Non-absoluteness of Hjorth's Cardinal Characterization

Philipp Lücke, Ioannis Souldatos

In [5], Hjorth proved that for every countable ordinal $\alpha$, there exists a complete ${\mathcal{L}}_{{\omega }_1,\omega }$-sentence ${\varphi }_{\alpha }$ that has models of all cardinalities less than or equal to…