(Edited:) I think you're conflating the natural numbers and the tape that the Turing machine runs on. Interpreting "nonstandard halting time," the way I think Eliezer is using the term, doesn't require changing our notion of what a tape is; it just requires translating the statement "this Turing machine is in state s at time t" into a statement in Peano arithmetic (where t is a natural number) and then interpreting it in a nonstandard model.