Suppose someone comes to you and claims to have a halting oracle. Is there any way for you to verify the truth or falsity of their claim in finite time? If so, what constraints on the proof process are there? Does the verification have to be interactive? Can you only prove it to a given probabilistic bound?
Update: Henning, below, suggested an oracle $A_F$ that will say "Halt" unless the TM in question can be proved to have infinite run-time by some formal system $F$. He then claimed that one cannot tell this oracle from a true halting oracle. I am not certain of this; I suspect there may be some sequence of questions one can ask this oracle to trip it up in a lie. Can anyone prove or disprove this statement?