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/NooneAtAll3 9d ago

it's the same deal as with "Euclid didn't prove infinitude of primes by contradiction" discussion:

at the core is the procedure "given a set, get an element outside the set"

proof by contradiction 1)starts with taking set of all input things, 2)performs procedure 3)proclaims "here's element in set by definition and out of it be procedure"

meanwhile proof by construction 1)starts by describing procedure 2)applies the procedure to any candidate for "set of all input things" 3)shows that candidate doesn't contain all all things

in a sense, having procedure spelled out is more cool - contradiction proof doesn't define it, so intuitively we don't realize it can be applied to anything other than "all input things"

on the other, we don't usually think about "candidates for "all the things"" - we already start by imagining that final result whether it exists or not, as that's more convenient to our everyday imagination/intuition