On first-order expressibility of satisfiability in submodels

Abstract

Let $\kappa ,\lambda$ be regular cardinals, $\lambda \le \kappa$, let $\phi$ be a sentence of the language ${\mathcal{L}}_{\kappa ,\lambda }$ in a given signature, and let $\vartheta (\phi )$ express the fact that $\phi$ holds in a submodel, i.e., any model $\mathfrak{A}$ in the signature satisfies $\vartheta (\phi )$ if and only if some submodel $\mathfrak{B}$ of $\mathfrak{A}$ satisfies $\phi$. It was shown in [1] that, whenever $\phi$ is in ${\mathcal{L}}_{\kappa ,\omega }$ in the signature having less than $\kappa$ functional symbols (and arbitrarily many predicate symbols), then $\vartheta (\phi )$ is equivalent to a monadic existential sentence in the second-order language ${\mathcal{L}}_{\kappa ,\omega }^2$, and that for any signature having at least one binary predicate symbol there exists $\phi$ in ${\mathcal{L}}_{\omega ,\omega }$ such that $\vartheta (\phi )$ is not equivalent to any (first-order) sentence in ${\mathcal{L}}_{\infty ,\omega }$. Nevertheless, in certain cases $\vartheta (\phi )$ are first-order expressible. In this note, we provide several (syntactical and semantical) characterizations of the case when $\vartheta (\phi )$ is in ${\mathcal{L}}_{\kappa ,\kappa }$ and $\kappa$ is $\omega$ or a certain large cardinal.