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

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Abstract: The Axiom of Dependent Choice

m a t h s f D C

and the Axiom of Countable Choice

m a t h s f {A C}_{o} m e g a

are two weak forms of the Axiom of Choice that can be stated for a specific set:

m a t h s f D C (X)

asserts that any total binary relation on

X

has an infinite chain, while

m a t h s f {A C}_{o} m e g a (X)

asserts that any countable collection of nonempty subsets of

X

has a choice function. It is well-known that

m a t h s f D C R i g h t a r r o w m a t h s f {A C}_{o} m e g a

. We study for which sets and under which hypotheses

m a t h s f D C (X) R i g h t a r r o w m a t h s f {A C}_{o} m e g a (X)

, and then we show it is consistent with

m a t h s f Z F

that there is a set

A s u b s e t e q m a t h b b R

for which

m a t h s f D C (A)

holds, but

m a t h s f {A C}_{o} m e g a (A)

fails.

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