Let $M$ be structure of cardinality $\kappa$ with language $\tau(M)\leq\kappa$. We say that is \textit{compactly expandable} if for every theory $T$ of language $\tau(T)$ with $\tau(M)\subseteq \tau(T)$ and $|T|\leq\kappa$ can be realised in an expansion of $M$, whenever every finite subset of $T$ can be realised in an expansion of $M$. We say that $M$ is \textit{expandable} if the requirement of finite satisfiability can be weakened to: $T$ is consistent with the complete theory Th$(M)$ of $M$. Clearly, expandable models are compactly expandable. The existence of a compactly expandable model which is not expandable was conjectured in \cite{95} and has been recently proved in \cite{19} using the notion of universal theory and the logic of the quantifier of cofinality $\omega$. We plan to present these results.
References
[1] E. Casanovas. Compactly expandable models and stability. The Journal of Symbolic Logic, 60:673–683, 1995.
[2] E. Casanovas and S. Shelah. Universal theories and compactly expandable models. The Journal of Symbolic Logic, 84:1215–1223, 2019.
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