There's no way to rule out a priori that the Jacobian conjecture is undecidable (in your favorite axiomatic system).
As I pointed out in my answer to another MO questionanother MO question, a proof that some statement is decidable would automatically mean that its status would be resolved up to a finite computation. You can be sure that if a conjecture as important as the Jacobian conjecture were reduced to a finite computation, then we would hear about it. This observation yields an informal "proof" that (almost) all the major open problems you care to name are not known to be decidable.
