- Axiom schema
In
mathematical logic , an axiom schema generalizes the notion ofaxiom .An axiom schema is a formula in the language of an
axiomatic system , in which one or moreschematic variable s 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 definedrecursive 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:
* Induction schema that is part ofPeano's axioms for the arithmetic of thenatural number s;
*Axiom schema of replacement that is part of the standardZFC axiomatization of set theory.It has been proved (first byRichard 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 ofvon Neumann-Bernays-Gödel set theory , but the latter is, quite surprisingly, finitely axiomatized. The set theoryNew Foundations can be finitely axiomatized, but only with some loss of elegance.Schematic variables in
first-order logic are usually trivially eliminable insecond-order logic , because a schematic variable is often a placeholder for anyproperty or 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.
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