A syntactic approach to Shelah's logic $L^1_\kappa$
Jouko Väänánen, Andrés Villaveces · arXiv (Cornell University) · 2021
In 2012 Shelah defined a new infinitary logic $L^1_\kappa$. We provide here an alternative approach to this logic, by defining a new logic $L^{1,c}_\kappa$, smaller than $L^1_\kappa$, whose closure under the $\Delta$-operator is $L^1_\kappa$ and which has a clearly defined, recursive syntax. We prove that it also satisfies a union lemma and the undefinability of well-order. We study its Lowenheim-Skolem-Tarski spectrum, its expressive power, and its model-theoretic potential. We provide a general proof of undefinability of well-order, in the context of abstract logics.