论文标题
正常迭代性的精细结构
Fine structure from normal iterability
论文作者
论文摘要
我们表明（i）从正常迭代性中遵循的标准优良结构特性（而经典的证明依赖于堆叠正常树的迭代性），以及（ii）每个在其项目上方有限生成的鼠标，都是其核心的迭代。 也就是说，让$ m $为整数，让$ m $为$ m $ sound，$（m，ω_1+1）$ - 价tobles。然后（i）$ m $是$（m+1）$ - 固体和$（M+1）$ - universal，$（m+1）$ condensation $ m $，如果$ m \ geq 1 $，则$ m $ then $ m $是super-doddd-sound，略微增强了dodd-sound。 （ii）如果在m $中有$ x \，则$ m $是$ \ mathrm {r}σ_{m+1} $ - $ρ_{m+1}^m \ cup \ {x \ {x \ {x \} $的参数的船体，那么$ c = \ mathfrak {c} _ {m+1}（m）$;实际上，有限长度的$ c $上有一个$ m $ -maximal迭代树$ \ Mathcal {t} $，因此$ m = m = m = m = m^{\ nathcal {t}} _ \ infty $，和$ i^{\ mathcal {\ mathcal {t}} _ {0 \ iffty is offty is offty is corecore。 应用事实（ii），我们证明，如果$ m \ mathrm {zfc} $是鼠标，而$ w \ subseteq m $是通过策略上$σ$ clucking using $ \ mathbb {p mathbb {p} \在w $中的$ m $的地面，则是$ m | \ m | \ aleph_1^m \ $ $ pribster w $ n y nys w $ hefter， $ \ aleph_1^m $在$ W $中），然后强迫是微不足道的；也就是说，$ m \ subseteq w $。 如果有可衡量的红衣主教，则有一个非固体预期。 通过Mitchell-steel索引进行首映的结果，使SuperStrong类型的扩展器出现在扩展器序列上。
We show that (i) the standard fine structural properties for premice follow from normal iterability (whereas the classical proof relies on iterability for stacks of normal trees), and (ii) every mouse which is finitely generated above its projectum, is an iterate of its core. That is, let $m$ be an integer and let $M$ be an $m$-sound, $(m,ω_1+1)$-iterable premouse. Then (i) $M$ is $(m+1)$-solid and $(m+1)$-universal, $(m+1)$ condensation holds for $M$, and if $m\geq 1$ then $M$ is super-Dodd-sound, a slight strengthening of Dodd-soundness. And (ii) if there is $x\in M$ such that $M$ is the $\mathrm{r}Σ_{m+1}$-hull of parameters in $ρ_{m+1}^M\cup\{x\}$, then $M$ is a normal iterate of its $(m+1)$-core $C=\mathfrak{C}_{m+1}(M)$; in fact, there is an $m$-maximal iteration tree $\mathcal{T}$ on $C$, of finite length, such that $M=M^{\mathcal{T}}_\infty$, and $i^{\mathcal{T}}_{0\infty}$ is just the core embedding. Applying fact (ii), we prove that if $M\models\mathrm{ZFC}$ is a mouse and $W\subseteq M$ is a ground of $M$ via a strategically $σ$-closed forcing $\mathbb{P}\in W$, and if $M|\aleph_1^M\in W$ (that is, the initial segment of $M$ of height $\aleph_1^M$ is in $W$), then the forcing is trivial; that is, $M\subseteq W$. And if there is a measurable cardinal, then there is a non-solid premouse. The results hold for premice with Mitchell-Steel indexing, allowing extenders of superstrong type to appear on the extender sequence.
