[Epistemic status: a little out of my depth. There might be subtleties I’m missing.]
An oracle machine with a halting oracle is a type of hypercomputer that can “solve” (by fiat, the known laws of physics do not permit such a thing) the halting problem of any conventional Turing machine, but then an analogous oracle-machine halting problem would appear which is undecidable by these halting oracle machines, so this doesn’t get rid of undecidable problems.
If we then suppose a second-order halting oracle, we can “solve” the oracle-machine halting problem, but then a new, harder, second-order oracle-machine halting problem would appear, and so on up to any order, in the arithmetical hierarchy.
Thus, we can never solve all undecidable problems, even with hypercomputers. Now, is your proposed hypercomputer model of equivalent power to a halting oracle machine? To the extent it is a well-defined model, I think so.
[Epistemic status: a little out of my depth. There might be subtleties I’m missing.]
An oracle machine with a halting oracle is a type of hypercomputer that can “solve” (by fiat, the known laws of physics do not permit such a thing) the halting problem of any conventional Turing machine, but then an analogous oracle-machine halting problem would appear which is undecidable by these halting oracle machines, so this doesn’t get rid of undecidable problems.
If we then suppose a second-order halting oracle, we can “solve” the oracle-machine halting problem, but then a new, harder, second-order oracle-machine halting problem would appear, and so on up to any order, in the arithmetical hierarchy.
Thus, we can never solve all undecidable problems, even with hypercomputers. Now, is your proposed hypercomputer model of equivalent power to a halting oracle machine? To the extent it is a well-defined model, I think so.
Specifically, I was trying to make a reflective oracle, shown here:
https://intelligence.org/2015/04/28/new-papers-reflective/
So it’s a non-standard oracle I’m trying to make.