  Let us begin with the notion of an axiom system.  To motivate this notion consider the manner in which justification traditionally proceeds in mathematics.  In reasoning about a given domain of mathematics (or, in fact, any domain) the question of justification is successively pushed back further and further until ultimately one reaches principles that do not admit more fundamental justification. The statements at this terminal stage are elected as axioms and the subject is then organized in terms of derivability from the base of axioms.  In the case of arithmetic this led to the axiom system PA (Peano arithmetic) and in the case of set theory it led to the axiom system ZFC (Zermelo-Frankel set theory with the Axiom of Choice).