I have a naive question: does there exist a Turing machine whose termination is true but unprovable by any natural, consistent and finitely axiomatizable theory? I ask for a mere existence proof rather than a specific example. This might have some connection with ordinal analysis. Indeed, for a Turing machine $M$, we can define $O(M)$ as the least ordinal of a consistent theory proving its termination (or the infimum of these ordinals). So I guess it would be equivalent to ask whether there exists $M$ such that $O(M) \geq \omega_1^{CK}$?
Turing machines whose termination is unprovable?
turing machines;proof theory;halting problem