Double weakness (Q2036606)

It is shown that (A) given a strongly inaccessible cardinal \(\kappa\), there is a generic extension in which \(\kappa\) is weakly, but not strongly, inaccessible, the sequence \(\langle 2^\theta : \theta < \kappa \rangle\) is not eventually constant, and the weak diamond fails at \(\kappa\), and (B) given a measurable cardinal \(\kappa\) of Michell order at least \(\kappa^+\), there is a generic extension in which \(\kappa\) is strongly inaccessible, \(\kappa^+ < 2^\kappa\), \(\diamondsuit_\kappa\) fails, but the following weaker, parametrized version of weak diamond holds: for any coloring \(c : {}^{< \kappa} 2 \rightarrow 2\), there is \(g_\eta : \kappa \rightarrow 2\) for \(\eta < \kappa^+\) with the property that for every \(f : \kappa \rightarrow 2\), one can find \(\eta\) such that \(\{ \alpha : c (f \vert \alpha) = g_\eta (\alpha) \}\) is stationary in \(\kappa\).