Axiom schema

An axiom schema is a rule specifying an infinite family of axioms, one for each permitted formula or instance. It lets a finite description stand for infinitely many axioms.

Substitution instance: A formula or expression obtained by replacing designated variables in a pattern with permitted expressions. Each permitted substitution into a schema yields one of its axiom instances.
Axiom: A statement accepted as a starting point within a formal theory or axiomatic system. A schema specifies many axioms, while an axiom is one statement in the resulting family.
Comprehension schema: A family of set-theoretic axioms asserting the existence of sets defined by formulas, subject to specified restrictions. It makes formula-by-formula set existence explicit rather than allowing unrestricted comprehension.
Finite axiomatization: A presentation of a theory using a finite set of axioms. A schema may describe infinitely many axioms even when its written presentation is finite.
Universal instantiation: An inference rule that derives a particular instance of a universally quantified statement. Some logical systems express universal instantiation as an axiom schema instead of a rule.
Axiomatic system: A formal system defined by axioms and rules for deriving theorems. Schemas provide compact ways to specify parts of an axiomatic system's starting assumptions.
Induction schema: A family of arithmetic axioms asserting induction for each eligible formula in a language. Each formula supplies a distinct induction axiom, producing infinitely many instances.
Second-order logic: A logic that quantifies over predicates, sets, or relations as well as individual objects. Some principles expressed by schemas in first-order logic can be stated with second-order quantifiers.
Formula: A well-formed expression built according to the formation rules of a formal language. The formulas permitted by a schema's substitution rules determine its instances.
Inference rule: A rule that licenses deriving conclusions from specified premises. An axiom schema supplies starting statements; an inference rule licenses steps from premises.

Linked from 7 pages