Fine structure from normal iterability

Farmer Schlutzenberg

Published 2020-11-19Version 1

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, $(m,\omega_1+1)$-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 $\mathrm{r}\Sigma_{m+1}$-hull of parameters in $\rho_{m+1}^M\cup\{x\}$, 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 $\sigma$-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.