1. With respect to a deductive system $\text{L}$, a property holding of $\text{L}$-theories $T$ when $T$ is the set of consequences of a finite set of formulae $\Gamma$. More formally, $T$ is finitely axiomatizable when it is the deductive closure of a finite set of formulae, i.e., there is a finite $\Gamma$ such that $T={\Gamma }^{{⊢}_{\text{L}}}$. As a finite set of formulae is inherently decidable, it follows that a finitely axiomatizable theory is also recursively axiomatizable. This does not entail that finitely axiomatizable theories are decidable. In the case of classical first-order logic, Robinson arithmetic $\text{Q}$ is axiomatized by a single axiom but is nevertheless undecidable.

2. A property of a deductive system $\text{L}$ that has a Hilbert-style calculus with finitely many axioms and rules.