= Dense order
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= Dense total order
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= Dense total orders
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A <total order> is dense when between every two distinct ordered points lies a third. Endpoints are allowed; “without endpoints” is an additional condition. A dense order with two points $a<b$ cannot be a <well-order>, since $\{x:a<x\}$ is nonempty and has no least member. Its first-order axioms are the total-order axioms plus $\forall x,y\,(x<y\Rightarrow\exists z\,(x<z<y))$. Countability and uncountability are not first-order axiomatizable restrictions, by the upward and downward witness constructions of the <Löwenheim-Skolem theorem>.
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