Abstract

This is the first part of a study on cardinals that are characterizable by Scott sentences. Building on previous work of Hjorth, Malitz, and Baumgartner, we study which cardinals are characterizable by a Scott sentence $\phi$, in the sense that $\phi$ characterizes $\kappa$, if $\phi$ has a model of size $\kappa$ but no models of size $\kappa^{+}$. We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions, and countable products. We also prove that if $\aleph_{\alpha}$ is characterized by a Scott sentence, at least one of $\aleph_{\alpha}$, $\aleph_{\alpha+1}$, or $$ is homogeneously characterizable. Based on an argument of Shelah, we give counterexamples that characterizable cardinals are not closed under predecessors or cofinalities