Let us consider Turing machines (or other Turing-complete model of computation) that, in addition to their regular input, are given some integer $H$, where $H$ is positive nonstandard. This means, in particular, that $H$ is greater than any standard integer. What would be the Turing degree of such a machine?

The machine can solve the halting problem for standard turing machines. To see this, just run a machine for $H$ steps. If it halts before $H$ steps, the machine halts, of course. If it has not halted by then, it never will (since a halting standard turing machine always halts after a standard number of steps).

To be more specific, when I say the Turing degree of such a machine, I mean the machine with an oracle that can simulate such a machine on any standard input (it may diverge on nonstandard input). We also only consider them able to solve decision problems.

Note: When I talk about nonstandard integers, I have something like [this][1] in mind.




  [1]: https://en.wikipedia.org/wiki/Hyperinteger?wprov=sfla1