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?