A homomorphic subset of a theory of logic cannot have its own interpretation that is not provably consistent with the theory of logic to which it maps. Any interpretation of the homomorphic subset must be consistent with the interpretation of the original theory, since the subset is a subset of the original theory and preserves its structure under a homomorphism.

However, it is possible for sentences to be written in the theory to which a homomorphic subset maps, that cannot be written in the homomorphic subset. This is because the homomorphic subset only includes those formulas and axioms that can be mapped to equivalent formulas and axioms in the original theory while preserving the relationships between them. Therefore, there may be formulas and axioms in the original theory that are not included in the homomorphic subset, and hence, cannot be written in the subset.

A homomorphic subset of a theory of logic cannot have its own interpretation that is not provably consistent with the theory of logic to which it maps. However, there may be formulas and axioms in the original theory that are not included in the homomorphic subset, and hence, cannot be written in the subset.