Definition:Axiom/Formal Systems

From ProofWiki
Jump to: navigation, search

Definition

Let $\mathcal L$ be a formal language.

Part of defining a proof system $\mathscr P$ for $\mathcal L$ is to specify its axioms.


An axiom of $\mathscr P$ is a well-formed formula of $\mathcal L$ that $\mathscr P$ approves of by definition.


Axiom Schema

An axiom schema is a well-formed formula $\phi$ of $\mathcal L$, except for it containing one or more variables which are outside $\mathcal L$ 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 $\mathscr P$.


Also see


Sources