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A dichotomy in classifying quantifiers for finite models

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Affiliation:
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond Safra Campus, Givat Ram, Jerusalem 91904, Israel Department of Mathematics, Hill Center-Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA, E-mail: shelah@math.huji.ac.il
Mor Doron
Affiliation:
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond Safra Campus, Givat Ram, Jerusalem 91904, Israel Department of Mathematics, Hill Center-Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA, E-mail: shelah@math.huji.ac.il
Corresponding

Abstract

We consider a family of finite universes. The second order existential quantifier Q means for each U Є 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, either Q is interpretable by quantifying over subsets of U and one to one functions on U both of bounded order, or the logic L(Q) (first order logic plus the quantifier Q) is undecidable.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2005

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References

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[4]Shelah, Saharon, There are just four second-order quantifiers, Israel Journal of Mathematics, vol. 15 (1973), pp. 282300.CrossRefGoogle Scholar
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