Preservation theorems for strong first-order logics

Abstract

We prove preservation theorems for ${\mathcal{L}}_{{\omega }_1,G}$, the countable fragment of Vaught's closed game logic. These are direct generalizations of the theorems of Łoś-Tarski (resp. Lyndon) on sentences of ${\mathcal{L}}_{{\omega }_1,\omega }$ preserved by substructures (resp. homomorphic images). The solution, in $ZFC$, only uses general features and can be extended to several variants of other strong first-order logic that do not satisfy the interpolation theorem; instead, the results on infinitary definability are used. This solves an open problem dating back to 1977. Another consequence of our approach is the equivalence of the Vopěnka principle and a general definability theorem on subsets preserved by homomorphisms.

Added acknowledgements. 8 pages. arXiv admin note: text overlap with arXiv:1906.09169