On forcing over $L(\mathbb {R})$

Abstract

Given that $L(\mathbb {R})\models {\text {ZF}}+ {\text {AD}}+{\text {DC}}$, we present conditions under which one can generically add new elements to $L(\mathbb {R})$ and obtain a model of ${\text {ZF}}+ {\text {AD}}+{\text {DC}}$. This work is motivated by the desire to identify the smallest cardinal $\kappa$ in $L(\mathbb {R})$ for which one can generically add a new subset $g\subseteq \kappa$ to $L(\mathbb {R})$ such that $L(\mathbb {R})(g)\models {\text {ZF}}+ {\text {AD}}+{\text {DC}}$.