Consistency and syntactic completeness

Fix a set signature $L$ and sentence theory $T$, with derivability as in Formal proofs from sentence theories. Fix the sentence $\mathrm{\perp }:=\mathrm{\exists }{v}_0\mathrm{\neg }(v_0=v_0)$.

The theory $T$ is consistent when $T⊬\mathrm{\perp }$. It is syntactically complete when for every $L$-sentence $\sigma$, $T⊢\sigma$ or $T⊢\mathrm{\neg }\sigma$. Deductive closure of a sentence theory means closure under its sentence consequences; it does not include open formulas as members.

If $L\subseteq L^{\mathrm{\prime }}$ and $T\subseteq U$ are sentence theories in their respective signatures, $U$ is conservative over $T$ when every $L$-sentence provable from $U$ is provable from $T$. The reverse preservation follows by retaining the same derivation in the expansion. These are syntactic definitions, including when $T$ is empty; no model existence is asserted.