Preservation theorems for strong first-order logics

2019/06/21 by Christian Espíndola, Espíndola, Christian
Computer Science · #Advanced Algebra and Logic #Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge

paper · pdf · doi:10.48550/arxiv.1906.09173

openalex publication_date 2019/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove preservation theorems for L_{ω1, 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 L_{ω1, ω} 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.

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