Axiom (mathematics and logic)

An axiom is a statement adopted as a starting point in a formal system, without proof within that system. Rules of inference combine axioms to derive theorems.

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Rule of inference: A rule of inference licenses a conclusion from specified premises. Inference rules turn axioms into theorems by specifying valid steps of deduction.
Formal language: A formal language is a precisely specified set of symbols and rules for forming expressions. Axioms must be expressible as well-formed statements in a system's language.
Euclid's Elements: Euclid's Elements is an ancient Greek treatise that organizes geometry through definitions, postulates, and propositions. Its geometric postulates became a lasting model for axiomatic mathematics.
Axiom schema: An axiom schema is a pattern that generates a potentially infinite collection of axioms. Some theories specify their starting principles through schemas rather than a finite list.
Gödel's incompleteness theorems: Gödel's incompleteness theorems show limitations on consistency and provability in sufficiently expressive formal theories. They constrain what any suitable arithmetic axiom system can prove about itself.
Formal system: A formal system consists of a language, axioms, and rules for deriving formulas. An axiom has its role only within the system that adopts it.
Deductive reasoning: Deductive reasoning derives conclusions that follow necessarily from stated premises. Axiomatic proofs use deduction to preserve truth from premises to conclusions.
Euclidean geometry: Euclidean geometry studies points, lines, and figures under the geometric principles associated with Euclid. Its parallel postulate became the classic example of a disputed axiom.
Logical axiom: A logical axiom is a formula treated as an axiom because of its logical form within a proof system. Logical axioms are distinguished from nonlogical axioms that express a theory's subject matter.
Gödel's second incompleteness theorem: Gödel's second incompleteness theorem states that a consistent, sufficiently expressive formal theory cannot prove its own consistency under standard conditions. It limits attempts to certify an axiom system's consistency using only that system.