# Definition:Axiom

## Contents

## 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.

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.

## 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

- -- 1937: Eric Temple Bell:

## Also see

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Axioms - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.) - 1993: M. Ben-Ari:
*Mathematical Logic for Computer Science*(1st ed.) ... (previous) ... (next): $\S 1.2$: Propositional and predicate calculus - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic Of Shape