Definition:Theory (Logic)
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Definition
Let $\mathcal{L}$ be a language.
An $\mathcal{L}$-theory $T$ is a set of $\mathcal{L}$-sentences.
The $\mathcal{L}$-theory of an $\mathcal{L}$-structure $\mathcal{M}$ is the $\mathcal{L}$-theory consisting of those $\mathcal{L}$-sentences $\phi$ such that $\mathcal{M}\models \phi$.
This theory is often denoted $\operatorname{Th}(\mathcal{M})$ when the language $\mathcal{L}$ is understood.
We say $T$ is complete if for every $\mathcal{L}$-sentence, either $T\models\phi$ or $T\models\neg\phi$.
We say $T$ is maximal if for every $\mathcal{L}$-sentence, either $\phi \in T$ or $\neg\phi \in T$.
