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The set $\Sigma$ is satisfiable if there is one $L$-structure $\mathcal{A}$ satisfying every first-order sentence in $\Sigma$ simultaneously. It is finitely satisfiable if each finite subset ${\Sigma }_0\subseteq \Sigma$ has a first-order model; that first-order model is allowed to depend on the chosen finite subset.

Thus the two statements have different quantifier orders:Satisfiability immediately implies finite satisfiability by using the same first-order structure for every finite subset. The converse is the substantive content of compactness theorem.