Well a consistent system that can't be computed can prove things about all Turing machines.

E.g. ZFC plus an infinite set of axioms of the sort "machine X halts" for all machines that halt. We can't compute all the axioms, but the system is perfectly consistent. We could compute the axioms if we had a halting Oracle.

It wouldn't be able to prove the halting behavior of some Turing machines that have halting oracles attached, though.