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Axiomatic system
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.
==Properties==
An axiomatic system is said to be consistent if it lacks contradiction, i.e. the ability to derive both a statement and its denial from the system's axioms.
In an axiomatic system, an axiom is called independent if it is not a theorem that can be derived from other axioms in the system. A system will be called independent if each of its underlying axioms is independent. Although independence is not a necessary requirement for a system, consistency is.
An axiomatic system will be called complete if for every statement, either itself or its negation is derivable.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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