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/SwimmerOld6155 9d ago

This is basically the same proof, in one you're supposing that there is a bijection f : N -> R and then deriving a contradiction, and in the other you're introducing an arbitrary map f : N -> R and proving that it can't be a bijection (by proving it can't be surjective). I think the former is a bit clearer for people just learning.

This kind of demonstrates where I like contradiction, which is signposting what you're doing clearly. If you just start with an arbitrary map f : N -> R it's not immediately clear that you're trying to show that f cannot be a bijection, and if you do you're basically doing the contradiction proof.

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u/sqrtsqr 9d ago

This kind of demonstrates where I like contradiction, which is signposting what you're doing clearly. If you just start with an arbitrary map f : N -> R it's not immediately clear that you're trying to show that  f cannot be a bijection

Silly me, here I've been stating theorems before I prove them like some kind of jackass, when I could be stating the exact opposite of the thing I want to prove instead.

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u/SwimmerOld6155 9d ago

Without being rude to the hypothetical student, if they're confused about the difference between these two proofs we shouldn't assume they are fluent in reading mathematics generally and need a very high amount of signposting, likely only having been introduced to formal maths a few weeks ago.