I think GP was drawing a parallel between Euclid’s fifth axiom of and results like Banach-Tarski since the fifth axiom is independent of the other four, and Bna

I think GP was drawing a parallel between Euclid’s fifth axiom of and results like Banach-Tarski since the fifth axiom is independent of the other four, and Bnach-Tarski follows from the Axiom of Choice which is independent of the rest of set theory. The pun was intended, by the way.