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https://en.wikipedia.org/wiki/Categorical_theory
Categorical theory
Not to be confused with Category theory.

In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism).[1] Such a theory can be viewed as defining its model, uniquely characterizing its 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 N.

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 Michael D. Morley (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.

Contents
1 History and motivation
2 Examples
3 Properties
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