Definition:Axiom/Formal Systems

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Let $\LL$ be a formal language.

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

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

Axiom Schema

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 $\mathscr P$.

Also known as

When $\LL$ is a logical language, then one also speaks of logical axioms.

Also see

  • Results about axioms can be found here.

Linguistic Note

The usual plural form of axiom is axioms.

However, the form axiomata can also sometimes be found, although it is sometimes considered archaic.