[throwaway587468]

The problem you’re referring to arose precisely due to lack of formalism. It was a problem back when mathematicians were toying with naive set theory, one not based on axioms, but instead on intuitive terminology. Moving to axiomatic, more formal, set theory solved it.

[auggierose]

The problem is the same, no matter if you look at it formally or informally. You could get your set theory axioms wrong, for example, but you would still be formal. Oh wait, you have a proof that set theory is consistent, right?