Finite Generation Problem and n-ary Quantifiers

An abstract logic,ℒ is finitely generated if it can be represented in the form ℒ ωω (Q) where Q is a finite set of Lindstrom quantifiers. This article is a survey of a fairly general method for proving that a given logic is not finitely generated. The main ingredients of this method are a back-and-forth charcterization of equivalence with respect to all n-ary quantifiers and constructions of non-isomorphic models for which this characterization applies.

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