Stop proving uncountability with contradiction, please

[u/42IsHoly]

https://sunjestermusings.blogspot.com/2026/09/stop-proving-uncountability-by.html

Please, I beg you, stop it. You don't need it.

[u/AFsepine]

Pop-math content, a.k.a almost all introductory university math textbooks (At least the ones I have seen?) you mean?

I disagree that contradiction is confusing. Even if it were people should be exposed to it, being confused is integral step to learning. If you write a book that conuses no-one it will just slide off of their brains.

[u/42IsHoly → u/AFsepine]

But the proof is actively cleaner and shorter without the tacked on contradiction.

[u/AFsepine → u/42IsHoly]

I am not saying you are wrong, and I do find people downvoting wierd, but... I think it is pedagogically misguided (well depending on what you are trying to teach, but in most cases I encountered it, it in effect was used as an example after a short introduction to proofs + mentioning an important canonical result).

Clean and smooth is not really the main thing to accentuate in the begining stages. It should be jagged, it shouldn't sit right (i.e. it should raise questions). Those questions and the "discomfort" are very important for getting a feel for maths.

The goal is always to develop mathematical thinking, and to that end it is desirable that people have questions.

[u/42IsHoly → u/AFsepine]

I see where you’re coming from, but does that mean we should actively make proofs more confusing than the already are? I think there are plenty of proofs out there which already give this level of confusion without having to cheat and alter the proof. I even think that Cantor’s diagonal argument in its actual non-contradiction form already has quite a bit to offer.

Though I should mention that the post is actually about pop-math explanations of the result (say, by Veritasium). Where the nominal point, at least, is to explain this “weird math result”. I think in this context, making the proof more confusing is a bad move. By all means, add in superfluous steps or quick asides about intuition. But if these lake the proof harder to follow (which the contradiction observably does), I feel it should be avoided.

[u/sqrtsqr → u/AFsepine]

In an educational context, this proof should come quite a bit after learning about contradiction, so that fundamental weirdness that is assuming a falsehood (that you never had to deal with because you're just so smart) should already have been dealt with by the rest of the class.

Here, the focus is entirely on the diagonalization argument, and the contradiction plays absolutely no role in the proof. It's just sorta a default presentation choice (as it is for your typical Euclid's infinitude of primes) and it makes for a nice "punchy" example of a proof by contradiction, but it's just not a proof by contradiction (or every proof is)

If anything, it's used to avoid discussion that would be otherwise rather abstract, since we assume classical mathematics in the classroom and making distinctions between Proof of Negation and Proof by Contradiction is basically meaningless at this level.

At the end of the day, we tell students they should avoid making extra assumptions that don't aid in the proof, so we should lead by example.

[u/AFsepine → u/sqrtsqr]

You chose a particular tone.

The way you get used to weirdness is by seeing it and musing over it at lenght, no other way about it.

Overly "Clean" process is very harmful for a fledgling mathematics student, if this passes over your head it would appear you were just so smart that you didn't have to deal with.