:::::: I am sorry for any communication gaffes- I am trying my best to be clear. Perhaps I am not using the best language, but I am saying something precise. The theorem is that (ZFC->X halts) imlies (ZF->X halts). 

 :::::: This can be proved by a textual transformation. If you are given a proof of a theorem (forall integers n  F^n(M) doesn't halt) in ZFC, where F is a function that takes a computer one processor step forward in time, and M is an integer that contains the computer's memory, you can translate the theorem to ZF. Replace every set S that occurs in every stage of the theorem with S_L, which just means the same S restricted to the Godel constructible universe. Then for every set which you used a choice function for, the L version has a choice function in ZF, so the proof goes through in L. Once the proof is finished, you end up with the theorem (forall L-integers n (F_L)^(n_L)(M_L) doesn't halt). But the L integers and the L-functions on the integers are the same as the ZF or ZFC integers, and so you've proved the same theorem. That's the translation. It works for any theorem about any absolute notions like integers and ordinals.