= Model theory
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<Model theory> studies mathematical <first-order structures> through the <first-order formulas> they satisfy. A theory specifies sentences, while a <model of a first-order theory> gives a domain and interpretations of the nonlogical symbols making those sentences true.
For example, dense <total orders> without endpoints have a common complete first-order theory, although a countable such order and the real line cannot be isomorphic. The <Löwenheim-Skolem theorem> and <compactness theorem> explain why first-order truth can leave cardinality and other global structure undetermined. <Quantifier elimination>, <complete types>, and <elementary embeddings> provide tools for describing exactly what a theory can distinguish.
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