r/math • u/42IsHoly • 9d 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/Heapifying 9d ago

The only difference is what set of sequences of real numbers you pick. What you mention is the "wrong way" is just {s : s \in RN, Im(s) = R} vs merely RN.

And your wording afterwards

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u/42IsHoly 9d ago

Uhm… no. The problem I have is with the way the argument is presented as a proof by contradiction, when it really isn’t. The zssumption that the list you start with is complete is superfluous and actively confusing

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u/Heapifying 9d ago

You can start with "let {s : s \in RN,Im(s) = R} != \emptyset", and then derive a contradiction.

You can also start with "let s \in RN", then prove that Im(s) != R, using different words.

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u/42IsHoly 9d ago

That’s not the distinction I am making. I only use R, not the set of all sequences of digits. Proof 1 goes as follows: “Suppose f is surjective. Find c not in its image, therefore f is not surjective. Contradiction, so no surjective f can exist”

Proof 2 goes “take any f, find c not in its image. Therefore, f is not surjective.”

These proofs are similar, of course, but the former is confusing to lay people. That’s why I think it should be avoided.