Revisiting the generalized Łoś-Tarski theorem

We present a new proof of the generalized Łoś-Tarski theorem (\(\mathsf{GLT}(k)\)) 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...

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Bibliographic Details
Published in:arXiv.org 2018-11
Main Author: Sankaran, Abhisekh
Format: Article
Language:English
Subjects:
Saturation
Sentences
Theorems
Online Access:Get full text
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Summary:We present a new proof of the generalized Łoś-Tarski theorem (\(\mathsf{GLT}(k)\)) 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 \(\mathsf{GLT}(k)\) 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 \(\Sigma^0_2\) 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. [1] 10.1016/j.apal.2015.11.001 ; [2] 10.1007/978-3-642-32621-9\_22
ISSN:2331-8422