Axiom schema
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2008-06-01T20:57:06Z
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In [[mathematical logic]], an '''axiom schema''' generalizes the notion of [[axiom]].
An axiom schema is a [[well-formed formula|formula]] in the language of an [[axiomatic system]], in which one or more [[schematic variable]]s appear. These variables, which are metalinguistic constructs, stand for any [[First-order logic#Formation_rules|term]] or [[first-order logic|subformula]] of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be [[free variable|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 [[recursive]]ly. 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:
* [[Mathematical induction|Induction]] schema that is part of [[Peano's axioms]] for the arithmetic of the [[natural number]]s;
* [[Axiom schema of replacement]] that is part of the standard [[ZFC]] axiomatization 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 [[ZFC]] are also theorems of [[von Neumann-Bernays-Gödel set theory]], but the latter is, quite surprisingly, finitely axiomatized. The set theory [[New Foundations]] can be finitely axiomatized, but only with some loss of elegance.
Schematic variables in [[first-order logic]] are usually trivially eliminable in [[second-order logic]], because a schematic variable is often a placeholder for any [[property]] or [[relation (mathematics)|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.
==References==
*http://plato.stanford.edu/entries/schema/
*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.
[[Category:Mathematical axioms|*]]
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