Why can't the length of a first order statement be a nonstandard number?

Question

If a first order statement could have a nonstandard length, and proofs could have nonstandard lengths, then you could prove that PA must have nonstandard numbers by proving there exists a number x=S(S(...0...)) for a nonstandard amount of S's. As PA has the natural numbers as a model this is a contradiction. So when describing the rules of first order logic, how are nonstandard length statements and proofs forbidden?