# Definition:Axiom

## Definition

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

That is, an axiom is a statement which is accepted as being true.

A statement that is considered an axiom can be described as being axiomatic.

### Logic

An axiom in logic is a statement which is taken as self-evident.

Note, however, that there has been disagreement for as long as there have been logicians and philosophers as to whether particular statements are true or not.

For example, the Law of Excluded Middle is accepted as axiomatic by philosophers and logicians of the Aristotelian school but is denied by the intuitionist school.

### Formal Systems

Let $\mathcal L$ be a formal language.

Part of defining a proof system $\mathscr P$ for $\mathcal L$ is to specify its axioms.

An axiom of $\mathscr P$ is a well-formed formula of $\mathcal L$ that $\mathscr P$ approves of by definition.

### Mathematics

The term axiom is used throughout the whole of mathematics to mean a statement which is accepted as true for that particular branch.

Different fields of mathematics usually have different sets of statements which are considered as being axiomatic.

So statements which are taken as axioms in one branch of mathematics may be theorems, or irrelevant, in others.

## Also known as

An axiom is also known as a postulate.

Among ancient Greek philosophers, the term axiom was used for a general truth that was common to everybody (see Euclid's "common notions"), while postulate had a specific application to the subject under discussion.

For most authors, the distinction is no longer used, and the terms are generally used interchangeably. This is the position of $\mathsf{Pr} \infty \mathsf{fWiki}$.

However, some believe there is a difference significant enough to matter:

... we shall use "postulate" instead of "axiom" hereafter, as "axiom" has a pernicious historical association of "self-evident, necessary truth", which "postulate" does not have; a postulate is an arbitrary assumption laid down by the mathematician himself and not by God Almighty.
-- 1937: Eric Temple Bell: Men of Mathematics: Chapter $\text{II}$: Modern Minds in Ancient Bodies