On a conjecture regarding the mouse order for weasels

Kruschewski, Jan; Schlutzenberg, Farmer

Abstract

We investigate Steel's conjecture in 'The Core Model Iterability Problem', 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 the cardinal successor of α in W is less or equal than the cardinal successor of α in R. We will show that the conjecture fails, assuming that there is an iterable premouse which models KP and which has a Σ1-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 δ, and P is a forcing poset with the δ-c.c. in M, and g is M-generic, but not necessarily Σ1-generic, M[g]  is a model of KP. Moreover, if M is such a mouse and T is a 0-maximal normal iteration tree on M such that T is non-dropping on its main branch, then MT∞ is again an admissible premouse with a largest regular and uncountable cardinal. We also answer another question from 'The Core Model Iterability Problem' regarding the S-hull property.