Page "Axiom of choice" Paragraph 43

The proof of the independence result also shows that a wide class of mathematical statements, including all statements that can be phrased in the language of Peano arithmetic, are provable in ZF if and only if they are provable in ZFC.

Statements in this class include the statement that P = NP, the Riemann hypothesis, and many other unsolved mathematical problems.

When one attempts to solve problems in this class, it makes no difference whether ZF or ZFC is employed if the only question is the existence of a proof.

It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.