[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.