Definition:Mathematical Theory

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.