aksjomaty

Aksjomaty are fundamental statements accepted without proof within a formal theory. They serve as the foundation from which theorems are derived by logical deduction. In contrast to theorems, axioms are not proven within the system; their choice determines the scope and strength of the theory. Axioms can be logical, providing the rules of inference, or mathematical, specifying properties of objects such as numbers, points, or sets. Often, axioms are chosen to be as simple and self-evident as possible, though what counts as self-evident can vary by context and culture.

Common examples: Euclid's postulates laid the groundwork for classical plane geometry; Peano's axioms formalize the natural numbers and support mathematical induction; Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) provides a widely used foundation for modern mathematics. In logic, propositional and predicate calculus have their own axiom systems and inference rules.

Axiomatization aims at achieving consistency (no contradictions) and, ideally, completeness (every statement is decidable) relative to the language and objects considered. In practice, most rich theories are incomplete: Gödel's incompleteness theorems show that any sufficiently powerful consistent system cannot prove all true statements about its objects. Model theory and proof theory study these aspects.

Aksjomaty are thus central to the formal study of mathematics, logic, and related disciplines, providing a shared starting ground and a framework for rigorous reasoning.