The Axiom of Determinacy Implies Dependent Choices in Mice

TL;DR

This paper demonstrates that the Axiom of Dependent Choices holds in certain models of set theory called mice, which are constructed over reals satisfying the Axiom of Determinacy, extending previous results and analysis.

Contribution

It proves that DC holds in countably iterable passive premice over reals satisfying AD, generalizing earlier arguments for models like L(R) using Steel's scale analysis.

Findings

DC holds in models constructed over reals satisfying AD.

The result applies to models M_n(A) with specific properties.

Extends Kechris's argument for L(R) to broader mice contexts.

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 $M_n(A) \vDash \operatorname{DC}$.