I always feel like these frameworks rely on a semantic sleight of hand that sounds plausible on the surface, but when you drill down a bit they render words like 'simulation' 'reality' or 'truth' as either unintelligible or trite, depending on how you define them.

They're defined relative to the axioms. In this case he is using the standard arithmetic & set theoretic constructions to define the terms & functions he's talking about. It's logically sound, whether it makes physical sense or not is another matter.