Notes on cardinals that are characterizable by a complete (Scott) sentence

TL;DR

This paper investigates which infinite cardinals can be uniquely characterized by Scott sentences, establishing closure properties and providing counterexamples related to their characterization.

Contribution

It identifies closure properties of cardinals characterized by Scott sentences and shows that such cardinals are not necessarily closed under predecessors or cofinalities.

Findings

Characterized cardinals are closed under successors, countable unions, and countable products.

At least one of an aleph and its successor is homogeneously characterizable.

Counterexamples show non-closure under predecessors and cofinalities.

Abstract

This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] 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 (cf. theorems 2.3, 3.4, and corollary 3.6). We also prove that if $\aleph_\alpha$ is characterized by a Scott sentence, at least one of $\aleph_alpha$ and $\aleph_alpha^+$ is homogeneously characterizable (cf. definition 1.3 and theorem 2.9). Based on Shelah's [8], we give counterexamples that characterizable cardinals are not closed under predecessors, or cofinalities.