Axiomas

An axiom is a statement that is assumed to be true within a formal theory and used as a starting point for deriving other statements. Axioms are not proven within the theory; they establish the basic concepts, language, and inference rules used to develop proofs. The selection of axioms shapes the scope and spirit of the mathematical framework.

In mathematics, theories are built through axiomatization. Historical examples include Euclid’s geometric postulates and common notions, which function as foundational assumptions for the geometry in his Elements. In modern practice, common foundational systems include the Peano axioms for the natural numbers, and the Zermelo-Fraenkel set theory with the axiom of choice (ZFC) for a broad theory of mathematics. Axioms may be organized into schemas, such as the axiom schema of induction or the axiom of extensionality, to cover infinitely many instances.

Key properties of axiom systems include independence, consistency, and, in some contexts, completeness. An axiom is independent of the others if it cannot be proved from them; a system is consistent if it does not derive a contradiction. Gödel’s incompleteness theorems show that any sufficiently powerful, consistent system cannot be complete. This has led to ongoing work in finding alternative axiom systems and exploring their relative consistency.

Axioms play a central role in logic and mathematics by providing the formal foundation for proofs and theories. They are chosen to be intuitive, minimal, or convenient for the problems at hand, and different but compatible axiom systems can describe the same mathematical landscape.