The Axiom of Determinacy Implies Dependent Choices in Mice

Abstract

We show that the Axiom of Dependent Choices, $\operatorname{DC}$, holds in countably iterable, passive premice $\mathcal{M}$ construced over their reals which satisfy the Axiom of Determinacy, $\operatorname{AD}$, in a $\operatorname{ZF}+\operatorname{DC}_{\mathbb{R}^{\mathcal{M}}}$ 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 $n \leq \omega$ and any countable set of reals $A$ so that $M_n(A) \cap \mathbb{R} = A$ and $M_n(A) \vDash \operatorname{AD}$, we have that $M_n(A) \vDash \operatorname{DC}$.