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The hierarchy theorem for generalized quantifiers

Published online by Cambridge University Press:  12 March 2014

Lauri Hella ,
Kerkko Luosto  and
Jouko Väänänen
Lauri Hella
Affiliation:
Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), 00014University of Helsinki, Finland, E-mail: hella@cc.helsinki.fi
Kerkko Luosto
Affiliation:
Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), 00014University of Helsinki, Finland, E-mail: kluosto@cc.helsinki.fi
Jouko Väänänen
Affiliation:
Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), 00014University of Helsinki, Finland, E-mail: jvaananen@cc.helsinki.fi
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Abstract

The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with a counting argument. We extend his method to arbitrary similarity types.

Keywords

generalized quantifierfinite model theoryabstract model theory
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 3 , September 1996 , pp. 802 - 817
Copyright
Copyright © Association for Symbolic Logic 1996

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