There are some interesting set-theory axioms under which the cardinality of the continuum, $2^{\aleph_0}$, is equal to $\aleph_2$. One example is Woodin's Strong $\Omega$ Conjecture, though I have heard there are others.
Under such an axiom, there is a single cardinal intermediate between $\aleph_0$ and $2^{\aleph_0}$, namely $\aleph_1$.
Under any such axiom, what properties of $\aleph_1$ are known? Are there known sets that it is the cardinality of?
Ideally, I'm looking for properties of $\aleph_1$ that don't require an extensive background in set theory to understand.
Edit: I'm particularly interested in simple-to-state theorems about $\aleph_1$ that hold in such a system, but are not known to hold in ZFC alone, or better yet are known to be false or unprovable in ZFC.
The order dimension of the Turing degrees is such a cardinal. See K. Higuchi, S. Lempp, D. Raghavan and F. Stephan, On the order dimension of locally countable partial orderings, Proc. Amer. Math. Soc., Vol. 148 (2020), no. 7, 2823--2833.