Supercompactness Can Be Equiconsistent with Measurability

Abstract

The main result of this paper, built on previous work by the author and T. Wilson, is the proof that the theory “${\mathsf{AD}}_{\mathbb{R}}+\mathsf{DC}$ $+$ there is an $\mathbb{R}$-complete measure on Θ” is equiconsistent with “$\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}$ $+$ there is a supercompact measure on ${\wp }_{{\mathit{\omega }}_1}(\wp (\mathbb{R}))+\mathrm{\Theta }$ is regular.” The result and techniques presented here contribute to the general program of descriptive inner model theory and in particular, to the general study of compactness phenomena in the context of $\mathsf{ZF}+\mathsf{DC}$.