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Kindly comments on From the "weird math questions" department... - Less Wrong
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Presumably you can't ask the oracle about the halting behavior of programs which call the oracle. Otherwise the standard proof that the halting problem is undecidable can be used to show that the oracle is fake: just have a program ask the oracle about itself, then do the opposite.
Ah, I didn't realize that. From looking over my program again, this appears to mostly be a slightly longer algorithm that does more or less what Thomblake said earlier. I didn't realize that was what I would end up with when I started trying to make one, but I guess it's a standard way of looking at it.