "nicht abzählbar" would be "not countable", right? $\endgroup$ Apr 12, 2022 at 6:06
  • 1
    $\begingroup$ @MichaelBächtold Correct. In my experience, use of überabzählbar ("more than can be counted") predominates in modern German usage in this context. $\endgroup$
    – njuffa
    Apr 12, 2022 at 6:13
  • $\begingroup$ @njuffa. This is great. $\endgroup$
    – TonyK
    Apr 12, 2022 at 17:39

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