# Definition:Axiom/Formal Systems/Axiom Schema

## Contents

## Definition

Let $\LL$ be a formal language.

Part of defining a proof system $\mathscr P$ for $\LL$ is to specify its **axiom schemata**.

An **axiom schema** is a well-formed formula $\phi$ of $\LL$, except for it containing one or more variables which are *outside* $\LL$ itself.

This formula can then be used to represent an infinite number of individual axioms in one statement.

Namely, each of these variables is allowed to take a specified range of values, most commonly WFFs.

Each WFF $\psi$ that results from $\phi$ by a valid choice of values for all the variables is then an axiom of $\mathscr P$.

## Examples

It was proved by Richard Merett Montague in 1957 that ZFC and Peano arithmetic require an **axiom schema**.

## Linguistic Note

The plural of **axiom schema** is correctly **axiom schemata**, but it is commonplace to see the word **schemas** used for **schemata**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**schema** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**schema**