Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explosion).
A good example is the relative consistency of absolute geometry with respect to the theory of the real number system.
Lines and points are undefined terms (also called primitive notions) in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems.
In their book Principia Mathematica, Alfred North Whitehead and Bertrand Russell attempted to show that all mathematical theory could be reduced to some collection of axioms.
More generally, the reduction of a body of propositions to a particular collection of axioms underlies the mathematician's research program.
The explication of the particular axioms used in a theory can help to clarify a suitable level of abstraction that the mathematician would like to work with.
For example, mathematicians opted that rings need not be commutative, which differed from Emmy Noether's original formulation.
Mathematicians decided to consider topological spaces more generally without the separation axiom which Felix Hausdorff originally formulated.
[5] Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.
Euclid of Alexandria authored the earliest extant axiomatic presentation of Euclidean geometry and number theory.
Once the axioms were clarified (that inverse elements should be required, for example), the subject could proceed autonomously, without reference to the transformation group origins of those studies.
Philosophers of mathematics sometimes assert that mathematicians choose axioms "arbitrarily", but it is possible that although they may appear arbitrary when viewed only from the point of view of the canons of deductive logic, that appearance is due to a limitation on the purposes that deductive logic serves.