# Definition:Mathematical Theory

## Definition

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 $\map \TT U$.