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 $W{\le }^{\ast }R$ 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.

30 pages. Changes to v2: Change of Title. New section about the S-hull property was added. Proposition 34 was weakened. Various minor corrections were made

References in corpus (1)

• Iterability for (transfinite) stacks