Descriptive set theory and uncountable model theory

In the early days of the development of model theory it was considered natural and was certainly beneficial to assume that the theories under investigation were in a countable language. The primary advantage of this assumption was the presence of the Omitting Types Theorem of Grzegorczyk, Mostowski, and Ryll-Nardzewski [1], which generalized arguments of Henkin [3] and Orey [8]. Following this, Vaught [13] gave a very pleasing analysis of the class of countable models of such a theory. This led to Morley’s categoricity theorem [7] for certain classes of uncountable models of theories in a countable language. The landscape was completely altered by the subsequent work of Shelah (see e.g. [11]). He saw that the salient features of Morley’s proof did not require the assumption of the language being countable. Indeed, many of notions that were central to Shelah’s work, including unstability, the fcp, the independence property and the strict order property, are local. That is, a theory possesses such a property if and only if some formula has the property. Consequently, the total number of formulas in the language is not relevant. Still other notions, such as superstability, are not local but can be described in terms of countable fragments of the theory. That is, a theory of any cardinality is superstable if and only if all of its reducts to countable fragments of the theory are superstable. Using a vast collection of machinery, Shelah was able to answer literally hundreds of questions about