Morley's categoricity theorem

Remarks

Michael Morley proved the theorem in 1965, answering a question of Jerzy Łoś. A key step shows that a theory categorical in an uncountable cardinal is $\omega$-stable: over each countable set of parameters there are only countably many complete types. Such a theory has a strongly minimal formula, possibly with parameters. Each model is determined up to isomorphism by the dimension of the set that this formula defines, as a vector space is determined by its dimension.

Example: the theory of algebraically closed fields of a fixed characteristic is categorical in every uncountable cardinal, with transcendence degree as the dimension. The theory of infinite vector spaces over $\Q$ is another example.

The countability of the signature matters in this form. Saharon Shelah extended the theorem: a complete theory $T$ categorical in some cardinal $\kappa > \abs{T}$ is categorical in every cardinal $\kappa > \abs{T}$. The theorem says nothing about $\aleph_0$. The theory of algebraically closed fields of characteristic $0$ is uncountably categorical but not $\aleph_0$-categorical. The theory of dense linear orders without endpoints is $\aleph_0$-categorical but not uncountably categorical.

Going deeper

For a student

The proof introduced tools that became the core of stability theory: Morley rank, $\omega$-stability, and prime models over sets. A countable complete theory with infinite models has one of four patterns of categoricity. It is categorical in all infinite cardinals, only in $\aleph_0$, only in the uncountable ones, or in none. Baldwin and Lachlan later showed that an uncountably categorical theory has either one countable model or countably infinitely many.