The Axiom of Determinacy Implies Dependent Choices in Mice

Abstract

We show that the Axiom of Dependent Choices, $\mathrm{DC}$, holds in countably iterable, passive premice $\mathcal{M}$ construced over their reals which satisfy the Axiom of Determinacy, $\mathrm{AD}$, in a $\mathrm{ZF}+{\mathrm{DC}}_{{\mathbb{R}}^{\mathcal{M}}}$ background universe. This generalizes an argument of Kechris for $L(\mathbb{R})$ using Steel's analysis of scales in mice. In particular, we show that for any $n\le \omega$ and any countable set of reals $A$ so that $M_n(A)\cap \mathbb{R}=A$ and $M_n(A)\models \mathrm{AD}$, we have that $M_n(A)\models \mathrm{DC}$.