Morley's Categoricity Theorem

Morley's categoricity theorem stands as a cornerstone in model theory, with many experts considering it the beginning of modern model theory. A complete theory $T$ in a countable language is $\kappa$-categorical if it has a unique (up to isomorphism) model of cardinality $\kappa$. Morley, with his PhD thesis "Categoricity in Power", published in 1962, positively answered the conjecture of Łoś stating that if $T$ is $\kappa$-categorical for some uncountable $\kappa $, then it is $\kappa$-categorical for all uncountable $\kappa$. This theorem is now known as the categoricity theorem. The ideas used to prove it now play a central role in model theory and still shape the direction of the field. We will follow a proof given by Lachlan and Baldwin, which presents many ideas and definitions that are still at the forefront of research, as presented in " Model Theory: An Introduction" by David Marker.