= Finite axiomatizability
A <first-order model> class is <finite axiomatizability>[finitely axiomatizable] in a fixed language if a <finite set> of sentences, equivalently one finite <conjunction>, has exactly that <first-order model>[model] class. Relative <finite axiomatizability>[finite axiomatizability] permits a fixed background <first-order theory>[theory] and asks for only finitely many additional <axioms>. The distinction matters in an infinite <scalar multiplication>[scalar] language, where even vector-space structure itself needs infinitely many named-symbol constraints.
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