So reading some other questions here, I noticed that the continuum hypothesis $2^{\aleph_0} = \aleph_1$ has some strange consequences for analysis. The negation of the Axiom of Symmetry is one, as well as this Fubini's theorem counterexample.
Now, I also know that $2^{\aleph_0}$ can be just about any aleph number, but are there any interesting or counterintuitive consequences of it specifically being $\aleph_2$? What about the more generalized version of this hypothesis, $2^{\aleph_\alpha} = \aleph_{\alpha+2}$? Is that even consistent?
I don't know any interesting consequences of $2^{\aleph_0}=\aleph_2$ or related statements alone. However, there are reasonably natural principles which imply $2^{\aleph_0}=\aleph_2$ and also imply many interesting statements, my favorite being the Proper Forcing Axiom (PFA) and its variants; see this paper by Moore.
Meanwhile, the principle "For all $\alpha$, $2^{\aleph_\alpha}=\aleph_{\alpha+2}$" is consistent (relative to large cardinals, necessarily), but that's far from obvious; see this paper of Magidor and Woodin, as well as this Mathoverflow question.
That said, again I don't know any interesting consequences from this alone. In general, my understanding is that simple arithmetic facts like these don't tend to have lots of interesting consequences on their own. (Obviously when I say "simple" I'm referring to the form of the statement; as the results mentioned above indicate, "$\forall\alpha(2^{\aleph_\alpha}=\aleph_{\alpha+2})$" is extremely complicated once we really dive intothings.)