Zermelo–Fraenkel Set Theory - Criticisms

Criticisms

For criticism of set theory in general, see Objections to set theory

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 of sets under ZFC is not closed under the elementary operations of the algebra of sets. Unlike von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory (MK), ZFC does not admit the existence of proper classes. These ontological restrictions are required for ZFC to avoid Russell's paradox, but critics argue these restrictions make the ZFC axioms fail to capture the informal concept of set. 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 MK.

There are numerous mathematical statements undecidable in ZFC. These include the continuum hypothesis, the Whitehead problem, and the Normal Moore space conjecture. Some of these conjectures are provable with the addition of axioms such as Martin's axiom, large cardinal axioms 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 (see projective determinacy). The Mizar system has adopted Tarski–Grothendieck set theory instead of ZFC so that proofs involving Grothendieck universes (encountered in category theory and algebraic geometry) can be formalized.