Definition:Axiom/Formal Systems/Axiom Schema
Jump to navigation Jump to search
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 $\mathscr P$.
- Results about axiom schemata can be found here.
The plural of axiom schema is correctly axiom schemata, but it is commonplace to see the word schemas used for schemata.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): schema
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): schema