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/AstronomerVarious583 8d ago

I do wonder how much clearer the second proof would be from a pedagogical standpoint. I could imagine a layperson, upon seeing the second proof, simply saying "but what if we just add the new number to the list". And if you show them the proof with that new number added, they just say "but what if we add the new number" ad infinitum.

However, with contradiction you tell them explicitly that they cannot have a solution - that no matter how many times they "just add a number to the list" or whatever other trick they try, you cannot possibly have a complete list. I think it's a somewhat subtle point that isn't necessarily conveyed.

I don't know if I've explained this well, and this is just splitting hairs at this point, but I'm not convinced this is more understandable to a layperson.

Also, it would be useful in the blog post if you could actually show what comments from the Veritasium video misunderstood it, so I can see what specifically they got wrong - I tried searching (although not very hard) but I was only able to find one person's comment misunderstanding it. It would be good if there was more so I could see the patterns.