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 $\map \TT U$.

Bourbaki Definition
The definition according to  is as follows:

The signs of a mathematical theory $\TT$ are:
 * $(1) \quad$ The logical signs: $\Box, \tau, \vee, \rceil$.
 * $(2) \quad$ The letters: uppercase and lowercase Roman letters, with or without accents, e.g. $A, A', A''$.
 * $(3) \quad$ 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.