= Definable set
{title2=$D=\{\mathbf a:M\models\varphi(\mathbf a,\mathbf b)\}$}
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= First-order definable set
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A <definable set> in a <first-order structure> $M$ is the set of tuples satisfying one <first-order formula>, possibly with a specified finite tuple of parameters $\mathbf b$ from the structure. A set definable over $B\subseteq M$ uses parameters from $B$. <Parameter-free definability> uses no extra named elements. The distinction matters: any singleton becomes definable if its own element is allowed as a parameter, while many singletons are not definable without parameters. Finite Boolean combinations of definable sets are definable, using <logical conjunction>, <logical disjunction> and <logical negation>.
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