Axiomsystem

Axiomsystem is a formal framework in which a language is equipped with a specified set of axioms and inference rules designed to derive theorems. It serves as the foundational structure for formal reasoning in mathematics, logic, and related disciplines.

Components of an axiomsystem include a formal language specifying symbols and formation rules; an axiom set (postulates) considered self-evident or accepted as true within the system; inference rules that determine how new statements can be derived from existing ones; a notion of theorems, proofs, models and semantics to interpret truth values. Key properties often discussed are consistency, soundness, and completeness, with decidability potentially applying to particular systems.

Common examples of axiomsystems include propositional calculus with a minimal axiom base and inference rules; first-order logic; set theory such as Zermelo–Fraenkel with Choice (ZFC); and Euclidean geometry as developed in the Hilbert tradition. Different axiomsystems may formalize the same concepts while varying in strength and expressive power.

Uses and significance of axiomsystems span formal verification, automated theorem proving, and foundational research in mathematics. They also play a role in computer science, philosophy, and linguistics. Gödel’s incompleteness theorems show that any sufficiently powerful, consistent axiomatic system cannot prove all truths about its own statements.

History and terminology: Modern emphasis on axiomatization arose in the 19th and 20th centuries with the work of Frege, Russell, Whitehead, and Hilbert, contributing to proof theory and model theory. The term “axiomsystem” is used in general discussions to refer to any such formal framework.

See also: formal logic, axiomatization, proof theory, model theory, automated theorem proving.