Lorenzo Notaro:
Separating \(\bmathsf{DC}(A)\) from \(\bmathsf{AC_\bomega}(A)\)
Time and place
Tuesday, 31.10.23, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Abstract
The axiom of dependent choice \(\bmathsf{DC}\) and the axiom of countable choice \(\bmathsf{AC_\bomega}\) are two weak forms of the axiom of choice that can be stated for a specific set: \(\bmathsf{DC}(X)\) assets that any total binary relation on \(X\) has an infinite chain; \(\bmathsf{AC_\bomega}(X)\) assets that any countable family of nonempty subsets of \(X\) has a choice function. It is well-known that \(\bmathsf{DC}\) implies \(\bmathsf{AC_\bomega}\). We show that it is consistent with \(\bmathsf{ZF}\) that there is a set \(A\bsubseteq \bmathbb{R}\) such that \(\bmathsf{DC}(A)\) holds but \(\bmathsf{AC_\bomega}(A)\) fails.\n\n