Axiom schema 764613 216477442 2008-06-01T20:57:06Z TeaDrinker 247414 /* References */ remove extraneous signature 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|*]] [[it:Schema di assiomi]] [[zh:公理模式]]