Editing Zermelo–Fraenkel set theory (section)

== Criticisms ==
{{see also|projective determinacy}}

ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and the [[universal set]].

Many mathematical theorems can be proven in much weaker systems than ZFC, such as [[Peano arithmetic]] and [[second-order arithmetic]] (as explored by the program of [[reverse mathematics]]). [[Saunders Mac Lane]] and [[Solomon Feferman]] have both made this point. Some of "mainstream mathematics" (mathematics not directly connected with axiomatic set theory) is beyond Peano arithmetic and second-order arithmetic, but still, all such mathematics can be carried out in ZC ([[Zermelo set theory]] with choice), another theory weaker than ZFC. Much of the power of ZFC, including the axiom of regularity and the axiom schema of replacement, is included primarily to facilitate the study of the set theory itself.

On the other hand, among [[axiomatic set theories]], ZFC is comparatively weak. Unlike [[New Foundations]], ZFC does not admit the existence of a universal set. Hence the [[Universe (mathematics)|universe]] of sets under ZFC is not closed under the elementary operations of the [[algebra of sets]]. Unlike [[von Neumann–Bernays–Gödel set theory]] (NBG) and [[Morse–Kelley set theory]] (MK), ZFC does not admit the existence of [[proper class]]es. A further comparative weakness of ZFC is that the [[axiom of choice]] included in ZFC is weaker than the [[axiom of global choice]] included in NBG and MK.

There are numerous [[list of statements undecidable in ZFC|mathematical statements independent of ZFC]]. These include the [[continuum hypothesis]], the [[Whitehead problem]], and the [[Moore space (topology)|normal Moore space conjecture]]. Some of these conjectures are provable with the addition of axioms such as [[Martin's axiom]] or [[large cardinal axiom]]s to ZFC.  Some others are decided in ZF+AD where AD is the [[axiom of determinacy]], a strong supposition incompatible with choice. One attraction of large cardinal axioms is that they enable many results from ZF+AD to be established in ZFC adjoined by some large cardinal axiom. The [[Mizar system]] and [[metamath]] have adopted [[Tarski–Grothendieck set theory]], an extension of ZFC, so that proofs involving [[Grothendieck universe]]s (encountered in category theory and algebraic geometry) can be formalized.