Definition:Axiomatization

Definition

Let $\LL$ be a logical language.

Let $\mathscr M$ be a formal semantics for $\LL$.

Let $\FF$ be an $\LL$-theory.

An axiomatization of $\FF$ is a subset $\AA \subseteq \FF$ such that:

$\FF = \set {\phi \in \LL: \AA \models_{\mathscr M} \phi}$

That is, all of $\FF$ is a semantic consequence of $\AA$.

Axiom

Let $\AA$ be an axiomatization of $\FF$.

Then a formula $\phi \in \AA$ is called an axiom.

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.5.4$: Definition $2.56$