Editing Categorical theory (section)

{{Short description|Type of theory in mathematical logic}}
{{redirect-distinguish2|Vaught's test|the [[Tarski–Vaught test]]}}
{{distinguish|Category theory}}
In [[mathematical logic]], a [[theory (mathematical logic)|theory]] is '''categorical''' if it has exactly one [[model (mathematical logic)|model]] ([[up to isomorphism]]).{{efn|Some authors define a theory to be categorical if all of its models are isomorphic.  This definition makes the inconsistent theory categorical, since it has no models and therefore vacuously meets the criterion.}} Such a theory can be viewed as ''defining'' its model, uniquely characterizing the model's structure.

In [[first-order logic]], only theories with a [[Finite set|finite]] model can be categorical.  [[Higher-order logic]] contains categorical theories with an [[Infinite set|infinite]] model.  For example, the second-order [[Peano axioms]] are categorical, having a unique model whose domain is the [[Set (mathematics)|set]] of natural numbers <math>\mathbb{N}.</math>

In [[model theory]], the notion of a categorical theory is refined with respect to [[cardinal number|cardinality]].  A theory is {{math|''κ''}}-'''categorical''' (or '''categorical in {{math|''κ''}}''') if it has exactly one model of cardinality {{math|''κ''}} up to isomorphism. '''Morley's categoricity theorem''' is a theorem of {{harvs|txt|authorlink=Michael D. Morley|first=Michael D. |last=Morley|year=1965}} stating that if a [[first-order theory]] in a [[countable]] language is categorical in some [[uncountable]] [[cardinality]], then it is categorical in all uncountable cardinalities.

{{harvs|txt|authorlink=Saharon Shelah|first=Saharon |last=Shelah|year=1974}} extended Morley's theorem to uncountable languages: if the language has cardinality {{math|''κ''}} and a theory is categorical in some uncountable cardinal greater than or equal to {{math|''κ''}} then it is categorical in all cardinalities greater than&nbsp;{{math|''κ''}}.