A regular uncountable cardinal $\kappa$ is said to be ineffable iff for every $\{A_\alpha\}_{\alpha<\kappa}$ such that $A_\alpha\subset \alpha$ there exists a stationary set $S$ verifying that for every $\alpha,\beta\in S$ ($\alpha$<$\beta$) then $A_\alpha=A_\beta\cap \alpha$.
I would like to ask regarding the following charaterization:
- $\kappa$ is ineffable iff $\kappa$ is inaccessible and $\Diamond(\kappa)$ holds whereas $\Diamond^*(\kappa)$ fails.
I have just proved that ineffability implies inaccessibility and $\Diamond(\kappa)$ but I am not able to see why it also implies $\neg\Diamond^*(\kappa)$ and also why the converse implication is true.
If something is willing to give me any idea I would be very grateful!
I can answer the first part of your question: why ineffability of $\kappa$ implies the failure of $\diamondsuit^*(\kappa)$. Suppose $\kappa$ is ineffable and $\{A_\alpha\}_{\alpha<\kappa}$ is a potential $\diamondsuit^*$ sequence. For each $\alpha$, let $B_\alpha$ be a set different from each set in $A_\alpha$ - of course this is possible and $|A_\alpha|=\alpha$. Using ineffability of $\kappa$, let $B\subset\kappa$ be such that $X=\{\alpha<\kappa:B_\alpha=B\cap\alpha\}$ is stationary. Then for any $\alpha\in X$, $B\cap\alpha=B_\alpha\notin A_\alpha$ so $B$ witnesses that $\{A_\alpha\}_{\alpha<\kappa}$ is not a $\diamondsuit^*$ sequence.