Definability In Abstract Elementary Classes

29.03.2023 15:00 - 16:30

J. N. Nájar Salinas (Fundación Universidad América, CO)

The development of definability in the context of Abstract Elementary Classes has been boosted by the recent work of Shelah and Villaveces in which they prove that for every AEC \(\mathcal{K}\) in a vocabulary \(\tau\), there is a sentence \(\psi\in\mathbb{L}_{\beth_2(\kappa)^{+++},\kappa}(\tau)\) axiomatizing where \(\kappa\) is the Löwenheim-Skolem number of the class. Vasey enlarges \(\tau\) to \(\tilde{\tau}\) and proves that if the AEC is tame and type-short, there is a bijection between the Galois Types of the AEC and the quantifier free types in an infinitary logic \(\mathbb{L}_{\lambda,\lambda}(\tilde{\tau})\) for some suitable \(\lambda\), the semantic–syntactic correspondence. We extend the ideas of Vasey to make a partial semantic–syntactic correspondence-like results between Galois types and some types of the logic \(\mathbb{L}_{\beth_2(\kappa)^{+++},\kappa}(\tau)\).

Part of this is joint work with Andrés Villaveces.




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