Zero is the answer to the question you asked. Because, unless you change the customary axioms of arithmetic, [math](1=2) [/math] is not provable. It could be true, for example, if the successor function were idempotent, i.e., [math]\forall n\in\mathbb{N} (n'' = n')[/math].

Infinitely many is the answer to the question you intended to ask. (But most of them will be very uninteresting, so you probably want to see some interesting, cleverly disguised cases.)