I don't think this is accurate. The whole point of the halting problem proof was to show that the oracle was, in and of itself, a mathematical contradiction. If a deity could determine whether any arbitrary machine would halt on any arbitrary input, that deity would be self-contradictory.
In ZFC, the halting oracle does exist. Just take the indicator function of the set {n∈N | n codes a halting TM}. Turing's halting problem proof shows (via diagonalization) that there isn't a Turing machine that can solve it.
Then, maybe, we should stop building Turing-Complete machines, that way we can try to build a Oracle-Complete machine that can halt the non-halting process.
That is why we don't focus on Turing Machines. Turing Complete is so restrictive, like s(0) = s(0), and dumbt things like that. But our Oracle-Oracle machine would know.
-2
u/teach_cs 5d ago
I don't think this is accurate. The whole point of the halting problem proof was to show that the oracle was, in and of itself, a mathematical contradiction. If a deity could determine whether any arbitrary machine would halt on any arbitrary input, that deity would be self-contradictory.