Aksiomatike

Aksiomatike, or axiomatics, refers to the branch of logic and foundations that studies axioms and the systematic development of theories from a chosen set of basic assumptions. An axiom is a statement accepted without proof and used as a basis for deriving theorems through formal rules of inference. Axiomatization is the process of selecting axioms and a formal language to capture a mathematical or logical theory, with aims such as clarity, simplicity, and internal consistency.

Historically, axiomatization began with Euclid’s Elements, which organized geometry from a small set of postulates and common notions. In the 19th and 20th centuries, mathematicians such as Frege, Peano, and Hilbert advanced formal axiom systems. Hilbert’s program sought a complete and consistent foundation for all mathematics by proving the consistency of axioms from finitary reasoning; Gödel’s incompleteness theorems later showed limits to this goal. In modern foundations, Zermelo–Fraenkel set theory (often with the axiom of choice, ZFC) serves as a standard framework, while alternative foundations include type theory and category theory.

Core concepts in axiomatics include axioms, rules of inference, and metatheory, as well as ideas of consistency, independence, and completeness. Key notions are that a system is consistent if no contradiction can be derived, independent when an axiom cannot be proved from others, and complete when every true statement is provable within the system. The method supports rigorous formulation of theories across mathematics and logic, from geometry and algebra to set theory and beyond.

See also: axiomatization, formal logic, foundations of mathematics, model theory, Hilbert’s program, ZFC.