ON A CONJECTURE REGARDING THE MOUSE ORDER FOR WEASELS

Abstract We investigate Steel’s conjecture in ‘The Core Model Iterability Problem’ [10], that if $\mathcal {W}$ and $\mathcal {R}$ are $\Omega +1$ -iterable, $1$ -small weasels, then $\mathcal {W}\leq ^{*}\mathcal {R}$ iff there is a club $C\subset \Omega $ such that for all $\alpha \in C$ , if $\alpha $ is regular, then $\alpha ^{+\mathcal {W}}\leq \alpha ^{+\mathcal {R}}$ . We will show that the conjecture fails, assuming that there is an iterable premouse M which models KP and which has a -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 a premouse which models KP with a largest, regular, uncountable cardinal $\delta $ , and $\mathbb {P} \in M$ is a forcing poset such that $M\models "\mathbb {P}\text { has the }\delta \text {-c.c.}"$ , and $g\subset \mathbb {P}$ is M -generic, then $M[g]\models \text {KP}$ . Additionally, we study the preservation of admissibility under iteration maps. At last, we will prove a fact about the closure of the set of ordinals at which a weasel has the S -hull property. This answers another question implicit in remarks in [10].