Aksiomeja

Aksiomeja refers to axioms, statements that are assumed without proof within a formal theory and used as the starting point for deriving theorems. They are not asserted as universal truths; they are chosen to be simple, noncontroversial, and sufficient to capture the intended structure of the subject. Theories combine axioms with rules of inference to produce a web of consequences. Axioms can be divided into logical axioms, which govern inference, and non-logical axioms, which constrain the objects of the theory. Some axiom systems use schemes, where an infinite family of statements is generated from a single rule, for example, the induction axiom in the Peano axioms for natural numbers.

Prominent examples include Euclid's postulates for geometry, Peano's axioms for arithmetic, and the Zermelo-Fraenkel set theory (often with the axiom of choice, ZFC) as a foundational framework for modern mathematics. The choice of axioms influences what can be proved and what cannot; independence refers to an axiom not being derivable from others, while consistency means that no contradictions can be derived.

Historically, axioms were gradually made more precise, beginning with Euclid and later formalized by mathematicians such as Hilbert. Gödel's incompleteness theorem shows that any sufficiently strong, consistent system is incomplete—there will always be true statements that cannot be proved within the system. Axioms remain central to logic, mathematics, and computer science, underpinning formal proofs, logical frameworks, and automated reasoning.