On the prewellordering associated to the dircted systems of mice

TL;DR

This paper explores the properties of prewellorderings derived from iterates of certain mice with multiple Woodin cardinals under AD, addressing open questions in the field.

Contribution

It provides new insights into the length of prewellorderings associated with minimal proper class mice with multiple Woodin cardinals, answering previously open questions.

Findings

Determined the length of prewellorderings for iterates of $ ext{M}_{2k+1}$.

Connected properties of mice with multiple Woodin cardinals to prewellordering lengths.

Resolved specific open questions from prior literature.

Abstract

Working under $AD$, we investigate the length of prewellorderings given by the iterates of $\M_{2k+1}$, which is the minimal proper class mouse with $2k+1$ many Woodin cardinals. In particular, we answer some questions from \cite{Hjorth01} (the discussion of the questions appears in the last section of \cite{HjorthD}).