An axiom is an irreducible primary.
It doesn't rest upon anything in order to be valid,
and it cannot be proven by any "more basic" premises.
A true axiom can not be refuted because the act of trying to refute it requires
that very axiom as a premise.
An attempt to contradict an axiom can only end in a contradiction.
The term "axiom" has been abused in many different ways, so it is important to distinguish
the proper definition from the others.
The other definitions amount to calling any arbitrary postulate an 'axiom'.
The famous example of this is Euclidean geometry.
Euclid was a Greek mathematician who applied deductive logic
to a few postulates, which he called axioms.
In this sense, "axiom" was used to mean a postulate which one was sure was true.
Later, though, it was shown that his postulates were sometimes false, and so the
conclusions he made were equally false.
The "axiom" he used was basing his geometry on a two dimensional plane.
When his work was applied to the surface of a sphere, though, it broke down.
A triangle's three angles add up to 180 degrees on a plane; they do not add up to 180 degrees on the surface of a sphere.
The point is that Euclid's "axioms" were actually postulates.
True axioms are more solid than that.
They are not statements we merely believe to be true;
they are statements that we cannot deny without using them in our denial.
Axioms are the foundation of all knowledge.
There are only a few axioms that have been identified.
These are: Existence Exists,
The Law of Identity, and
(This page mirrored from Importance of Philosophy.com)