[eigenket]

This is very very strongly not a concern for physics. Banach-Tarski and similar "paradoxes" require dividing things up into unmeasurable sets. These sets are pretty wild and aren't ever going to turn up when you're doing physics.

Speaking as a physicist, we don't care at all about stuff like Banach-Tarski, and there is essentially zeo expectation that stuff like ZF vs ZFC will have any impact on physics.