# Definition:Mathematical Theory

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## 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.

## Sources

- 1965: A.M. Arthurs:
*Probability Theory*... (previous) ... (next): Chapter $2$: Probability and Discrete Sample Spaces: $2.1$ Introduction - 1968: Nicolas Bourbaki:
*Theory of Sets*... (previous) ... (next): Chapter $\text I$: Description of Formal Mathematics: $1$. Terms and Relations: $1$. Signs and Assemblies