  For example, when L is the first-order  language  L of arithmetic, Gödel originally used as coding structure the standard model of arithmetic ℕ and as coding map the well-known function obtained from the prime factorization theorem for natural numbers. The recursive enumerability of  Val(L)  then means simply that the set of codes (“Gödel numbers”) of members of  Val(L) is definable in  ℕ  by an  L-formula  of the form ∃yφ(x, y), where φ(x, y) is a recursive formula.