Cardinal arithmetic does not seem to open its way to the existence of $\aleph_1$ that is not $2^{\aleph_0}$, as any operation on $\aleph_0$ would lead to $\aleph_0$ or $2^{\aleph_0}$ and $2^{2^{\aleph_0}}$ or so forth.
According to my knowledge, choice does not determine whether continuum hypothesis is right or wrong, so what is going on with this?