Maximal Prikry Sequences

TL;DR

This paper explores the structure of the Jensen-Steel core model under the absence of inner models with Woodin cardinals, demonstrating the existence of maximal Prikry sequences and their implications for covering properties.

Contribution

It establishes the existence of maximal Prikry sequences in the core model and analyzes their role in covering properties under anti-large cardinal hypotheses.

Findings

Existence of maximal Prikry sequences in the core model.

Covering of small subsets of 9 by sets in K[C].

Optimality of the covering results.

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.