Definition 2.2.3 (Recursive / decidable language). A language $L$ is recursive if there is an algorithm that decides whether a string of symbols is an $L$-formula.

An $L$-theory $T$ is recursive if membership in $T$ is decidable (for $L$-sentences).

An $L$-theory $T$ is decidable if there is an algorithm for deciding whether $T\models \phi$.