= Infinite-cardinal categoricity forces completeness
If a <first-order theory>[theory] without finite <first-order model>[models] is categorical in an infinite cardinal at least the language size, it is complete. Otherwise its two <consistent first-order theory>[consistent] extensions deciding a sentence oppositely have infinite <first-order model>[models]. The <Löwenheim-Skolem theorems> produce <first-order model>[models] of both of the specified <cardinality>, contradicting their forced <isomorphism of structures>[isomorphism]. Categoricity includes existence at that <cardinality>, so it is not vacuous.
Back to article page
