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 -categorical (or categorical in ) if it has exactly one model of cardinality 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 and a theory is categorical in some uncountable cardinal greater than or equal to then it is categorical in all cardinalities greater than . History and motivation 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 -categoricity, which asks: for which cardinals is there exactly one model of cardinality 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 -categorical at some : T is totally categorical, i.e. T is -categorical for all infinite cardinals . T is uncountably categorical, i.e. T is -categorical if and only if is an uncountable cardinal. T is countably categorical, i.e. T is -categorical if and only if is a countable cardinal. In other words, he observed that, in all the cases he could think of, -categoricity at any one uncountable cardinal implied -categoricity at all other uncountable cardinals. This observation spurred a great amount of research into the 1960s, eventually culminating in Michael Morley's famous result that these are in fact the only possibilities. The theory was subsequently extended and refined by