The Smooth 4-dimensional Poincare conjecture ($S4PC$) states that any closed smooth 4-dimensional manifold which is homeomorphic to $S^4$ will be diffeomorphic to $S^4$.
My question is wether this statement could in principle be independent of the usual ZFC set theory, just like the continuum hypothesis.
The same question has been asked for the Goldbach conjecture (In Peano arithmetic) and the Riemann hypothesis (In ZFC), where the answer was (If I understood correctly): It is in both cases possible that they are independent of Peano arithmetic/ZFC, but in that case they would both be true in the standard models of Peano arithmetic/ZFC. Is the same thing true for $S4PC$?
I would guess that if we could translate $S4PC$ into a statement involving natural numbers then it's the same as for the Riemann hypothesis. But I do not know if that is possible. Maybe someone does?