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DOES
$\mathsf {DC}$ IMPLY 
${\mathsf {AC}}_\omega $, UNIFORMLY?
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 06 May 2024, pp. 1-25
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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.
