mathematical logic, an axiom schema generalizes the notion of axiom.
An axiom schema is a formula in the language of an
axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term.
Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is
countably infinite, an axiom schema stands for a countably infinite set of axioms. This set can usually be defined recursively. A theory that can be axiomatized without schemata is said to be "finitely axiomatized". Theories that can be finitely axiomatized are seen as a bit more metamathematically elegant, even if they are less practical for deductive work.
Two very well known instances of axiom schemas are the:
* Induction schema that is part of
Peano's axiomsfor the arithmetic of the natural numbers;
Axiom schema of replacementthat is part of the standard ZFCaxiomatization of set theory.It has been proved (first by Richard Montague) that these schemata cannot be eliminated. Hence Peano arithmetic and ZFC cannot be finitely axiomatized. This is also the case for quite a few other axiomatic theories in mathematics, philosophy, linguistics, etc.
All theorems of
ZFCare also theorems of von Neumann-Bernays-Gödel set theory, but the latter is, quite surprisingly, finitely axiomatized. The set theory New Foundationscan be finitely axiomatized, but only with some loss of elegance.
Schematic variables in
first-order logicare usually trivially eliminable in second-order logic, because a schematic variable is often a placeholder for any propertyor relation over the individuals of the theory. This is the case with the schemata of "Induction" and "Replacement" mentioned above. Higher-order logic allows quantified variables to range over all possible properties or relations.
*Corcoran, J. 2006. Schemata: the Concept of Schema in the History of Logic. "Bulletin of Symbolic Logic" 12: 219-40.
*Mendelson, Elliot, 1997. "Introduction to Mathematical Logic", 4th ed. Chapman & Hall.
*Potter, Michael, 2004. "Set Theory and its Philosophy". Oxford Univ. Press.
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