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 , in the sense that characterizes , if has a model of size but no models of size .
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 is characterized by a Scott sentence, at least one of , , or 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.
"Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence." Notre Dame J. Formal Logic 55 (4) 533 - 551, 2014. https://doi.org/10.1215/00294527-2798727