Earlier quoted context omitted.
This "proof" seems to be popular, and it always bothers me that it's invalid. It uses self-reference in an invalid way. Allow me to formalize; we take as a rigorous definition of an "interesting number" that a number has a unique property. Specifically, a number n is interesting if there is some predicate P(x) which is true only for n. In formal first order logic, n is interesting if there exist a predicate P and a n…
> Specifically, a number n is interesting if there is some predicate P(x) which is true only for n. Well, if you read the "every number is interesting" "proof", this clearly doesn't capture the proof's criteria of interestingness. I see it as analogous to Berry's paradox - the proof isn't "wrong" per se, but the relevant notion of interestingness is not well defined
You can express whatever idea of "interestingness" you like in this framework by finding a predicate that expresses it.