Axiomatic system

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.

Properties of axiomatic systems

An axiomatic system is said to be consistent if it lacks contradiction, i.e the the ability to prove a statement and its negative of being both true based on the same set of 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 all of its underlying axioms are independent.

Independence is not a necessary requirement for a system, yet consistency is necessary. An axiomatic system will be called complete if no additional axiom can be added to the system without making the new system either dependent or inconsistent.

Models for axiomatic systems

A mathematical model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system. The existence of a concrete model* proves the consistency of a system.

Models can also be used to show the independence of an axiom in the system. By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem.

Two models are said to be isomorphic if one-to-one correspondence can be found between their elements, in a manner that preserves their relationship. An axiomatic system for which every model is isomorphic to another is called categorial, and the property of categoriallity ensures the completeness of a system.

* A model is called concrete if the meanings assigned are objects and relations from the real world, as opposed to an abstract model which is based on other axiomatic systems.

The first axiomatic system was Euclidean geometry.

See also

• Axiomatization
• Model theory
• Gödel's incompleteness theorem