Revisiting the Generalized Łoś-Tarski Theorem

Abstract

We present a new proof of the generalized Łoś-Tarski theorem ($\mathsf {GLT}({k})$) from [6], over arbitrary structures. Instead of using $\lambda$-saturation as in [6], we construct just the “required saturation” directly using ascending chains of structures. We also strengthen the failure of $\mathsf {GLT}({k})$ in the finite shown in [7], by strengthening the failure of the Łoś-Tarski theorem in this context. In particular, we prove that not just universal sentences, but for each fixed k, even $\varSigma ^0_2$ sentences containing k existential quantifiers fail to capture hereditariness in the finite. We conclude with two problems as future directions, concerning the Łoś-Tarski theorem and $\mathsf {GLT}({k})$, both in the context of all finite structures.