The definability of $\mathbb{E}$ in self-iterable mice

TL;DR

This paper proves that extender sequences in certain mice are definable within the mice themselves, leading to implications for the structure of the universe and uniqueness of certain inner models.

Contribution

It establishes the definability of extender sequences in self-iterable mice and generalizes this to models with iteration strategies, impacting inner model theory.

Findings

Extender sequences are definable in self-iterable mice.

If a mouse knows enough of its strategy, its universe models V=HOD.

No certain types of iterable bicephali exist, ensuring uniqueness in L[E] constructions.

Abstract

Let $M$ be a fine structural mouse and let $F\in M$ be such that $M\models$``$F$ is a total extender'' and $(M||\mathrm{lh}(F),F)$ is a premouse. We show that it follows that $F\in\mathbb{E}^M$, where $\mathbb{E}^M$ is the extender sequence of $M$. We also prove generalizations of this fact. Let $M$ be a premouse with no largest cardinal and let $\Sigma$ be a sufficient iteration strategy for $M$. We prove that if $M$ knows enough of $\Sigma\upharpoonright M$ then $\mathbb{E}^M$ is definable over the universe $\lfloor M\rfloor$ of $M$, so if also $\lfloor M\rfloor\models\mathrm{ZFC}$ then $\lfloor M\rfloor\models$``$V=\mathrm{HOD}$''. We show that this result applies in particular to $M=M_{\mathrm{nt}}|\lambda$, where $M_{\mathrm{nt}}$ is the least non-tame mouse and $\lambda$ is any limit cardinal of $M_{\mathrm{nt}}$. We also show that there is no iterable bicephalus $(N,E,F)$ for which $E$ is type $2$ and $F$ is type $1$ or $3$. As a corollary, we deduce a uniqueness property for maximal $L[\mathbb{E}]$ constructions computed in iterable background universes.