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Axiom
 For the algebra software named Axiom, see Axiom computer algebra system. For the 1970s Australian rock music group, see Axiom (band) .
In epistemology, an axiom is a selfevident truth upon which other knowledge must rest, from which other knowledge is built up. Not all epistemologists agree that any axioms, understood in that sense, exist.
In mathematics, axioms are not selfevident truths. They are of two different kinds: logical axioms and nonlogical axioms. Axiomatic reasoning is today most widely used in mathematics.
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Etymology
The word axiom comes from the Greek word αξιωμα (axioma), which means that which is deemed worthy or fit or that which is considered selfevident. The word comes from αξιοειν (axioein), meaning to deem worthy, which in turn comes from αξιος (axios), meaning worthy. Among the ancient Greek philosophers an axiom was a claim which could be seen to be true without any need for proof.
Mathematics
In the field of mathematical logic, a clear distinction is made between two notions of axioms: logical axioms and nonlogical axioms.
Logical axioms
These are formulas which are valid, i.e., formulas that are satisfied by every model (a.k.a. structure) under every variable assignment function. More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values.
Now, in order to claim that something is a logical axiom, we must know that it is indeed valid. That is, it might be necessary to offer a proof of its validity (truth) in every model. This might challenge the very classical notion of axiom; this is at least one of the reasons why axioms are not regarded as obviously true or selfevident statements.
Logical axioms, being mere formulas, are devoid of any meaning; but the point is that when they are interpreted in any universe, they will always hold no matter what values are assigned to the variables. Thus, this notion of axiom is perhaps the closest to the intended meaning of the word: that axioms are true, no matter what.
Examples
An example, used in virtually every deductive system , is the:
Axiom of equality.
In this example, for this not to fall into vagueness and a neverending series of "primitive notions", either a precise notion of what we mean by "=" (or for all what matters, "to be equal") has to be well established first, or a purely formal and syntactical usage of the symbol "=" has to be enforced  and mathematical logic does indeed that, properly delegating the meaning of "=" to axiomatic set theory.
Another, more interesting example, is that of:
Axiom of universal instantiation. Given a formula in a first order language , a variable and a term that is substitutable for in , the formulais valid.
This axiom simply states that if we know for some property , and is particular term in the language (i.e., it stands for a particular object in our structure), then we should be able to claim .
Likewise, we have the:
Axiom of existential generalization. Given a formula in a first order language , a variable and a term that is substitutable for in , the formulais valid.
Nonlogical axioms
Nonlogical axioms are formulas that play the role of theoryspecific assumptions. Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the nonlogical axioms aim to capture what is special about a particular structure (or set of structures, such as algebraic groups). Thus nonlogical axioms, unlike logical axioms, are not tautologies. Another name for a nonlogical axiom is postulate.
Almost every modern mathematical theory starts from a given set of nonlogical axioms, and it was thought that in principle every theory could be axiomatized in this way and fomalized down to the bare language of logical formulas. This turns out to be impossible and proved to be quite a story.
This is the role of nonlogical axioms, they simply constitute a starting point in a logical system. Since they are so fundamental in the development of a theory, it is often the case that they are simply referred to as axioms in the mathematical discourse, but again, not in the sense that they are true propositions nor as if they were assumptions claimed to be true. For example, in some groups, the operation of multiplication is commutative; in others it is not.
Thus, an axiom is an elementary basis for a formal logic system that together with the rules of inference define a deductive system.
Examples
Arithmetic, Euclidean geometry, linear algebra, real analysis, topology, group theory, set theory, projective geometry, symplectic geometry, von Neumann algebras, ergodic theory, probability, etc. All these theories are based on their respective set of nonlogical axioms.
Arithmetic
In all this formalism, the Peano axioms constitute the most widely used axiomatization of arithmetic; these are a set of nonlogical axioms strong enough to prove several relevant facts of number theory and they allowed Gödel to establish his second incompleteness theorem
The language is where is a constant symbol and is a unary function. The postulates are:
 for any formula with one free variable.
There is a standard structure is where is the set of natural numbers, is the successor function and is naturally interpreted as the number 0.
Geometry
Probably the most famous very early set of axioms is the 4 + 1 postulates of Euclid. This turns out to be incomplete, and many more postulates are necessary to completely characterize his geometry (Hilbert used 23).
"4 + 1" because for nearly two millennia the fifth (parallel) postulate (through a point outside a line there is exactly one parallel) was suspected of being derivable from the first four. Ultimately, the fifth postulate was found to be independent of the first four. Indeed, one can assume that no parallels through a point outside a line exist, that exactly one exists, or that infinitely many exist. These choices give us alternative forms of geometry in which the interior angles of a triangle add up to less than, exactly or more than a straight line respectively and are known as elliptic, Euclidean and hyperbolic geometries.
Real analysis
The real numbers, the standard numbers of "real analysis", are described by the axioms of a complete real closed Archimedean field, which define them uniquely up to isomorphism. However, expressing these properties as axioms requires use of secondorder logic. The LöwenheimSkolem theorems tell us that if we restrict ourselves to firstorder logic, any axiom system for the reals admits other models, including both models that are smaller than the reals and models that are larger. Some of the latter are studied in nonstandard analysis.
Deductive systems
The formal issue arises in the need to derive what logicians call a deductive system, which consists of a set Λ of logical axioms, a set Σ of nonlogical axioms and a set {(Γ,φ)} of rules of inference. Gödel's completeness theorem establishes that every deductive system with a consistent set of nonlogical axioms is complete,
if then
i.e., for any statement that is a logical consequence of Σ there actually exists a deduction of the statement from Σ. Again, more simply, anything that is true from a given set of axioms can be proved from those axioms (with reasonable rules of inference).
Note the subtle difference between this and the later and equally celebrated Gödel's first incompleteness theorem, which states that no set of recursive, consistent, nonlogical axioms Σ of the Theory of Arithmetic is complete, in the sense that there will always exist a true arithmetic statement φ such that neither φ nor can be proved (the later is not the same as φ being disproved  it simply means what it says, that there cannot be a deduction from Σ to ) from the given set of axioms.
There is thus, in one hand, the notion of completeness of a deductive system and on the other hand that of completeness of a set of nonlogical axioms.
The moral is, any fact that we can derive from a set of axioms (logical or nonlogical) is not needed as an axiom. Anything that we cannot derive from the axioms and for which we also cannot derive the negation might reasonably be added as an axiom.
Further discussion
Early mathematicians regarded axiomatic geometry as a model of physical space, and obviously there could only be one such model. The idea that alternative mathematical systems might exist was very troubling to mathematicians of the 19th century and the developers of systems such as Boolean algebra made elaborate efforts to derive them from traditional arithmetic. Galois showed just before his untimely death that these efforts were largely wasted, but that the grand parallels between axiomatic systems could be put to good use, as he algebraically solved many classical geometrical problems. Ultimately, the abstract parallels between algebraic systems were seen to be more important than the details and modern algebra was born.
See also
 Axiomatic system
 Peano axioms
 Axiom of choice
 Axiom of countability
 Axiomatic set theory
 Parallel postulate
 Continuum hypothesis
 Axiomatization
 List of axioms
External links
 Metamath axioms page http://metamath.planetmirror.com/mpegif/mmset.html#axioms