The lean proof seens correct, I can't read lean well, but it appears to just be a formalization of the standard proof. It at least constructs the set needed for the proof.
>Proof assistants use much more powerful systems in order to gain the same expressibility as the natural language which is used in informal proofs. (Which is kind of strange: In using natural language mathematicians don't seem to be using some advanced type theory.)
Look up "Curry-Howard isomorphism".
>Overall, this highlights a major problem with formalization of existing proofs. There is no way (at least no obvious way) to "prove", that a formal proof X actually is a formalization of some informal proof Y.
And? This seems not relevant to the question at hand. Surely some textbook contains an insufficient proof, which couldn't be formalized without more steps. Why does that matter. The question still is whether "folk theorems" are real. Certainly Cantor's theorem is no such folk theorem, as you have just been given a fully formal proof.
>Though they, unlike Cantor, don't talk about real numbers here
Cantors theorem usually refers to the existence of uncountable sets, which can be proven by showing that there is no bijection between N and R, but also by showing that there is no bijection between a set and its powerset.
>Every natural number sequence (with a decimal point) corresponds to some real number, but real numbers don't necessarily have just one decimal expression, e.h. 0.999...=1. So real numbers and sequences can't straightforwardly be identified.
This is only relevant if you say that the real numbers are "infinte strings of digits", which is not a formal definition and therefore not used in mathematics.
