  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.
