On a Conjecture Regarding the Mouse Order for Weasels

Abstract

We investigate Steel's conjecture in 'The Core Model Iterability Problem', that if $W$ and $R$ are $\Omega+1$-iterable, $1$-small weasels, then $W\leq^{*}R$ iff there is a club $C\subset\Omega$ such that for all $\alpha\in C$, if $\alpha$ is regular, then the cardinal successor of $\alpha$ in $W$ is less or equal than the cardinal successor of $\alpha$ in $R$ . We will show that the conjecture fails, assuming that there is an iterable premouse which models $KP$ and which has a $\Sigma_{1}$-Woodin cardinal. On the other hand, we show that assuming there is no transitive model of $KP$ with a Woodin cardinal the conjecture holds. In the course of this we will also show that if $M$ is an iterable admissible premouse with a largest, regular, uncountable cardinal $\delta$, and $\mathbb{P}$ is a forcing poset with the $\delta$-c.c. in $M$, and $g$ is $M$-generic, but not necessarily $\Sigma_{1}$-generic, $M[g]$ is a model of $KP$. Moreover, if $M$ is such a mouse and $T$ is maximal normal iteration tree on $M$ such that $T$ is non-dropping on its main branch, then $M_{\infty}^{T}$ is again an iterable admissible premouse with a largest regular and uncountable cardinal. At last we answer another open question from 'The Core Model Iterability Problem' regarding the S-hull property.