So, I ask whether from the ZFC axioms one can prove X that *every uncountable set has strictly more than continuum many subsets*, or whether X is independent of the ZFC axioms. Note that (within ZFC) the continuum hypothesis implies X and hence "not X" is not provable in ZFC if ZFC is consistent.

The above question arose from this MO-question to which an answer is given by Problem 10 on page 99 in Richard M. Dudley's book *Real Analysis and Probability*, Wadsworth 1989. However, in view of Problem 9 on page 387 *loc.cit.* it seems that some additional hypothesis (e.g. CH or the weaker X above) is missing in Dudley's problem setting. If X is provable within ZFC, then without any additional assumptions we have the result that *every Borel map from any separable metric space to a metric space has separable range*. If X is not provable, then Dudley's hints for a possible proof seem to require some ad hoc assumption for the proof to succeed.