The Hierarchy Theorem for Generalized Quantifiers

The concept of a generalized quantiier of a given similarity type was deened in Lin66]. Our main result says that on nite structures diierent similarity types give rise to diierent classes of generalized quantiiers. More exactly, for every similarity type t there is a generalized quantiier of type t which is not deenable in the extension of rst order logic by all generalized quantiiers of type smaller than t. This was proved for unary similarity types by Per Lindstrr om Wes] with a counting argument. We extend his method to arbitrary similarity types.