Building Models of Determinacy from Below

Abstract

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