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Finite model theory studies first-order structures restricted to finite domains, and asks which properties can be expressed in specified formal languages. The restriction changes the available tools: ordinary first-order compactness does not hold when only finite models are admitted.

For example, the sentences saying there are at least $n$ distinct elements, one for every positive integer $n$, have a finite model for each finite subset but no common finite model. This is a direct failure of finite compactness. Logical games, locality, and bounded-variable fragments then compare the expressive power of formulas on finite databases and graphs, linking definability with computational complexity theory.