An \(L_{\omega_1,\omega}\) sentence characterizes a cardinal \(\kappa\) if it has a model of size \(\kappa\) but no model in \(\kappa^+\). We study the known examples of complete sentences that characterize \(\aleph_1\) and observe several notable phenomena about them. Our goal is to understand the mechanisms that make a sentence characterize \(\aleph_1\). This is related to some recent developments:

(1) Hjorth showed that if there is a counterexample to Vaught's conjecture, there is also one that characterizes \(\aleph_1\). So it is tempting to try proving Vaught's conjecture by showing that every counterexample must have a model in \(\aleph_2\) (which moreover a result of Harrington's suggests). This however turns out to be a red herring.

(2) While we know the notion of a complete sentence having a model in kappa is absolute for \(\kappa=\aleph_1\) and non-absolute for \(\kappa=\aleph_3\), even assuming GCH, this is still an open issue for \(\kappa=\aleph_2\).