Diamonds in mice

Abstract

Let $M$ be a $(0,{\omega }_1+1)$-iterable mouse with no largest cardinal. Let $\gamma \le \kappa$ be uncountable cardinals of $M$, with $\kappa$ regular in $M$. Then $M$ satisfies ${♢}_{\kappa \gamma }^{+}$, and if $\kappa$ is non-ineffable in $M$ then $M$ satisfies ${♢}_{\kappa \kappa }^{+}$. Suppose that either ${\omega }_1^M={\omega }_1$ or $M$ satisfies "I am $(0,{\omega }_1+1)$-iterable". Let $\kappa \ge {\omega }_1^M$ be a cardinal of $M$. Then $M$ satisfies ${♢}_{\kappa {\omega }_1^M}^{\ast }$.

arXiv comment6 pages