Actually, if understand Kurt Godel correctly (one of the most significant logicians of all time), he showed that a combination of simple self-evident axioms demands that we acknowledge the existence of true ‘supernatural’ theorems whose truth can never be proven directly. If true, Godel has demonstrated that science can indeed study certain aspects of the supernatural. And as far as I know, Godel has not been disproven.
Sure it can. Ghost hunters use thermal vision and thermometers to look for ghosts.
But, since there is no such thing ghosts, they have yet to come up with any scientific proof.
No, he didn't.