Editing Categorical theory (section)

==Properties==

Every categorical theory is [[complete theory|complete]].{{sfn|Monk|1976|p=349}} However, the converse does not hold.<ref>{{cite web |url=https://math.stackexchange.com/q/933632 |title=Difference between completeness and categoricity |last=Mummert |first=Carl |date=2014-09-16}}</ref>

Any theory ''T'' categorical in some infinite cardinal {{math|''κ''}} is very close to being complete. More precisely, the [[Łoś–Vaught test]] states that if a satisfiable theory  has no finite models and is categorical in some infinite cardinal {{math|''κ''}} at least equal to the cardinality of its language, then the theory is complete.  The reason is that all infinite models are first-order equivalent to some model of cardinal {{math|''κ''}} by the [[Löwenheim–Skolem theorem]], and so are all equivalent as the theory is categorical in {{math|''κ''}}. Therefore, the theory is complete as all models are equivalent.  The assumption that the theory have no finite models is necessary.<ref>Marker (2002) p. 42</ref>