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 $\kappa^* = |\{\lambda|\kappa < \lambda < 2^\kappa\}|$

  • $\begingroup$ 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. $\endgroup$ – Yair Hayut Jun 18 at 16:24

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