mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...

, 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 any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains ...

, 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 sen ...

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.

Finite axiomatization

Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is

countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...

, 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.

Examples

Two well known instances of axiom schemata are the: * induction schema that is part of

Peano's axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearl ...

for the arithmetic of the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

s; * axiom schema of replacement that is part of the standard ZFC axiomatization of

set theory Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concern ...

. Czesław Ryll-Nardzewski 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 formaliz ...

proved that ZFC cannot be finitely axiomatized.Czesław Ryll-Nardzewski 1952; Richard Montague 1961. 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.

Finitely axiomatized theories

All theorems of ZFC are also theorems of von Neumann–Bernays–Gödel set theory, but the latter can be finitely axiomatized. The set theory

New Foundations 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''. Quine first proposed NF in a 1937 article titled "New Foundatio ...

can be finitely axiomatized, but only with some loss of elegance.

In higher-order logic

Schematic variables in

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

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, r ...

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.

Notes

References

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