On the derived models of self iterable universes

TL;DR

This paper proves that in self-iterable universes with an inaccessible limit of Woodin cardinals, certain determinacy and regularity properties hold in the derived model, using a novel, fine-structure free approach.

Contribution

It introduces a new, more general proof that the determinacy and regularity of a in derived models hold under specific large cardinal assumptions, without relying on fine-structure theory.

Findings

AD_R + "a is regular" holds in the derived model at a.

The proof is fine-structure free and more general than previous results.

The approach simplifies understanding of derived models under large cardinal assumptions.

Abstract

We show that if the universe is self-iterable and $\kappa$ is an inaccessible limit of Woodin cardinal then $AD_R + "\Theta$ is regular" holds in the derived model at $\kappa$. The proof is fine-structure free, and only assumes basic knowledge of iteration trees and iteration strategies. Our proof can be viewed as the fine-structure free version of the well-known fact that $AD_R + "\Theta$ is regular" is true in the derived models of hod mice that have inaccessible limit of Woodin cardinals (see for example [6]). However, the proof uses a different set of ideas and is more general.