On the absoluteness of generalized quantifiers with certain closure properties

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发表日期：

2026年4月25日

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Nathanael Ackerman,
Cameron Freer,
Mary Leah Karker

摘要

We consider generalized quantifiers which are either closed under embeddings or closed under monomorphisms or consist of structures that contain a substructure in a fixed $\Sigma ^1_2$ $Σ_{2}^{1}$ -definable set (which we call a $\Sigma ^1_2$ $Σ_{2}^{1}$ -quantifier). We study the relativization of these quantifiers with respect to these closure properties. We also consider when formulas built from such quantifiers are absolute with respect to these closure properties. We show that formulas built from embedding-closed or monomorphism-closed quantifiers need not be absolute (with respect to these definitions), whereas formulas built from $\Sigma ^1_2$ $Σ_{2}^{1}$ -quantifiers are absolute (with respect to the $\Sigma ^1_2$ $Σ_{2}^{1}$ -definitions). The latter result makes use of an absoluteness result, which may be of independent interest.