 In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. The theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. The formal theory typically means an axiomatic system, for example formulated within model theory. The formal proof is a complete rendition of a mathematical proof within a formal system.