# Definition:Theorem

## 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.

### 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 $\LL$ be a formal language.

Let $\mathscr P$ be a proof system for $\LL$.

A theorem of $\mathscr P$ is a well-formed formula of $\LL$ 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".