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 $\mathcal T \left({U}\right)$ be the set of all logical formulas $P$ such that $P$ is a logical consequence of $U$.

That is, let $\mathcal T \left({U}\right) = \left\{{P: U \models P}\right\}$.

Then $\mathcal T$ is called the (mathematical) theory of $U$.

The elements of $\mathcal T \left({U}\right)$ are called theorems of $U$.

The elements of $U$ are called the axioms of $\mathcal T \left({U}\right)$.

Bourbaki Definition
The definition according to Bourbaki's is as follows:

The signs of a mathematical theory $\mathcal T$ are:
 * 1) The logical signs: $\Box, \tau, \vee, \rceil$.
 * 2) The letters: uppercase and lowercase Roman letters, with or without accents, e.g. $A, A', A''$.
 * 3) The specific signs which depend on the theory under consideration.

A mathematical theory also contains:
 * a series of rules which lets us determine whether particular assemblies are either terms or relations of the theory;
 * another series of rules which lets us determine whether particular assemblies are theorems of the theory.