Axiomatic system

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (March 2013) (Learn how and when to remove this message)

This article needs attention from an expert in Mathematics. The specific problem is: Few citations despite the degree of detail. WikiProject Mathematics may be able to help recruit an expert. (January 2025)

In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemma or theorems. A proof within an axiom system is a sequence of deductive steps that establishes a news statement as a consequence of the axioms. An axiom system is called complete with respect to a property if every formula with the property can be derived using the axioms. The more general term theory is at times used to refer to an axiomc system and all its derived theorems.

In its pure form, an axiom system is effectively a syntactic construct and does not by itself refer to (or depend on) a formal structure, although axioms are often defined for that purpose. The more modern field of model theory refers to mathematical structures. The relationship between an axiom systems and the models that correspond to it is often a major issue of interest.