Axiom Systems

Abstract

An axiom system or postulate system consists of some undefined terms and a list of statements, called axioms or postulates, concerning the undefined terms. One obtains a mathematical theory by proving new statements, called theorems, using only the axioms and previous theorems. Definitions are made in the process in order to be more concise. Aesthetically it may be preferable to give the list of axioms all at once. This may be impractical, however, as some of the axioms often depend on definitions and theorems resulting from earlier axioms. Usually one does not construct an axiom system from scratch. It is common to assume at least a language, a logic, and some set theory.