Definition:Axiom

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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.


Logical Axiom

A logical axiom is an axiom which does not stand in the context of a wider subject matter.

That is, it is a statement which is considered 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.


Nonlogical Axiom

An axiom which deals with some specific subject matter is known as a nonlogical axiom.

Hence it is a statement which is accepted as true for a particular branch of mathematics.


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.


Formal Systems

Let $\LL$ be a formal language.

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


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


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


Also see

  • Results about axioms can be found here.


Linguistic Note

The usual plural form of axiom is axioms.

However, the form axiomata can also sometimes be found, although it is sometimes considered archaic.


Sources

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