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: \(\mathsf{DC}(X)\) assets that any total binary relation on \(X\) has an infinite chain; \(\mathsf{AC_\omega}(X)\) assets that any countable family of nonempty subsets of \(X\) has a choice function. It is well-known that \(\mathsf{DC}\) implies \(\mathsf{AC_\omega}\).
We discuss and sketch the proof of the following theorem: it is consistent with \(\mathsf{ZF}\) that there is a set \(A\subseteq \mathbb{R}\) such that \(\mathsf{DC}(A)\) holds but \(\mathsf{AC_\omega}(A)\) fails.
This is joint work with Alessandro Andretta.