> ZFC is not regarded as sound by some because of these Paradoxes. Can you give relevant pointers? I'm curious because while I do know a couple of arguments fo

> ZFC is not regarded as sound by some because of these Paradoxes. Can you give relevant pointers? I'm curious because while I do know a couple of arguments for the inconsistency of ZFC, none of these are related to the Banach–Tarski paradox or similar nonintuitive results. Also I'd like to stress that ZFC and ZF are exactly as consistent: If ZF is consistent, then so is ZFC (and vice versa). This meta-result is proven in a very weak finitary logical meta-system ("PRA"), hence can be trusted even if one is wary of set-theoretic infinities. The keyword is "Gödel's constructible hierarchy". In this line, there are even more astonishing meta-theorems, all provable in PRA: * From any ZFC-proof of a purely number-theoretical statement (a statement which refers only to natural numbers), any usage of the axiom of choice can be mechanically eliminated. That is, any ZFC-proof of such a result can be transformed into a ZF-proof. * From any ZFC-proof of a statement of the form "for all numbers n, there exists a number m such that ...", where in "..." no more quantifiers ("for all", "there exist") appear, any usage of the law of excluded middle ("any statement is either true or not") can be mechanically eliminated. That is, any such ZFC-proof can be transformed into a proof in IZF ("Intuitionistic Zermelo–Fraenkel"). The keyword is "double negation translation". These meta-theorems indicate that the axiom of choice and the law of excluded middle can be regarded as "mathematical phantoms". Just as the complex numbers work wonders for us but can be compiled away to pairs of real numbers, these set-theoretic principles promise to work wonders for us and their usage can be compiled away if we wish so.