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A dichotomy in classifying quantifiers for finite models

Mor Doron·Saharon Shelah
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Abstract

We consider a family U of finite universes. The second order quantifier Q_R, means for each u in U quantifying over a set of n(R)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Q_R, ever Q_R is interpretable by quantifying over subsets of u and one to one functions on u both of bounded order, or the logic L(Q_R) (first order logic plus the quantifier Q_R) is undecidable.

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Mor Doron, Saharon Shelah. 2004-05-06. A dichotomy in classifying quantifiers for finite models. https://arxiv.org/abs/math/0405091

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