TITLE: An inconsistent version of Jensen's principle $\diamondsuit$
QUESTION [10 upvotes]: In page 93 of Kunen's book, exercise 55 says: 

Show that the following version of $\diamondsuit$  is inconsistent: There are $A_\alpha \subset \alpha$ for $\alpha < \omega_1$, such that for all stationary $A\subset \omega_1$, $\exists \alpha \in A (A \cap \alpha=A_\alpha)$.

I know that there is a version of this principle taking closed unbounded sets, but why we can't use stationary sets? Why is this version of Jensen's principle $\diamondsuit$ inconsistent?

REPLY [4 votes]: Funnily enough, such a sequence does actually exist. Just let $A_\alpha = \emptyset$ for every $\alpha$, then for any stationary set, the least element makes the right guess.
So we need to assume some kind of nontriviality, say, $A_\alpha \neq \emptyset$ for stationarily many $\alpha$. 
Now then, by Fodor's Lemma, the least element of $A_\alpha$ is constant for stationarily many $\alpha$. Say, $\min{A_\alpha} = \beta$ when $\alpha \in S$, and $S$ is stationary. 
But $S \setminus \{\beta\}$ is stationary as well, yet cannot be guessed by  $A_\alpha$ when $\alpha \in S$.