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Concept
Categorical theory
Summary
In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). 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 model can be categorical. Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers
In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of 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.
extended Morley's theorem to uncountable languages: if the language has cardinality κ and a theory is categorical in some uncountable cardinal greater than or equal to κ then it is categorical in all cardinalities greater than κ.
Oswald Veblen in 1904 defined a theory to be categorical if all of its models are isomorphic. It follows from the definition above and the Löwenheim–Skolem theorem that any first-order theory with a model of infinite cardinality cannot be categorical. One is then immediately led to the more subtle notion of κ-categoricity, which asks: for which cardinals κ is there exactly one model of cardinality κ of the given theory T up to isomorphism? This is a deep question and significant progress was only made in 1954 when Jerzy Łoś noticed that, at least for complete theories T over countable languages with at least one infinite model, he could only find three ways for T to be κ-categorical at some κ:
T is totally categorical, i.e. T is κ-categorical for all infinite cardinals κ.
T is uncountably categorical, i.e. T is κ-categorical if and only if κ is an uncountable cardinal.
T is countably categorical, i.e.
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