TITLE: What is the first cardinal number which is greater than continuum?
QUESTION [6 upvotes]: What is the first cardinal number which is greater than continuum? We denote it by? Thanks very much.

REPLY [4 votes]: The axioms of ZFC can prove that there exists $\alpha$ such that $\frak c=\aleph_\alpha$, but the axioms themselves are insufficient in order to prove much on that $\alpha$. We can prove that $\alpha\neq 0$ and that $\alpha$ does not have cofinality of $\omega$, whatever that might be.
But besides these two facts we can't say anything intelligible on what exactly is $\alpha$. We know it is possible to have a universe of set theory where $\alpha=1$ and another where $\alpha=2$.
Therefore we cannot really say much on what is $\frak c^+$, or $\aleph_{\alpha+1}$, since we don't know what is $\aleph_\alpha$ in this case. We can, however, prove that it exists and give it a symbol such as:

$\left(2^{\aleph_0}\right)^+$;
$\frak c^+$;
$\left(\beth_1\right)^+$

And so on.