Axiomatisation

Axiomatisation (also spelled axiomatization) is the process of formulating a formal system by specifying a set of axioms from which all theorems of the system can be derived. It is a central practice in logic, mathematics, and computer science, and is used in some sciences to clarify foundational assumptions. In an axiomatised theory, axioms are chosen to be simple, self-evident, or at least widely accepted starting points, and a deductive apparatus (rules of inference) is used to derive more complex results. Axioms may be independent (no axiom follows from the others) and the theory seeks to be consistent (no contradictions).

In mathematics and logic, axiomatisation aims to provide a rigorous foundation for a subject. Famous examples include Euclid's postulates for geometry, Peano's axioms for the natural numbers, and the Zermelo–Fraenkel axioms (often with the axiom of choice) for set theory. Many theories employ axiom schemas, which describe infinitely many axioms in a compact form.

Important limiting notions accompany axiomatisation. Completeness, whereby every statement expressible in the system is either provable or refutable, is not guaranteed in sufficiently strong theories, as shown by Gödel's incompleteness theorems. Independence questions ask whether an axiom can be derived from the rest. Consistency must be established, often relative to a stronger system.

In practice, axiomatisation reflects aims and convictions about a domain: it clarifies assumptions, enables formal analysis, and supports automated reasoning, but it may abstract away empirical or contextual factors. Critics note that axiomatised theories can become detached from real-world phenomena if the chosen axioms do not capture essential features.