||This article includes a list of references, related reading or
external links, but its sources remain unclear because it lacks
inline citations. (March 2013)| 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. A mathematical
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; usually though, the effort
towards complete formalisation brings diminishing returns in
certainty, and a lack of readability for humans. A formal theory
typically means an axiomatic system, for example formulated
within model theory. A formal proof is a complete rendition of a
mathematical proof within a formal system. An axiomatic system
is said to be consistent if it lacks contradiction, i.e. the
ability to derive both a statement and its negation from the
system's axioms. In an axiomatic system, an axiom is called
independent if it is not a