Term model

For a complete consistent theory with the Henkin witness property, the term model has closed terms modulo provable equality as elements. Functions are evaluated by forming terms, and atomic relations hold exactly when the corresponding atomic sentences belong to the theory. Provable equality makes these interpretations well defined.

A sentence with closed-term parameters holds in a term model if and only if it belongs to the complete consistent theory defining that model. Prove this by structural induction: atoms are the definition, Boolean steps use completeness and consistency, and existential sentences use the Henkin witness property. Conversely any existential witness represented by a term gives the existential sentence by logical inference.