In If P=NP, could we obtain proofs of Goldbach's Conjecture etc.? it talks about the hypothetical world where P=NP and using the proof of it to prove a problem/theorem assuming that it has a short proof.
But what exactly is the restriction imposed on problems/theorems? Do they have to be describable in $\Pi_1$ to use Levin's universal search?
If so, can the question like Navier-Stokes existence and smoothness problem be statble in $\Pi_1$?