Three red herrings around Vaught’s conjecture

Abstract

We give a model theoretic proof that if there is a counterexample to Vaught’s conjecture there is a counterexample such that every model of cardinality $\aleph _1$ is maximal (strengthening a result of Hjorth’s). In the process we analyze three examples of a sentence characterizing $\aleph _1$. We also give a new proof of Harrington’s theorem that any counterexample to Vaught’s conjecture has models in $\aleph _1$ of arbitrarily high Scott rank below $\aleph _2$.