Does $\mathsf{DC}$ imply ${\mathsf{AC}}_{\omega }$, uniformly?

Abstract

The Axiom of Dependent Choice $\mathsf{DC}$ and the Axiom of Countable Choice ${\mathsf{AC}}_{\omega }$ are two weak forms of the Axiom of Choice that can be stated for a specific set: $\mathsf{DC}(X)$ asserts that any total binary relation on $X$ has an infinite chain, while ${\mathsf{AC}}_{\omega }(X)$ asserts that any countable collection of nonempty subsets of $X$ has a choice function. It is well-known that $\mathsf{DC}\Rightarrow {\mathsf{AC}}_{\omega }$. We study for which sets and under which hypotheses $\mathsf{DC}(X)\Rightarrow {\mathsf{AC}}_{\omega }(X)$, and then we show it is consistent with $\mathsf{ZF}$ that there is a set $A\subseteq \mathbb{R}$ for which $\mathsf{DC}(A)$ holds, but ${\mathsf{AC}}_{\omega }(A)$ fails.

23 pages

References in corpus (1)

• Iterating Symmetric Extensions