The role of matroids in model theory
The role of matroids in model theory
The role of matroids in model theory
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112 CategoricityDef<strong>in</strong>itionA <strong>theory</strong> T is categorical if any two <strong>model</strong>s <strong>of</strong> T are isomorphic.<strong>The</strong>orem (Löwenheim-Skolem)Let T be a countable complete first-order <strong>theory</strong>.If T has <strong>model</strong>s <strong>of</strong> size κ for some <strong>in</strong>f<strong>in</strong>ite card<strong>in</strong>al κ,then T has <strong>model</strong>s <strong>of</strong> size κ for all <strong>in</strong>f<strong>in</strong>ite card<strong>in</strong>als κ.CorollaryLet T be a countable complete first-order <strong>theory</strong>.T is categorical if and only if T has a f<strong>in</strong>ite <strong>model</strong>.
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