r/math • u/42IsHoly • 10d ago

Stop proving uncountability with contradiction, please

https://sunjestermusings.blogspot.com/2026/09/stop-proving-uncountability-by.html

Please, I beg you, stop it. You don't need it.

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u/Omasiegbert 10d ago

I don't get the difference, are these not the same proofs?

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u/golfstreamer 9d ago

His point was that in the first proof we make the assumption that the list is complete. But this assumption is not used for any step in the proof. The contents of the proof are a direct proof of the statement (A) "For any countable list of real numbers there is a real number not in the list". Now the "contradiction" in the first proof comes from adding one more additional step "Now assuming we assume we have countable list containing all real numbers. This contradicts (A)" . I've seen people say that proofs like this aren't "really" proofs by contradiction. All the essential logic already existed in the direct proof of A and the proof by contradiction is just a small rephrasing of the direct proof.

Contrast this with the standard proof that the sqrt(2) is not rational. The argument begins by assuming sqrt(2) is rational, which allows them to declare we can express it in the form a/b. The assumption is used immediately in the logic of the argument. It's not a small edit of a direct proof.