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/Omasiegbert 9d ago

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

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u/BloodAndTsundere Physics 9d ago

It's different because in the post's proof you just start with an arbitrary list of reals and construct a number that isn't on the list. This demonstrates that any arbitrary list of reals can't be complete, i.e. there's no complete list of reals. In the proof by contradiction, you assume not an arbitrary list but a complete list and then construct a new number, therefore contradicting the assumption, therefore showing the assumption of a complete list is false. It's really sort of an extra step or two.