# 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 Bourbaki's Theory of Sets is as follows:

The signs of a mathematical theory $\mathcal T$ 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.