Comment by davrosthedalek

I am not sure I follow? But maybe I misunderstood your quote. It seems any given TM (And for a fixed N, there are only finite many), there are only the two option: it halts, or it doesn't halt. There is no third option? Of course, the question whether you can proof that it doesn't halt, in ZF or ZFC, is a different question. (You can always proof that it halts if you know that it halts. You run it until it halts)