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

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 $\varphi$, 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 (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 ${\aleph }_{a}lpha$ and ${\aleph }_{a}lph{a}^{+}$ 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.

Version 2 replaces version 1 of the same paper (with the same title), but version 2 contains only half of the content of version 1. The second half of version 1 will be posted by itself

References in corpus (1)

• Linear Orderings and Powers of Characterizable Cardinal

Cited by in corpus (1)

• Kurepa trees and spectra of ${\mathcal{L}}_{{\omega }_1,\omega }$-sentences