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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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@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. – Arturo Magidin Nov 8 '10 at 20:06
Is it $\aleph_1$ (assuming continuum hypothesis), not $\aleph$? – kennytm Nov 8 '10 at 20:12
@KennyTM: $2^{\aleph_0} = \aleph_1$ is the continuum hypothesis ($2^{\aleph_0} = \mathfrak{c}$ holds regardless). – Arturo Magidin Nov 8 '10 at 20:13
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". – AD. Nov 8 '10 at 20:23
$\aleph$ 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. – Andrés E. Caicedo Nov 8 '10 at 23:54
up vote 4 down vote accepted

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 ...".

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

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