Building Models of Determinacy from Below

Abstract

We present an $L$-like construction that produces the minimal model of ${\mathsf{AD}}_{\mathbb{R}}+$"$\hat{I}˜$ is regular". In fact, our construction can produce any model of ${\mathsf{AD}}^{+}+{\mathsf{AD}}_{\mathbb{R}}+V=L(P(\mathbb{R}))$ in which there is no hod mouse with a measurable limit of Woodins.