With the cardinality of the natural numbers as $|\mathbb{N}| = \aleph_0$ and its powerset as $|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}$, the continuum hypothesis and the axiom of choice says that there is no set such that
$$ \aleph_0 < |S| < 2^{\aleph_0} = \aleph_1 $$
The whole hierarchy of aleph numbers can be constructed from $\aleph_n = \mathcal{P}(\aleph_{n-1})$. Each aleph is constructed from powerset iterations of the natural numbers. Is there any non-finite set $S$ where this is not true?
$$ |\aleph_0| < |S| \notin \{\aleph_0, \aleph_1, \aleph_2, \ldots \} $$