[camccann]

Definitions have nothing to do with truth, they serve only to clarify communication by establishing a common language; and mathematics as a whole makes no claim regarding the truth of axioms, only what a set of axioms do and do not entail.

There may be disagreements in matters of taste as to what Choice of axioms (har har, little joke there) one makes, but given a basis of axioms to work from--rarely more than a handful--the rest is, for most purposes, not at all open to interpretation.