On A Conjecture Regarding The Mouse Order For Weasels

Abstract

We investigate Steel’s conjecture in ‘The Core Model Iterability Problem’ [10], that if W and R are Ω + 1-iterable, 1-small weasels, then W ≤* R iff there is a club C ⊂ Ω such that for all α ∈ C, if α is regular, then α^{+W} ≤ α^{+R}. We will show that the conjecture fails, assuming that there is an iterable premouse M which models KP and which has a boldface-Σ_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 a premouse which models KP with a largest, regular, uncountable cardinal δ, and P ∈ M is a forcing poset such that M |= “P has the δ-c.c.”, and g ⊂ P is M-generic, then M[g] |= 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].