# Definition:Theorem

## Contents

## General Definition

In all contexts, the definition of the term **theorem** is by and large the same.

That is, a **theorem** is a statement which has been proved to be true.

## Mathematics

The term **theorem** is used throughout the whole of mathematics to mean a statement which has been proved to be true from whichever axioms relevant to that particular branch.

Note that statements which are taken as axioms in one branch of mathematics may be theorems in others.

It is possible (and this is the ultimate aim of $\mathsf{Pr} \infty \mathsf{fWiki}$) to justify basing the whole of mathematics on a handful of axioms.

## Logic

A **theorem** in logic is a statement which can be shown to be the conclusion of a logical argument which depends on *no* premises except axioms.

A sequent which denotes a theorem $\phi$ is written $\vdash \phi$, indicating that there are no premises.

In this context, $\vdash$ is read as:

**It is a theorem that ...**

## Formal Systems

Let $\mathcal L$ be a formal language.

Let $\mathscr P$ be a proof system for $\mathcal L$.

A **theorem of $\mathscr P$** is a well-formed formula of $\mathcal L$ which can be deduced from the axioms and axiom schemata of $\mathscr P$ by means of its rules of inference.

That a WFF $\phi$ is a **theorem** of $\mathscr P$ may be denoted as:

- $\vdash_{\mathscr P} \phi$

## Also defined as

Some sources make more of this term than is perhaps merited.

For example, Gary Chartrand: *Introductory Graph Theory* has:

*Theorems are true implications which are usually of special interest.*

No distinction is made on $\mathsf{Pr} \infty \mathsf{fWiki}$ between theorems that are or are not of "special interest".

## Also see

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S 1.1$: Constants and variables - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs