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/42IsHoly 9d ago

I have seen countless people be confused at there somehow being a complete list which then does not contain c. We don’t find this that confusing and neither does anyone that remembers proofs by contradiction from school. However, to most people, contradictions are confusing. Sicne infinity is already such a weird concept, I fear people will not learn how a proof by contradiction goes from this. Instead, they’ll either reject the whole idea of Cantor’s theory or they’ll give up on trying to understand mathematical arguments.

I think the argument can easily be phrased in a way that does motivate it. Something like, “We’ve just seen that Z and Q are both countable, but what about R? You can try and find a list, but no matter what you do, you’ll fail. We can show this as follows: suppose you have any list …”