# Can the cardinality of the set of all intervening cardinals between sets and their power sets be always singular?

This is a question that I've posted to Mathematics Stack Exchange, that was un-answered.

To re-iterate it here:

Is the following known to be consistent relative to some large cardinal assumption?

$$\forall \kappa [\kappa >2 \to \kappa < \kappa^* < 2^\kappa \wedge singular(\kappa^*)]$$

where $$\kappa$$ is a cardinal and $$<"$$ is strict cardinal smaller than, and $$\kappa^* = |\{\lambda|\kappa < \lambda < 2^\kappa\}|$$

• In the paper "On Foreman's Maximality Principle" by Golshani and myself, we construct a model in which $2^\kappa$ is singular. More precisely, $2^\kappa$ is $\aleph_{\kappa^+}$ for Prikry point or successor of Prikry point and $\aleph_{\rho^+}$ where $\rho$ is the next Prikry point otherwise. I think that this method can be adjusted to get $2^\kappa = \aleph_{\aleph_{\kappa^+}}$ at Prikry points and their successors and $2^\kappa = \aleph_{\aleph_{\rho^{+}}}$ for other $\kappa$-s, where $\rho$ is the next Prikry point. – Yair Hayut Jun 18 at 16:24