[tarre]

I still remember Kaisa telling in primary school, that Euclid's proof of infinite number of prime numbers is often considered as the most beautiful proof in mathematics.

[osivertsson]

Interesting, can you give more context on when and where this was?

[tarre]

We were both in a very small school in the countryside and when I was on 5th and she on 6th grade, we had wonderful teacher, Harri Ketamo (Google him and you'll end up wondering, what on earth he was doing in that school). Harri was especially interested in teaching mathematics. As there was only about ten pupils in the class, he had possibility to teach more advanced pupils further covering among other things programming and prime numbers.

I don't remember if it was in school or after school, when we had with Kaisa a debate over whether or not there was infinite number of prime numbers. The next day or so she presented, that it has been proven and here it is. Back in those days I didn't get it completely, but she was already there.

[soberhoff]

I personally prefer Cantor's diagonal argument. Especially when considering how many useful results have come from it.

[jacobolus]

Can you define “useful”? Arguably anything requiring infinite steps is essentially a pure thought experiment with no physical-world ramifications.

Any mathematical theory we can apply to physical-world science/engineering can be recast to not have anything to do with Cantor’s set theory.

But maybe you mean useful to making analysis proofs... in which case fair enough.

[soberhoff]

I think infinities also have use in the real world. Take the insolubility of the quintic. You can reword this as saying that none of the infinitely many candidate solutions for quintics work. Its real world utility is straightforward: there's no point to continue looking for a solution.

[ice109]

every result about undecidability uses Cantor's diagonalization.

[zem]

millay would agree: https://genius.com/Edna-st-vincent-millay-euclid-alone-has-l...