Revisiting the generalized Łoś-Tarski theorem
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We present a new proof of the generalized Łoś-Tarski theorem (GLT(k)) introduced in [1], over arbitrary structures. Instead of using λ-saturation as in [1], we construct just the "required saturation" directly using ascending chains of structures. We also strengthen the failure of GLT(k) in the finite shown in [2], 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 Σ^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 GLT(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
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