Double weakness

英文摘要：

We prove that, consistently, there exists a weakly but not strongly inaccessible cardinal $λ$ for which the sequence $\langle 2^θ:θ<λ\rangle$ is not eventually constant and the weak diamond fails at $λ$. We also prove that consistently diamond fails but a parametrized version of weak diamond holds at some strongly inaccessible $λ$.