a self-evident truth that requires no proof.
a universally accepted principle or rule.
Logic, Mathematics. a proposition that is assumed without proof for the sake of studying the consequences that follow from it.

1475–85; < Latin axiōma < Greek: something worthy, equivalent to axiō-, variant stem of axioûn to reckon worthy + -ma resultative noun suffix

1. adage, aphorism, apothegm, axiom, maxim, proverb ; 2. assumption, axiom, premise, presumption. Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2014.
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World English Dictionary
axiom (ˈæksɪəm)
1.  a generally accepted proposition or principle, sanctioned by experience; maxim
2.  a universally established principle or law that is not a necessary truth: the axioms of politics
3.  a self-evident statement
4.  logic, maths Compare assumption a statement or formula that is stipulated to be true for the purpose of a chain of reasoning: the foundation of a formal deductive system
[C15: from Latin axiōma a principle, from Greek, from axioun to consider worthy, from axios worthy]

Collins English Dictionary - Complete & Unabridged 10th Edition
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Word Origin & History

late 15c., from M.Fr. axiome, from L. axioma, from Gk. axioma "authority," lit. "that which is thought worthy or fit," from axioun "to think worthy," from axios "worthy, worth, of like value, weighing as much," from PIE adj. *ag-ty-o- "weighty," from base *ag- "to drive, draw, move" (cf. Gk. agein "weigh,
Online Etymology Dictionary, © 2010 Douglas Harper
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American Heritage
Science Dictionary
axiom   (āk'sē-əm)  Pronunciation Key 
A principle that is accepted as true without proof. The statement "For every two points P and Q there is a unique line that contains both P and Q" is an axiom because no other information is given about points or lines, and therefore it cannot be proven. Also called postulate.
The American Heritage® Science Dictionary
Copyright © 2002. Published by Houghton Mifflin. All rights reserved.
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American Heritage
Cultural Dictionary
axiom [(ak-see-uhm)]

In mathematics, a statement that is unproved but accepted as a basis for other statements, usually because it seems so obvious.

Note: The term axiomatic is used generally to refer to a statement so obvious that it needs no proof.
The American Heritage® New Dictionary of Cultural Literacy, Third Edition
Copyright © 2005 by Houghton Mifflin Company.
Published by Houghton Mifflin Company. All rights reserved.
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Computing Dictionary

AXIOM definition

A commercially available subset of the Scratchpad, symbolic mathematics system from IBM.
["Axiom - The Scientific Computing System", R. Jenks et al, Springer 1992].
[Relationship with AXIOM*?]

axiom definition

A well-formed formula which is taken to be true without proof in the construction of a theory.
Compare: lemma.
The Free On-line Dictionary of Computing, © Denis Howe 2010
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Example sentences
If we omit this last axiom, the remaining axioms give either Euclidean or
  hyperbolic geometry.
Forget the old axiom it's not whether you win or lose, it's how you play the
It used to be an axiom that solar power grew steadily cheaper as time passed.
Another axiom of science is dispassionate observation.
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