# Definition:Mathematical Theory

Jump to navigation
Jump to search

This page has been identified as a candidate for refactoring of medium complexity.In particular: This source comes from a very different direction (probability theory) than all the others on Definition:Theory. Determine if and how this can be aligned.Until this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

## 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 $\map \TT U$.

## Sources

- 1965: A.M. Arthurs:
*Probability Theory*... (previous) ... (next): Chapter $2$: Probability and Discrete Sample Spaces: $2.1$ Introduction