A Survey of the Uncountable Spectra of Countable Theories

Let I(T, κ) denote the number of non-isomorphic models of T of size k, and let the spectrum of T denote the map κ → I(T, κ). As the determination of the spectrum of a theory for all cardinals k involves settling Vaught’s conjecture, we concentrate on the uncountable spectrum, i.e., restricting the spectrum to the class of uncountable cardinals κ. In this paper we survey the classification of the uncountable spectra of complete theories in a countable language, culminating with Theorem 3.6, which enumerates the possible uncountable spectra of such a theory. In the process, we exhibit a family of countable configurations which determine the uncountable spectrum of any countable theory T. That is, the uncountable spectrum of T is determined by which of these configurations are embeddable in a model of T.