mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of

axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

Formal definition

An axiom schema is a

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

in the

metalanguage In logic and linguistics, a metalanguage is a language used to describe another language, often called the ''object language''. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quota ...

of an

axiomatic system In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems. A proof within an axiom system is a sequence of deductive steps that establishes ...

, in which one or more

schematic variable In logic, a metavariable (also metalinguistic variable or syntactical variable) is a symbol or symbol string which belongs to a metalanguage and stands for elements of some object language. For instance, in the sentence :''Let A and B be two sent ...

s appear. These variables, which are metalinguistic constructs, stand for any term or

subformula In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbol (formal), symbols from a given alphabet (computer science), alphabet that is part ...

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.

Examples

Two well known instances of axiom schemata are the: * induction schema that is part of Peano's axioms for the arithmetic of the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

s; *

axiom schema of replacement In set theory, the axiom schema of replacement is a Axiom schema, schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image (mathematics), image of any Set (mathematics), set under any definable functional predicate, mappi ...

that is part of the standard ZFC axiomatization of

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

Czesław Ryll-Nardzewski Czesław Ryll-Nardzewski (; 7 October 1926 – 18 September 2015) was a Polish mathematician. Life and career Born in Wilno, Second Polish Republic (now Vilnius, Lithuania), he was a student of Hugo Steinhaus. At the age of 26 he became profe ...

proved that Peano arithmetic cannot be finitely axiomatized, and

Richard Montague Richard Merritt Montague (September 20, 1930 – March 7, 1971) was an American mathematician and philosopher who made contributions to mathematical logic and the philosophy of language. He is known for proposing Montague grammar to formalize th ...

proved that ZFC cannot be finitely axiomatized. Hence, the axiom schemata cannot be eliminated from these theories. This is also the case for quite a few other axiomatic theories in mathematics, philosophy, linguistics, etc.

Finite axiomatization

Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is infinite, an axiom schema stands for an infinite

class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...

or set of axioms. This set can often be defined recursively. A theory that can be axiomatized without schemata is said to be finitely axiomatizable.

Finitely axiomatized theories

All theorems of ZFC are also theorems of

von Neumann–Bernays–Gödel set theory In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collec ...

, but the latter can be finitely axiomatized. The set theory

New Foundations In mathematical logic, New Foundations (NF) is a non-well-founded, finitely axiomatizable set theory conceived by Willard Van Orman Quine as a simplification of the theory of types of ''Principia Mathematica''. Definition The well-formed fo ...

can be finitely axiomatized through the notion of stratification.

In higher-order logic

Schematic variables in

first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...

are usually trivially eliminable in

second-order logic In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...

, because a schematic variable is often a placeholder for any

property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, re ...

relation Relation or relations may refer to: General uses * International relations, the study of interconnection of politics, economics, and law on a global level * Interpersonal relationship, association or acquaintance between two or more people * ...

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.

Notes

References

* . * * . * . * . * . {{Mathematical logic Formal systems *