Baughn comments on The genie knows, but doesn't care - Less Wrong
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Moreover, the halting problem doesn't show that the set of programs you can't decide halting for is in any way interesting.
It's a constructive proof, yes, but it constructs a peculiarly twisted program that embeds its own proof-checker. That might be relevant for AGI, but for almost every program in existence we have no idea which group it's in, and would likely guess it's provable.