On First-Order Expressibility of Satisfiability in Submodels

Abstract

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

The work was supported by grant 16-11-10252 of Russian Science Foundation and was carried out at Steklov Mathematical Institute of Russian Academy of Sciences.