Aksiomsett

Aksiomsett, in Norwegian usage, refers to a collection of axioms that form the foundational basis of a formal theory in logic and mathematics. An axiom is a proposition assumed without proof; the axioms, together with a specified set of inference rules, are used to derive the theorems of the theory.

Axioms can be concrete statements or axiom schemes, the latter representing infinitely many individual axioms obtained by substitution of formulas. Well-known examples include the Peano axioms for the natural numbers and the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). In propositional and predicate logic, axiom schemes encode general inference rules, such as patterns like A → (B → A) and (A → (B → C)) → ((A → B) → (A → C)).

The properties of an axiom set are central to its usefulness. Consistency means that no contradiction can be derived; independence means that no axiom is a logical consequence of the others; completeness concerns whether every true statement in the theory’s language is derivable. Gödel's incompleteness theorems show that any sufficiently expressive, consistent axiom set cannot prove all truths about its subject and is therefore incomplete.

Axioms underpin formal theories, enabling rigorous proofs, formal reasoning in computer science, and foundations for mathematics. The specific choice of axioms and language affects what can be proven, and different axiom sets can often formalize the same mathematical body in different ways.

See also: Axiom, Formal system, Gödel's incompleteness theorems, Peano axioms, ZFC.