Axiom
An axiom is a foundational statement accepted as a starting assumption in a formal theory. Its role depends on the theory in which it is adopted.
Formal system: A formal system consists of symbols, formation rules, axioms, and inference rules for deriving statements. An axiom is one of the ingredients that makes a formal system.
First-order logic: First-order logic is a formal system for reasoning about objects, properties, and relations using quantifiers. Many mathematical axioms are expressed and studied in first-order logic.
Euclid's Elements: Euclid's Elements is an ancient Greek mathematical treatise organizing geometry and number theory through definitions, postulates, and proofs. Its geometric postulates became a lasting model for axiomatic mathematics.
Postulate: A postulate is a foundational assumption adopted within a particular theory, especially in mathematics. The word often emphasizes geometric assumptions, while axiom is the broader formal term.
Formal language: A formal language is a set of symbol strings defined by precise rules of formation. Axioms must be well-formed statements in the language of their theory.
Deductive system: A deductive system is a set of formal rules for deriving conclusions from premises. Its rules specify how axioms yield theorems.
Euclid's postulates: Euclid's postulates are the foundational geometric assumptions stated at the start of the Elements. They exemplify axioms whose consequences structure an entire mathematical theory.
Definition: A definition assigns a precise meaning to a term within a language or theory. Unlike an axiom, a definition introduces terminology rather than asserting a starting claim.
Inference rule: An inference rule specifies how to derive a conclusion from one or more premises. Inference rules determine what a theory can prove from its axioms.
Soundness: A deductive system is sound when every statement it proves is true in all models of its premises. Soundness connects derivations from axioms to truth in their models.