In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. To say the least, not all epistemologists agree that axioms, understood in that sense, exist.
As the word axiom is understood in mathematics, an axiom is not a proposition that is self-evident. Rather, it simply means a starting point in a logical system. For example, in some rings, the operation of multiplication is commutative, and in some it is not; those rings in which it is are said to satisfy the "axiom of commutativity of multiplication." Another name for an axiom is postulate. An axiom is an elementary basis for a formal logic system that together with the rules of inference define a logic.
Using these axioms, and defining the customary short names 1, 2, 3, and so on for inc(0), inc(inc(0)), inc(inc(inc(0))) respectively, we can show that:
1 + 2 = 1 + inc(1) Expansion of abbreviation (2 = inc(1))
1 + 2 = inc(1) + 1 Axiom 4
1 + 2 = 2 + 1 Abbreviation (2 = inc(1))
1 + 2 = 2 + inc(0) Expansion of abbreviation (1 = inc(0))
1 + 2 = inc(2) + 0 Axiom 4
1 + 2 = 3 Axiom 3 and Use of abbreviation (inc(2) = 3)
Any fact that we can derive from the axioms need not be an axiom. Anything that we cannot derive from the axioms and for which we also cannot derive the negation might reasonably added as an axiom.
Probably the most famous very early set of axiom are the 4+1 postulates of Euclid. These turn out to be fairly incomplete, actually, and many more postulates are necessary to completely characterize his geometry (Hilbert used 23).
I say 4+1 since the 5th postulate (through a point outside a line there is exactly one parallel) was suspected to be derivable from the first 4 for nearly two millennia. 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. The general theory of relativity is essentially a claim that mass gives space hyperbolic geometry.
The fact that alternative forms of geometry might exist was very troubling to mathematicians of the 19th century and in similar developments, say Boolean Algebra, there were generally elaborate efforts taken to derive the system from normal arithmetic systems. 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 algebraicly 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.
In the twentieth century, Gödel's Incompleteness Theorem showed that no explicit list of axioms sufficiently large for ordinary mathematics could be both (1) complete (i.e. every statement can be either proved or disproved) and (2) consistent (i.e. no statement can be both proved and disproved).