Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence

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 $\varphi$, in the sense that $\varphi$ characterizes $\kappa$, if $\varphi$ 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 (see Theorems 3.3 and 4.6 and Corollary 4.8). We also prove that if ${\aleph }_{\alpha }$ is characterized by a Scott sentence, at least one of ${\aleph }_{\alpha }$, ${\aleph }_{\alpha +1}$, or $({\aleph }_{\alpha +1},{\aleph }_{\alpha })$ is homogeneously characterizable (see Definitions 1.3 and 1.4 and Theorem 3.19). Based on an argument of Shelah, we give counterexamples that characterizable cardinals are not closed under predecessors or cofinalities.