Indescribable cardinals without diamonds
Recall the combinatorial principle \(\diamondsuit_{\kappa,E}\): There is a sequence \(\langle S_ \alpha: \alpha\in E\rangle\) such that \(\forall\alpha\in E\), \(S_ \alpha\subseteq\alpha\), and \(\forall S\subseteq\kappa\), \(\{\alpha\in E:S_ \alpha=S\cap\alpha\}\) is stationary in \(\kappa\), where \(E\) is a stationary subset of some regular cardinal \(\kappa\). Large cardinals carry \(\diamondsuit\) sequences, for example R. Jensen has shown that \(ZFC\vdash\forall\kappa(\kappa\hbox{ subtle}\to\diamondsuit_{\kappa,Reg\cap\kappa})\) , where \(Reg\) is the class of all regular cardinals. Assuming the consistency of the existence of a \(\Pi^ m_ n\) \((m\geq 1\), \(n\geq 1)\) indescribable cardinal, we construct a model in which \(\diamondsuit_{\kappa,Reg\cap\kappa}\) fails at some indescribable cardinal \(\kappa\). Moreover, the model satisfies the Generalized Continuum Hypothesis. The construction uses iterated forcing and master condition arguments together with an elementary embedding characterization of \(\Pi^ m_ n\) indescribability.
scientific article; zbMATH DE number 2023
scientific article