April 12
Boban Velickovic,
University of Paris
On some infinitary logics
Lindstrom Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the downward Lowenheim-Skolem Theorem. Despite intensive efforts, no such model theoretic characterizations were obtained for infinitary logics, until Shelah introduced his logic $L^1_\kappa$. We define a new class of logics $L^1_{\kappa,\alpha}$ generalizing Shelah's logic and examine their expressive power. If $\kappa$ is a fixed point of the $\beth$-function these logics coincide with $L^1_\kappa$. We given a different version of Lindstrom's theorem in terms of the $\phi$-submodel relation.
We also discuss algebraic characterization of the elementary equivalence relation for these logics for suitable $\kappa$.
Joint work with Jouko Väänänen.