On the derived models of self-iterable universes

Abstract

We show that if the universe is self-iterable and κ \kappa is an inaccessible limit of Woodin cardinal then A D R + “ Θ \mathsf {AD}_{\mathbb {R}}+“\Theta is regular” holds in the derived model at κ \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 A D R + “ Θ \mathsf {AD}_{\mathbb {R}}+“\Theta is regular” is true in the derived models of hod mice that have inaccessible limit of Woodin cardinals (see for example Sargsyan [Mem. Amer. Math. Soc. 236 (2015), p. viii+172]). However, the proof uses a different set of ideas and is more general.