In mathematical logic, an axiom schema generalizes the notion of axiom. To meet Wikipedias quality standards, this article or section may require cleanup. ... This article does not cite its references or sources. ...

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. In logic, WFF is an abbreviation for well-formed formula. ... In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ... The word term refers to either a word unit or a time unit with specified boundaries or limits. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...

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. In mathematics the term countable set is used to describe the size of a set, e. ... See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page — a list of pages that otherwise might share the same title. ...

Two very well known instances of axiom schemas are the:

• Induction schema that is part of Peano's axioms for the arithmetic of the natural numbers;
• Axiom schema of replacement that is part of the standard ZFC axiomatization of set theory.

It has been proved 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. Look up induction in Wiktionary, the free dictionary. ... In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...

All theorems of ZFC are also theorems of Von Neumann-Bernays-Godel 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. The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... In foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiom system for set theory designed to yield the same results as Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC), but with only a finite number of axioms, that is without axiom schemas. ... In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. ...

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 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. First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... In mathematical logic, second-order logic is an extension of either propositional logic or first-order logic which contains variables in predicate positions (rather than only in term positions, as in first-order logic), and quantifiers binding them. ... Property designates those things that are commonly recognized as being the possessions of a person or group. ... In mathematics, a finitary relation is defined by one of the formal definitions given below. ...

References

• Mendelson, Elliot, 1997. Introduction to Mathematical Logic, 4th ed. Chapman & Hall.
• Potter, Michael, 2004. Set Theory and its Philosophy. Oxford Univ. Press.