Consistency of Hewitt’s theorem with ZF

Determine whether there exists a model of Zermelo–Fraenkel set theory (ZF) in which Hewitt’s theorem—asserting that a Tychonoff space X is realcompact if and only if every z-ultrafilter with the countable intersection property in X is fixed—is false. Equivalently, ascertain whether the equivalence in Hewitt’s theorem can fail in some model of ZF.

Background

Hewitt’s theorem is classically proved in ZFC and characterizes realcompactness via z-ultrafilters with the countable intersection property. The authors establish ZF-valid variants and show that Hewitt’s theorem holds in every model of ZF that satisfies the countable axiom of multiple choice (CMC). They also prove a ZF modification using a weaker intersection property. Despite these advances, the independence of the full theorem from ZF remains unresolved.