The Nonabsoluteness of Model Existence in Uncountable Cardinals for Lω1,ω

Abstract

For sentences $\varphi$ of $L_{{\omega }_1,\omega }$, we investigate the question of absoluteness of $\varphi$ having models in uncountable cardinalities. We first observe that having a model in ${\aleph }_1$ is an absolute property, but having a model in ${\aleph }_2$ is not as it may depend on the validity of the continuum hypothesis. We then consider the generalized continuum hypothesis (GCH) context and provide sentences for any $\alpha \in {\omega }_1\setminus \{0,1,\omega \}$ for which the existence of a model in ${\aleph }_{\alpha }$ is nonabsolute (relative to large cardinal hypotheses). Finally, we present a complete sentence for which model existence in ${\aleph }_3$ is nonabsolute.