 This result is guaranteed by the assumption of a fixed domain of quantification. Carnap proves that MFC is sound, that is, all its theorems are L-true, and raises the question of completeness for both FC and MFC. Gödel proved completeness for the first order predicate calculus with identity, but the notion of validity employed was truth in every non-empty domain of quantification, including finite domains. Carnap instead adopts one unique denumerable domain of quantification. The adoption of a fixed denumerable domain of individuals generates some additional validities already at the pre-modal level which jeopardize completeness, for example “There are at least two individuals”, \((\exists x)(\exists y)(x\ne y)\), turns out to be valid (1946: 53).