  This question of Łoś was a tremendous stimulus to research, and it led to a classic paper of Michael Morley in 1965 which showed that Łoś’s three possibilities are in fact the only ones. One central idea of Morley’s analysis was that models of an uncountably categorical theory have the smallest possible number of types of element; this led directly to the branch of model theory called stability theory, which studies theories that have a limited number of element types. These theories have the remarkable property that every infinite indiscernible sequence in any of their models is indiscernible under any linear ordering whatever; so these sequences are a kind of generalisation of bases of vector spaces. Another idea implicit in Morley’s work, but much clarified by later work of William Marsh, John Baldwin and Alistair Lachlan, was that in any model of an uncountably categorical theory there is a central core (called a strongly minimal set) which carries a dependence relation obeying similar laws to linear dependence in vector spaces. In terms of this dependence relation one can define a dimension for the model, and what remains of the model outside the core is so closely tied to the core that the dimension determines the model up to isomorphism.