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The usual symbol for the cardinality of the continuum (i.e. the real numbers) is Fraktur $\mathfrak{c}$. However, I recall some sources also using $\aleph$ (with no subscript). This usage is not mentioned in Wikipedia or Mathworld, but I found some support for it over Google.

Is the $\aleph$ notation standard?

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    $\begingroup$ @Yuval: I'm not a set theorist, but I had never seen it before you posted it. In my experience $\aleph$ is usually considered a kind of "ordinal function", where for each ordinal $\alpha$ you get the cardinal $\aleph_{\alpha}$. Jech's book, for instance, never seems to use $\aleph$ without an index. $\endgroup$ – Arturo Magidin Nov 8 '10 at 20:06
  • $\begingroup$ Is it $\aleph_1$ (assuming continuum hypothesis), not $\aleph$? $\endgroup$ – kennytm Nov 8 '10 at 20:12
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    $\begingroup$ @KennyTM: $2^{\aleph_0} = \aleph_1$ is the continuum hypothesis ($2^{\aleph_0} = \mathfrak{c}$ holds regardless). $\endgroup$ – Arturo Magidin Nov 8 '10 at 20:13
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    $\begingroup$ I have seen just $c$ - but I agree to Asaf Karagila, it more clear if you just state it "Let us denote by $c$ the cardinality of the continuum". $\endgroup$ – AD. Nov 8 '10 at 20:23
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    $\begingroup$ $\aleph$ is not standard notation for the continuum within the set-theoretic or set-theoretic topology communities. You can use it if you want, of course (after telling the reader what you mean), but I would suggest to use ${\mathfrak c}$ instead. $\endgroup$ – Andrés E. Caicedo Nov 8 '10 at 23:54
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I have seen the use of $\aleph$ and $\mathfrak{c}$ as well the explicit $2^{\aleph_0}$.

If you're uncertain, it's best just to add "We denote the cardinality of the continuum by ...".

(Edit: I should perhaps clarify, that $\aleph$ is not uncommonly used in Israel in basic set theoretic courses. Some of which are taught by respected set theorists, although not all of them. It is true, however, that in a more advanced capacity this usage disappears.)

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I have never seen $\aleph$ used without a subscript in any treatise of Set Theory.

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  • $\aleph_0$ is the cardinality of the set $\{0,1,2,3,\ldots\}$ of all finite cardinalities. These are linearly ordered in a way that gives each of them only finitely many predecessors.
  • $\aleph_1$ is the cardinality of the set of all countable ordinals.
  • $\frak{c}$ is the same as $2^{\aleph_0}$ and is the cardinality of the set of all real numbers. The notation $a^b$ means the cardinality of the set of all mappings from a set of size $b$ into a set of size $a$. Hence $2^{\aleph_0}.$

These notations were introduced by Georg Cantor in the 19th century. Cantor proved that $\aleph_0<\aleph_1$ and that $\aleph_0 < 2^{\aleph_0}.$ He showed that $\aleph_1 \le 2^{\aleph_0}$ using (what would later be recognized as) the axiom of choice, and he conjectured that those are equal. Much later it was shown that standard axioms of set theory do not give enough information to determine whether they are equal. In set theory without the axiom of choice is is possible that neither is greater than the other but they are nonetheless not equal.

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