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.

Therefore discussion of axiomatic systems is normally only semi-formal. 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 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. This is very difficult to achieve, however, and as shown by the combined works of Kurt Gödel and Paul Cohen, impossible for axiomatic systems involving infinite sets.

So, along with consistency, relative consistency is also the mark of a worthwhile axiom system. This is when the undefined terms of a first axiom system are provided definitions from a second such that the axioms of the first are theorems of the second.

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