Revisiting the generalized {\L}o\'s-Tarski theorem

We present a new proof of the generalized {\L}o's-Tarski theorem ($\mathsf{G}\mathsf{L}\mathsf{T}(k)$) introduced in [1], over arbitrary structures. Instead of using $\lambda$-saturation as in [1], we construct just the “required saturation” directly using ascending chains of structures. We also strengthen the failure of $\mathsf{G}\mathsf{L}\mathsf{T}(k)$ in the finite shown in [2], by strengthening the failure of the {\L}o's-Tarski theorem in this context. In particular, we prove that not just universal sentences, but for each fixed $k$, even ${\mathrm{\Sigma }}_2^0$ sentences containing $k$ existential quantifiers fail to capture hereditariness in the finite. We conclude with two problems as future directions, concerning the {\L}o's-Tarski theorem and $\mathsf{G}\mathsf{L}\mathsf{T}(k)$, both in the context of all finite structures. [1] 10.1016/j.apal.2015.11.001 ; [2] 10.1007/978-3-642-32621-9_22