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 $\phi$, 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. We (...) also prove that if $\aleph_{\alpha}$ is characterized by a Scott sentence, at least one of $\aleph_{\alpha}$, $\aleph_{\alpha+1}$, or $$ is homogeneously characterizable. Based on an argument of Shelah, we give counterexamples that characterizable cardinals are not closed under predecessors or cofinalities. (shrink)

In [5], examples of incomplete sentences are given with maximal models in more than one cardinality. The question was raised whether one can find similar examples of complete sentences. In this paper, we give examples of complete ‐sentences with maximal models in more than one cardinality. From (homogeneous) characterizability of κ we construct sentences with maximal models in κ and in one of and more. Indeed, consistently we find sentences with maximal models in uncountably many distinct cardinalities.

We define some variations of the Scott rank for countable models and obtain some inequalities involving the ranks. For mono-unary algebras we prove that the game rank of any subtree does not exceed the game rank of the whole model. However, similar questions about linear orders remain unresolved.

We show that every ℵα can be characterized by the Scott sentence of some countable model; moreover there is a countable structure whose Scott sentence characterizes ℵ1 but whose automorphism group fails the topological Vaught conjecture on analytic sets. We obtain some partial information on Ulm type dichotomy theorems for the automorphism group of Knight's model.