axiomatisk

Axiomatic, in logic and mathematics, refers to a formal way of building a theory from a specified set of axioms using rules of inference. An axiomatic system consists of a formal language, a set of axioms, and inference rules. The axioms are assumed true within the system, and theorems are statements proven from these axioms.

The aim of axiomatization is to clarify foundations, ensure rigor, and separate basic assumptions from derived results. Classic examples include Euclidean geometry, developed from a small collection of postulates; Peano arithmetic for the natural numbers; and Zermelo-Fraenkel set theory (often with the axiom of choice) as a foundation for much of modern mathematics. Hilbert’s program sought a finite, complete set of axioms that could prove all mathematical truths and establish consistency; Gödel’s incompleteness theorems later showed limits to this goal by proving that any sufficiently powerful, consistent system cannot be both complete and able to prove its own consistency from within.

Key properties of axiomatic systems include consistency (no contradictions can be derived), independence (no axiom is a logical consequence of the others), and (relative) completeness, though full completeness is generally unattainable for strong systems. In practice, axiomatic methods are used not only in pure mathematics and logic but also in computer science (formal verification, type theory) and philosophy, to provide clear, reproducible foundations for reasoning.