# The definability of the extender sequence $\mathbb{E}$ from $\mathbb{E}{\upharpoonright}\aleph_1$ in $L[\mathbb{E}]$

## TL;DR

This paper demonstrates that in certain models of set theory, the entire extender sequence can be defined from its restriction to the first uncountable ordinal, leading to insights about the structure of mice and models where V=HOD.

## Contribution

It proves the definability of the extender sequence from initial segments in models with short extenders and establishes conditions under which the universe is equal to HOD.

## Key findings

- The extender sequence is definable from its restriction to leph_1 in models with short extenders.
- Models with a Woodin limit of Woodin cardinals satisfy V=HOD.
- Various local and generic extension versions of the definability result are established.

## Abstract

Let $M$ be a short extender mouse. We prove that if $E\in M$ and $M$ satisfies "$E$ is a countably complete short extender whose support is a cardinal $\theta$ and $\mathcal{H}_\theta\subseteq\mathrm{Ult}(V,E)$", then $E$ is in the extender sequence $\mathbb{E}^M$ of $M$. We also prove other related facts, and use them to establish that if $\kappa$ is an uncountable cardinal of $M$ and $\kappa^{+M}$ exists in $M$ then $(\mathcal{H}_{\kappa^+})^M$ satisfies the Axiom of Global Choice. We prove that if $M$ satisfies the Power Set Axiom then $\mathbb{E}^M$ is definable over the universe of $M$ from the parameter $X=\mathbb{E}^M\upharpoonright\aleph_1^M$, and $M$ satisfies "every set is $\mathrm{OD}_{\{X\}}$". We also prove various local versions of this fact in which $M$ has a largest cardinal, and a version for generic extensions of $M$. As a consequence, for example, the minimal proper class mouse with a Woodin limit of Woodin cardinals models "$V=\mathrm{HOD}$". This adapts to many other similar examples. We also describe a simplified approach to Mitchell-Steel fine structure, which does away with the parameters $u_n$.