\begin{definition}[Definition:Mathematical Theory]
A '''mathematical theory''', or just '''theory''', is a concept in mathematical logic.
Let $U$ be a set of logical formulas.
Let $\map \TT U$ be the set of all logical formulas $P$ such that $P$ is a semantic consequence of $U$.
That is, let $\map \TT U = \set {P: U \models P}$.
Then $\TT$ is called '''the (mathematical) theory of $U$'''.
The elements of $\map \TT U$ are called theorems of $U$.
The elements of $U$ are called the axioms of $\map \TT U$.
\end{definition}