A characterization of extenders of HOD

TL;DR

This paper characterizes extenders on HOD under AD+V=L(R), showing they are generated by countably complete measures, and connects this to Woodin's results and the structure of H in models of determinacy.

Contribution

It provides a new characterization of extenders on HOD with critical point κ as generated by countably complete measures, extending to models of determinacy.

Findings

Extenders on HOD with critical point κ are generated by countably complete measures.

The characterization applies to all cutpoint measurable cardinals in models of determinacy.

Provides a simple proof that successor members of the Solovay sequence are cutpoints in H.

Abstract

Assume $AD+V=L(\mathbb{R})$. Let $\kappa=\utilde{\delta}^2_1$, the supremum of all $\utilde{\Delta}^2_1$ prewellorderings. We prove that extenders on the sequence of $\H$ that have critical point $\kappa$ are generated by countably complete measures. This provides a partial reversal of Woodin's result that the $<\Theta$-strongness of $\kappa$ in $\H$ is witnessed by $\kappa$-complete ultrafilters on $\k$. The aforementioned characterization of extenders works in a more general setting for all cutpoint measurable cardinals of $\H$ in all models of determinacy where the fine structural analysis of $\H$ has been carried out. For example, it holds in the minimal model of the Largest Suslin Axiom. It also gives a simple proof of a theorem of Steel that the successor members of the Solovay sequence are cutpoints in $\H$ (in models where $\H$ analysis is carried out).