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

@article{Souldatos2014NotesOC,
title={Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence},
author={Ioannis A. Souldatos},
journal={Notre Dame J. Formal Log.},
year={2014},
volume={55},
pages={533-551}
}
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…
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