On a Conjecture Regarding the Mouse Order for Weasels

TL;DR

This paper examines Steel's conjecture on the mouse order for weasels, demonstrating its failure under certain large cardinal assumptions and confirming it under others, while also exploring related forcing and iteration properties.

Contribution

It provides the first proof of the conjecture's failure assuming a Woodin cardinal and confirms its validity assuming no such model exists, also addressing related forcing and iteration questions.

Findings

The conjecture fails with a Woodin cardinal assumption.

The conjecture holds if no transitive model of KP with a Woodin cardinal exists.

Established new properties of models under forcing and iteration.

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.