Relation between successor cardinals and power sets

Question

What are the known relation between successor cardinals $\kappa^+$ and power sets $2^\kappa$ (when GCH is not assumed)?

For example, is it true that $\kappa^+ \le 2^\kappa \le \kappa^{++}$?

In general, I am interested in rules and tricks for transforming an inequality involving one into an inequality involving the other. Is there a good (preferably online) reference that covers how the successors and power sets relate to one another?

Answer 1

Since $\kappa<2^\kappa$ it is always true that $\kappa^+\leq2^\kappa$.

Anything beyond that is unprovable completely by $\sf ZFC$. To see why, recall Solovay's theorem that given any $\lambda$ it is consistent that $\lambda\leq 2^{\aleph_0}$. Now simply take $\lambda\geq\kappa^{+++}$ and then we have that $\kappa^{++}<2^{\aleph_0}\leq2^\kappa$.

If you want to omit the axiom of choice, then we can prove there is always a surjection from $2^\kappa$ onto $\kappa^+$, but that's about it. We cannot prove there is an injection in the other direction.

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In general, we know nowadays that the power set operation is a wild card. It cannot be controlled without additional axioms such as $\sf GCH$.

Answer 2

It’s always true that $\kappa^+\le 2^\kappa$. Beyond that, however, you can’t say very much: if $\lambda$ is any cardinal whose cofinality is greater than $\kappa$, it’s consistent with $\mathsf{ZFC}$ that $2^\kappa=\lambda$.