Maximal Prikry Sequences

Abstract

In this paper we investigate the covering machinery of the Jensen-Steel core model $K$, under the hypothesis that there is no inner model with a Woodin cardinal. In an earlier work, Mitchell and the first author showed that if $\nu>\omega_2$ is a regular cardinal in $K$ but a singular ordinal in $V$, then $\nu$ is a measurable cardinal in $K$. In this article, we further show that under certain circumstances, there exists a maximal Prikry sequence $C$ for a measure on $\nu$ in $K$. The first author shows that the anti-large cardinal hypothesis is necessary. In a more restrictive setting, we prove that every subset of $\nu$ with size $<|\nu|$ can be covered by a set in $K[C]$ with size $<|\nu|$. Benhamou and the first author show that the result is optimal.