From Z(ω) or CMC(ℵ0,∞) to van Douwen’s Countable Choice Principle

Determine whether the principle Z(ω) implies vDCP(ω) and whether CMC(ℵ0,∞) implies vDCP(ω), in either ZF or ZFA.

Background

The authors investigate the relative strength of several weak choice principles related to countable choice and multiple choice. Although they establish many implications and separations, they explicitly state that the status of deriving van Douwen’s Countable Choice Principle vDCP(ω) from Z(ω) or from CMC(ℵ0,∞) remains unknown in both ZF and ZFA.