I'm wondering about the following (I've written a couple set theory papers but don't consider myself a set theorist, so please keep this in mind when answering):
Is it consistent w/ ZFC that there is a cardinal $\kappa$ such that $\aleph_0<\kappa<2^{\aleph_0}$, yet for every cardinal $\beta\geq2^{\aleph_0}$, there are no cardinals properly between $\beta$ and $2^\beta$? Any references would be most appreciated.
Well, by Easton's theorem, we immediately get the following:
It is consistent that $2^{\aleph_0}=\aleph_{17}$ (say) and $2^\kappa=\kappa^+$ for every regular cardinal $\kappa\ge\aleph_{17}$.
In principle, singular cardinals could pose a problem, and indeed this comes up when trying to make the continuum function $\kappa\mapsto 2^\kappa$ do more complicated things. However, it's easy to see that they don't: it's not hard to check that adding $\aleph_{17}$-many Cohen reals to a model of GCH does the job.
In fact, this is actually much easier to prove than Easton's theorem; I mostly mentioned Easton above for context (and because it's cool).
Here are the details:
Let our ground model $V$ satisfy GCH, and let $\mathbb{P}$ be the forcing adding $\aleph_{17}$-many Cohen reals. The trick is nice names - if $x$ is a set in $V$, a nice name for a subset of $x$ is a family $(A_i)_{i\in x}$ of $x$-many maximal antichains in $\mathbb{P}$, together with maps $e_i: A_i\rightarrow 2$ for each $i\in x$.
It's not hard to see that every subset $\alpha$ of $x$ in the generic extension $V[G]$ has a corresponding nice name $\nu_\alpha=((A_i)_{i\in x}, (e_i)_{i\in x})$ - that is, such that $$i\in\alpha\iff \exists a\in A_i\cap G[e_i(a)=1].$$ So counting the subsets of $x$ in $V[G]$ amounts to counting the nice names. Since $\mathbb{P}$ has size $\aleph_{17}$, there are at most $2^{\aleph_{17}}=\aleph_{18}$-many maximal antichains; for any $\kappa\ge\aleph_{17}$, there are $\aleph_{18}^\kappa=2^{(\aleph_{17}})^\kappa=2^{\aleph_{17}\cdot \kappa}=2^\kappa=\kappa^+$-many $\kappa$-sequences of antichains; and similarly, only $\kappa^+$-many sequences of appropriate maps.
So we get, for $\kappa\ge\aleph_{17}$, that there are at most $\kappa^+$-many nice names for subsets of $\kappa$, and clearly there are at least $\kappa^+$-many nice names for subsets of $\kappa$; so $(2^\kappa)^{V[G]}=\kappa^+$.
(Note that we also need a nice name argument to say that $2^{\aleph_0}=\aleph_{17}$ in $V[G]$; all the forcing gives us trivially is that the continuum is at least $\aleph_{17}$, and weird things can happen (if you add $\aleph_\omega$-many Cohens to a model of GCH, you get $\aleph_{\omega+1}$-many reals). However, the argument is basically the same.)
