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

中文翻译：

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双重弱点

我们证明，一直以来，存在一个弱但不是强不可及的基数\（\ lambda \），对于该基数\（\ langle 2 ^ \ theta：\ theta <\ lambda \ rangle \）最终不是恒定的，并且弱菱形在\（\ lambda \）处失败。我们还证明了钻石始终会失败，但是参数化版本的弱钻石在某些不可访问的\（\ lambda \）处成立。