$\star$-translation for Varsovian models

TL;DR

This paper advances the theory of $igstar$-translation, especially for Varsovian models, to analyze determinacy theories and large cardinals, providing detailed development and verifying key hypotheses.

Contribution

It offers a detailed development of $igstar$-translation, including a new variant for Varsovian models, and applies it to verify hypotheses and establish self-iterability.

Findings

Strengthened the large cardinal level in $igstar$-translation development.

Developed a variant of $igstar$-translation for Varsovian models.

Verified the star-translation hypothesis and deduced self-iterability facts.

Abstract

The technique of $\star$-translation is important in arguments calibrating the strengths of determinacy theories against large cardinals, for example in [9] and [1] in the paper's bibliography. It has also been used in analysing the internal theory of mice, for example in [5], [3], [6]. We give a detailed development of $\star$-translation, slightly strengthening the large cardinal level in the development of [1]. We then develop a variant of $\star$-translation for Varsovian models, in which certain extenders overlapping the Woodin cardinal are total, and used to directly induce extenders on the sequence of the $\star$-translation (details for this component are missing from this draft). This variant was used in [3] and [6], where it was outlined but not developed in detail. We use the material to verify the star-translation hypothesis of [5] and deduce some self-iterability facts.