The diagonolisation technique is utilized to prove that the halting problem is undecidable. However, I kind of sense that it is making the assumption that decidable sets should be countable. Is this true? If not, which part am I missing?
2026-04-01 12:43:44.1775047424
decidability and countability
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The original proof of the undecidability of the halting problem assumes that there exists a program that can determine whether any other program halts and then explicitly constructs a counterexample that the program will give the wrong answer for. No assumptions of countability are necessary for this. However, there are only countably many computable sets because there are only countably many programs of finite length.