论文标题
关于鼠标鼠标鼠标订单的猜想
On a Conjecture Regarding the Mouse Order for Weasels
论文作者
论文摘要
我们调查了钢铁在“核心模型迭代性问题”中的猜想，如果$ w $和$ r $是$ω+1 $ - 可允许的，$ 1 $ -small牛油，则$ w \ w \ leq^{*} r $ iff club $ c \ c \subsetΩ$ in $ cub in c $ in $ sire in $ ins $ ins $ rascinal in priondic或lod $α$ rad $ rad $α\ n y rates n in n coldical或wor n pressinal或lods $α\α\比$ r $ $α$的红衣主教继任者。我们将证明猜想失败了，假设有一个值得建模型$ kp $，并且具有$σ_{1} $ -Woodin Cardinal。另一方面，我们表明，假设猜想的木质枢机主教没有$ kp $的传递模型。 在此过程中，我们还将表明，如果$ m $是一个可允许的预期，则具有最大，常规，无数的红衣主教$δ$，而$ \ mathbb {p} $是带有$δ$ -C.C.-C.C.-C.c.c.c.c.在$ m $中，$ g $是$ m $ - 生成的，但不一定是$σ_{1} $ - 通用，$ m [g] $是$ kp $的型号。此外，如果$ m $是这样的鼠标，而$ t $是$ m $上的最大普通迭代树，这样$ t $在其主分支上是不可接受的，那么$ m _ {\ infty}^{t} $又是最大的可容纳的预期，具有最大的常规和无数的红衣主教。最后，我们回答了有关S-Hull属性的“核心模型迭代性问题”的另一个开放问题。
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.
