ON THE DERIVED MODELS OF SELF-ITERABLE UNIVERSES

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 Sargsyan [Mem. Amer. Math. Soc. 236 (2015), p. viii+172]). However, the proof uses a different set of ideas and is more general.

Automated summary

We prove that in a self-iterable universe, if kappa is an inaccessible limit of Woodin cardinal, then AD(R) + Theta holds in the derived model at kappa. The proof is fine-structure free and only requires basic knowledge of iteration trees and iteration strategies. Our proof can be seen as a fine structure free version of the well-known fact that AD(R) + Theta is true in the derived models of hod mice with inaccessible limit of Woodin cardinals (see for example Sargsyan [Mem. Amer. Math. Soc. 236 (2015), p. viii+172]). However, the proof uses different ideas and is more general.