Why did Cantor (and others) use $\mathfrak{c}$ for the continuum?

Question

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

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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?

Answer 1

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.

Answer 2

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.

Answer 3

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 :

• Gregory Moore, Zermelo's Axiom of Choice : Its Origin Development and Influence (1982 - also Dover reprint), page 42.

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)."

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According to :

• Michael Hallett, Cantorian set theory and limitation of size (1986), page 68 :

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.

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Possible sources :

• William Henry Young & Grace Chisholm Young, The Theory of Sets of Points (1906), page 46 :

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.

• Edward V.Huntington, The Continuum, and Other Types of Serial Order: With an Introduction to Cantor's Transfinite Numbers (1917 - also Dover reprint), page 80 :

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]

$$c=2^{\aleph_0}$$

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

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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.