Aksiomien

Aksiomien (axioms) are statements assumed without proof within a formal theory. They function as foundational starting points from which theorems are derived using rules of inference. They are not proven within the theory; different theories may adopt distinct axiom systems to formalize the same domain. In mathematics and logic, axioms are distinguished from theorems; postulates are historical or domain-specific assertions, while axioms are general assumptions of the framework.

Aksiomien can be logical (the basic laws of propositional and first-order logic) or mathematical (structure-specific, such as axioms for numbers, groups, geometry, or sets). Examples include Euclid’s postulates for plane geometry, Hilbert’s axioms that formalize Euclidean geometry, Peano axioms for the natural numbers, and the Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC) used as a foundation for modern mathematics. Some axiomien systems use schemes, such as the induction axiom in Peano arithmetic or the axiom schemas of ZFC.

Key properties sought from axiomien systems are consistency (no contradictions), independence (no axiom is derivable from the others), and, in powerful theories, incompleteness in the Gödel sense. The choice of axiomien shapes what can be proven and even whether certain questions are decidable. Different axiomien systems can yield the same mathematical structures, while altering an axiom—such as the parallel postulate—produces non-Euclidean geometries.

In short, aksiomien are the accepted starting assumptions that define a formal theory and underpin its entire body of mathematical or logical results.