Axiom schema

In mathematical logic, an axiom schema (plural: axiom schemata) generalizes the notion of axiom.

Formal definition

An axiom schema is a formula in the metalanguage 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.

Finite axiomatization

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.

Examples

Two very well known instances of axiom schemata 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 (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.

Finitely axiomatized theories

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.

In higher-order logic

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.

Notes

References

Corcoran, John (2008-09-21). "Schema". Stanford Encyclopedia of Philosophy.
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.

Set theory
Axioms	Choice countable dependent Constructibility (V=L) Determinacy Extensionality Infinity Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement De Morgan's laws Disjoint union Intersection Power set Set difference Symmetric difference Union
Concepts Methods	Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram
Set types	Countable Empty Finite (hereditarily) Fuzzy Infinite Recursive Subset · Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem
Set theorists	Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Skolem Willard Quine

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