Categoricity

One of the primary questions that motivated the development of model theory was: Which structures are determined by their first-order properties? While Theorem A.1 (located in the appendix) shows that all structures with finite domains are so determined, even if their language is infinite, all structures with infinite domains are never determined by their first-order properties. If a theory in a countable language has a countable model, then it not only has both countable and uncountable models, but models of arbitrary infinite cardinality. Models with domains of different cardinalities are never isomorphic—thus, if by counting we mean assigning a cardinal number to some collection of objects, then there are too many nonisomorphic models to count. The situation changes if the size of the domain is fixed. In this chapter we will discuss two essential cases: models with countable domains and models with domains the size of the set of real numbers. To avoid some set-theoretic complications, we will only consider theories in finite or countable languages.