Axiomensystem

Axiomensystem, or axiomatic system, is a formal framework consisting of a formal language, a set of axioms, and a set of inference rules from which theorems are derived. The language defines the symbols and formation rules; the axioms are statements accepted without proof; the inference rules specify valid steps to generate new statements; the theorems are those statements that can be derived from the axioms.

Common types include Hilbert-style systems, natural deduction, and sequent calculi. In propositional logic, axiom schemes and a few inference rules suffice. In first-order logic, axiom schemes for logical connectives combined with quantifier rules are used. Examples include Hilbert-style systems for arithmetic such as Peano arithmetic, and foundational theories like ZFC in set theory. Axiomatic methods underpin Euclidean geometry, Peano arithmetic, and much of modern mathematics by providing a precise starting point for proofs.

Key properties include consistency, if no contradiction can be derived; soundness, every derivable statement is valid in all intended interpretations; and completeness, every statement true in every model is derivable (Gödel's completeness theorem for first-order logic). However Gödel's incompleteness theorems show that any consistent, effectively axiomatizable theory that includes enough arithmetic is incomplete.

History and use: The axiomatic method developed in the formalist tradition, notably by Hilbert in the early 20th century, influencing foundations of mathematics. Axiomatic systems are central in logic, mathematics, and computer science, providing a precise basis for formal proofs, automated theorem proving, and formal verification.