In https://cs.stackexchange.com/a/45524/28999, they do a binary search using an oracle for an NP-Complete problem. They show that the original problem can be reduced to that NP-Complete problem, implying that the original problem is in NP.
Why is that allowed? Why are you allowed to continue after the oracle returns false? Doesn't this defeat the whole idea of NPC versus co-NPC? You could then simply use an oracle for a NP-complete problem to show that NPC = co-NPC?
I would think that the whole reduction is wrong and that the original problem is probably not in NP. Or aren't they implying that?