> "x is in a non-empty subset of S" could be _expressed_ as a FOL language > expression: x e S' /\ Ay[ y e S' -> y e S]. > > On the other hand, in "x is proven to be in a non-empty subset of S", > the _meta phrase_ "is proven" can not be expressed by a FOL language: > "is proven" pertains to a meta truth, which in turns can't be equated > to a language expression: truth and semantics aren't the same.
"x is in a non-empty subset of S" can be expressed in the language of a first order theory with a binary predicate e. The intended meaning of e is given by the non-logical axioms of that theory.
What reason is there to suppose that "x is proven" cannot be expressed in the language of a first order theory with a unary predicate p (say)? The intended meaning of p would then be given by the non-logical axioms of that theory.
Note that set theory can express its own provability predicate.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting