Categoricity
In mathematics, a theory in the sense of model theory is categorical if it has one and only one model up to isomorphism. A theory is κ-categorical if it has one and only one model of cardinality κ up to isomorphism.
History and motivation
The notion of categoricity was introduced by Oswald Veblen in 1904 to describe a theory with a structure strong enough to force all of its models to be 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. T is κ-categorical if and only if κ is a countable cardinal.
In other words, he observed that, in all the cases he could think of, κ-categoricity at any one uncountable cardinal implied κ-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 Saharon Shelah in the 1970s and beyond, leading to stability theory and Shelah's more general programme of classification theory.