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.
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