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Inmathematical logic, anaxiom schema (plural:axiom schemata oraxiom schemas) generalizes the notion ofaxiom.
An axiom schema is aformula in themetalanguage of anaxiomatic system, in which one or moreschematic variables appear. These variables, which are metalinguistic constructs, stand for anyterm orsubformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables befree, or that certain variables not appear in the subformula or term[citation needed].
Two well known instances of axiom schemata are the:
Czesław Ryll-Nardzewski proved that Peano arithmetic cannot be finitely axiomatized, andRichard Montague proved that ZFC cannot be finitely axiomatized.[1][2] Hence, the axiom schemata cannot be eliminated from these theories. This is also the case for quite a few other axiomatic theories in mathematics, philosophy, linguistics, etc.
Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is infinite, an axiom schema stands for an infiniteclass or set of axioms. This set can often bedefined recursively. A theory that can be axiomatized without schemata is said to befinitely axiomatizable.
All theorems ofZFC are also theorems ofvon Neumann–Bernays–Gödel set theory, but the latter can be finitely axiomatized. The set theoryNew Foundations can be finitely axiomatized through the notion ofstratification.
Schematic variables infirst-order logic are usually trivially eliminable insecond-order logic, because a schematic variable is often a placeholder for anyproperty orrelation over the individuals of the theory. This is the case with the schemata ofInduction andReplacement mentioned above. Higher-order logic allows quantified variables to range over all possible properties or relations.
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