According to the Wikipedia article on the Continuum Hypothesis, the statement $\aleph_1 = 2^{\aleph_0}$ is independent of ZFC.
I'm curious how many distinct cardinals there can be between $\aleph_1$ and $2^{\aleph_0}$, let's call this cardinal $\lambda$.
So, the Continuum Hypothesis holds if and only if $\lambda$ is zero.
$ \lambda = 1 $ if and only if $\aleph_1 = 2^{\aleph_0}$ and, more generally, $\aleph_{(\lambda - 1)} = 2^{\aleph_0}$.
I'm curious what is known about the size of the possible "gap" between $\aleph_0$ and $2^{\aleph_0}$, since most of what I've seen while reading a little about the continuum hypothesis just seems to concern whether a gap exists at all or not.