Yep, your ordered pair scenario is what I'm saying. The correct set is undecidable in that it would take an infinitely large TM to correctly identify all the p

Yep, your ordered pair scenario is what I'm saying. The correct set is undecidable in that it would take an infinitely large TM to correctly identify all the pairs, so it is not possible to characterize the set with a finite TM. The arbitrary set is labeled with a random oracle, or perhaps with an oracle that outputs HALT for every program, or it just says "I don't know." This seems coherent and to capture the notion of a PHO. All of this to point out that the fact humans cannot figure out every math problem does not mean they can be reduced to some sort of finite TM, so that standard objection against the halting oracle idea fails. Here's a proof of the idea, demonstrating a partial halting oracle can violate the law of information non-growth: https://www.am-nat.org/site/law-of-information-non-growth/ https://www.am-nat.org/site/law-of-information-non-growth/