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

@article{Schlutzenberg2019TheDO, title={The definability of the extender sequence \$\mathbb\{E\}\$ from \$\mathbb\{E\}\{\upharpoonright\}\aleph\_1\$ in \$L[\mathbb\{E\}]\$.}, author={Farmer Schlutzenberg}, journal={arXiv: Logic}, year={2019} }

Let $M$ be an iterable fine structural mouse.
We prove that if $E\in M$ and $M\models$``$E$ 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…

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