Kontinuum is German for continuum, but Cantor used $\mathfrak{c}$.

Revision. J.W.Perry questions whether or not Cantor ever in fact used the symbol $\mathfrak{c}$. I must admit I just assumed that he did from statements that say that Cantor proved the cardinality of $\mathfrak{c}$ is larger than that of $\aleph_0$ (and he certainly introduced the $\aleph$ notation).

So the proper question is now:

Who introduced the symbol $\mathfrak{c}$ for the continuum, and when?


3 Answers 3


Cantor did not invent the term, it goes back to antiquity. "Latin was a lingua franca, the learned language for scientific and political affairs, for more than a thousand years, being eventually replaced by French in the 18th century and English in the late 19th". In Latin, French and English continuum starts with "c".

Cantor saw himself as confronting the dogma of medieval scholasts "infinitum actu non datur" (actual infinity is not given), which originates in Aristotle, and the arguments supporting it, such as "annihilation of numbers" by infinity. From Dauben's book:

"Cantor condemned this kind of argument, however, on the grounds that it was fallacious to assume that infinite numbers must exhibit the same arithmetic characteristics as did finite numbers... Having dealt with Aristotle and the scholastics, Cantor undertook an investigation of other works by some of the most impressive thinkers of the seventeenth century, a century that witnessed serious and often profound analysis of the nature of infinity. He suggested that anyone interested in such things would do well to consult Locke, Descartes, Spinoza, and Leibniz, while Hobbes and Berkeley were highly recommended as additional reading."

Not many of Cantor's contemporaries were interested in the subtleties of actually infinite (one exception is Dedekind), so most of Cantor's intellectual companions wrote, or were translated into, Latin.

EDIT: After J.W. Perry's comment I looked through Medvedev's book Early History of the Axiom of Choice, where he quotes Cantor's set theoretic papers and letters to Dedekind from 1872 to 1899, and also did not find any instance of him using $\mathfrak{c}$. The earliest usage Medvedev quotes is from Bernstein's paper Über die Reihe der transfiniten Ordnungszahlen in Mathematische Annalen, v.60 (1905), 187-193, where he writes (my translation, direct link to the paper, see p.192):

"Although it remains very likely that $2^{\aleph_0}=\mathfrak{c}=\aleph_1$, so far nobody managed to prove that $2^{\aleph_0}>2^{\aleph_1}$ [sic!]. Therefore, it is not ruled out that $2^{\aleph_0}=2^{\aleph}$, where $\aleph$ is any aleph. In that case $2^{\aleph_0}$ would contain all alephs as subsets..."

In his 1901 dissertation Untersuchungen aus der Mengenlehre, later published in Mathematische Annalen, v.61 (1905), 117-155, but circulated among experts earlier, Bernstein uses Latin transcription for "c" instead, writing "Bezeichnet $c$ de Mächtigkeit des Kontinuums..." (Let $c$ refer to the cardinality of the continuum), despite "Kontinuum" with a "K". Direct link, see p. 133.

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    $\begingroup$ I am pretty sure Cantor did not ever even use this symbol after poring through every original German source I could find where he might (or would have reason to) have used it. I'd love to be struck wrong here, but I am pretty sure I am right. I would need an original source to call me wrong here, and I just read through a boatload of them; no $\mathfrak{c}$ anywhere to be found. $\endgroup$ Commented Jun 8, 2015 at 4:58
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    $\begingroup$ Conifold, you did not even explicitly state that he used this symbol. You owe no apologies. On the contrary, thanks is due to you for coming off of some of the most incredibly thorough and well researched posts on this site, including this one. That is precisely why I put the comment here. I had no post of my own, just a few hours of speed reading in all the right places to arrive at this conclusion. It made the most sense to post the finding here as comment at the time. I am pleased to see it helped motivate further honing of the entire Q&A. $\endgroup$ Commented Jun 9, 2015 at 3:46
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    $\begingroup$ @ J. W. Perry Thank you, this really means a lot. I was so focused on "why" that I ended up "explaining" something that did not even happen. But if Cantor did use $\mathfrak{c}$ that would be why. :-) $\endgroup$
    – Conifold
    Commented Jun 9, 2015 at 17:44
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    $\begingroup$ Incidentally, the statement $2^{\aleph_0}=2^{\aleph_1}$ is known as Lusin's hypothesis. (For 3 more google search possibilities, substitute "Luzin" for "Lusin" and/or substitute "axiom" for "hypothesis".) $\endgroup$ Commented Sep 14, 2015 at 18:04
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    $\begingroup$ @J.W. Perry: I can confirm your result. In fact it seems, to my surprise, that Cantor never used C for the cardinality of the continuum. He used $\mathfrak{o}$ or $\aleph$ or $\aleph_1$ in his papers and letters. The first and only use of c for the cardinality of the continuum in Cantors collected works is a footnote added by Zermelo 1932. $\endgroup$
    – Franz Kurz
    Commented Jun 22, 2017 at 18:00

Latin loan words in German usually retained their "c" until about the end of the 19th century. The modern spellings like Kontinuum (for Continuum) and Zentrum (for Centrum) result from a fairly recent spelling reform.


Long comment

I agree with Perry's comment.

In Cantor's Beiträge zur Begründung der transfiniten Mengenlehre (1) Mathematische Annalen, 46:481–512 (1895), page 488 (and see the English (1915) translation, page 96 ) the symbol used for $2^{\aleph_0}$ is clearly not $\mathfrak c$.

In previous pages $\mathfrak a, \mathfrak b$ and $\mathfrak c$ are used for cardinal numbers.

See also :

Some early sources :

  • Felix Hausdorff, Set theory (English transl of the 3rd German ed : 1937), page 44, uses $\aleph$ [without subscript] for "the cardinality of the continuum".

  • Wacław Sierpiński, Cardinal and ordinal numbers (English transl of the 3rd Polish ed : 1928), page 372 : "The power of the set $Z_2$ of all numbers of the 2-nd class is denoted by $\aleph_1$ (read aleph-one)."

According to :

the aleph notation was first introduced to the mathematical public in Cantor's Beiträge I [1895]. $\aleph_0$ is characterized as 'the power of the natural numbers', i.e. of the [first number] class (I). [...] Thus, in Beiträge II [1897], $\aleph_1$ is introduced as the power of [the second number class] (II). The aleph notation appears to have been referred to first in a letter to Vivanti of 13 December 1893, though here Cantor has $\aleph_1, \aleph_2$, etc. in place of the later $\aleph_0, \aleph_1$ etc.

Possible sources :

We shall now see that, in order to measure a perfect set we only have to take the linear continuum itself as unit; in this way we obtain a new potency, which is more than countable, viz. the potency $c$ of the linear continuum.

Perhaps the most famous result in this algebra [tha algebra of the cardinal numbers] is the formula [ref in footnote to Beiträge I, page 488]


where $c$ stands for the cardinal number of the continuum [...].

Finally, see : Patrick Suppes, Axiomatic set theory (1960 - also Dover reprint), page 193 :

The symbol $\mathfrak c$ is the standard one [sic !] for the cardinality of the continuum.


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