I am recently exploring set-theoretical postulates that contradict GCH. One particularly interesting one is the proposition "$2^\kappa$ is singular for each infinite cardinal $\kappa$". However I could not prove that this proposition is at all consistent with ZFC.
It seems that a slightly weakened form "$2^\kappa$ is singular for each infinite regular cardinal $\kappa$" is indeed consistent with ZFC, since the function $\kappa\mapsto\aleph_{\kappa^+}$ satisfies the conditions of Easton's theorem and there therefore exists a model where $2^\kappa=\aleph_{\kappa^+}$ for each regular $\kappa$. Can this result also be extended to all singular cardinals?
Lemma 3.4 in the following paper by Golshani & Hayut implies that this is consistent relative to the existence of a strong cardinal.
Golshani, Mohammad; Hayut, Yair, On Foreman’s maximality principle, J. Symb. Log. 81, No. 4, 1344-1356 (2016). ZBL1387.03037.