Translation procedures in descriptive inner model theory

TL;DR

This paper introduces a new translation method in descriptive inner model theory that converts hod mice to mice without assuming AD, and uses it to derive the existence of models with strong large cardinal properties.

Contribution

The paper develops a coarse translation procedure for hod mice to mice, extending prior methods and applying it to establish the existence of models with Woodin and strong cardinals.

Findings

A new translation procedure for hod mice to mice that does not assume AD.

Application of the translation to show models with proper class of Woodin and strong cardinals.

Answer to Trevor Wilson's question about the existence of certain large cardinal models.

Abstract

We develop a basic translation procedure that translates a hod mouse to an equivalent mouse. Unlike the translation procedure used by Steel and Zhu, our procedure works in a coarse setting without assuming AD. Nevertheless, the procedure resembles the one developed by Steel. We use the translation procedures to answer a question of Trevor Wilson. Namely, we show that if there is a stationary class of lambda such that lambda is a limit of Woodin cardinals and the derived model at lambda satisfies AD the first member of the Solovay sequence is is less than Theta then there is a transitive model M such that M contains the ordinals and satisfies there is a proper class of Woodin cardinals and a strong cardinal.