Most pitches for Everett branches (which are a specific enough concept that it's silly to get all technical, but whatever) hold that there is an amplitude distr

Most pitches for Everett branches (which are a specific enough concept that it's silly to get all technical, but whatever) hold that there is an amplitude distribution over the entire configuration space of the universe. While the proposed mechanism of decoherence implies a sort of discreteness, every single possible configuration would have non-zero amplitude. Of course there's no specific evidence for this over, say, collapse theories (or something stranger, like Bohmian mechanics), but there's certainly no philosophical issue. Not sure what you think ZFC has to do with physics, but an empty set with members is a tad nonexistent.