Preservation theorems for strong first-order logics

TL;DR

This paper establishes preservation theorems for a countable fragment of Vaught's closed game logic, extending classical results to a stronger logic and solving a long-standing open problem in model theory.

Contribution

It generalizes classical preservation theorems to a strong infinitary logic without relying on the interpolation property, using definability results instead.

Findings

Proves preservation theorems for $ ext{L}_{ ext{ω}_1, G}$ in ZFC.

Extends classical theorems of Łoś-Tarski and Lyndon to a stronger logic.

Shows the equivalence of the Vopěnka principle with a definability theorem.

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 \L{}o\'s-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 Vop\v{e}nka principle and a general definability theorem on subsets preserved by homomorphisms.