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.
Read more about Axiomatic System: Properties, Relative Consistency, Models, Axiomatic Method
Famous quotes containing the words axiomatic and/or system:
“It is ... axiomatic that we should all think of ourselves as being more sensitive than other people because, when we are insensitive in our dealings with others, we cannot be aware of it at the time: conscious insensitivity is a self-contradiction.”
—W.H. (Wystan Hugh)
“He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one.”
—Henry David Thoreau (1817–1862)