关于Weasel的Mouse序的一个猜想
On a Conjecture Regarding the Mouse Order for Weasels
AI总结：
研究Steel关于Ω+1-可迭代1-小型weasels的Mouse序猜想，证明其在存在满足KP且有Σ₁-Woodin基数的可迭代premouse时不成立，并解决相关开放问题。
摘要：
我们研究Steel在《核心模型迭代性问题》中的猜想：若W和R是Ω+1-可迭代的1-小型weasels，则W≤*R当且仅当存在club C⊂Ω，使得对所有α∈C，若α是正则的，则W中α的基数后继小于等于R中α的基数后继。我们将证明该猜想不成立，假设存在一个可迭代的premouse，其满足KP模型且有Σ₁-Woodin基数。此外，我们还证明：若M是可迭代的可容premouse，有最大的正则不可数基数δ，P是M中具有δ-c.c.的力迫偏序，g是M-泛型（不一定是Σ₁-泛型），则M[g]是KP模型；若M是此类mouse，T是M上的极大正规迭代树且主分支无丢弃，则M∞^T仍是具有最大正则不可数基数的可迭代可容premouse。最后，我们回答了《核心模型迭代性问题》中关于S-hull性质的另一个开放问题。
英文摘要：
We investigate Steel's conjecture in 'The Core Model Iterability Problem', that if $W$ and $R$ are $Ω+1$-iterable, $1$-small weasels, then $W\leq^{*}R$ iff there is a club $C\subsetΩ$ such that for all $α\in C$, 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 $Σ_{1}$-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 an iterable admissible premouse with a largest, regular, uncountable cardinal $δ$, and $\mathbb{P}$ is a forcing poset with the $δ$-c.c. in $M$, and $g$ is $M$-generic, but not necessarily $Σ_{1}$-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.
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