Technology
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First, there’s no “somehow magically” about it, the entire logic of the halting problem’s proof relies on being able to set up a contradiction. I’ll agree that returning undecidable doesn’t solve the problem as stated because the problem as stated only allows two responses.
My wider point is that the Halting problem as stated is a purely academic one that’s unlikely to ever cause a problem in any real world scenario. Indeed, the ability to say “I don’t know” to unsolvable questions is a hot topic of ongoing LLM research.