Hostname: page-component-7c8c6479df-p566r Total loading time: 0 Render date: 2024-03-27T20:33:58.882Z Has data issue: false hasContentIssue false

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

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baldwin, John T., Definable second order quantifiers, Model theoretic logics (Barwise, J. and Feferman, S., editors), Perspectives in Mathematical Logic, Springer-Verlag, 1985, pp. 446478.Google Scholar
[2]Gaifman, Haim, On local and nonlocal properties, Logic Colloquium '81 (Stern, J., editor). North Holland, 1982, pp. 105135.Google Scholar
[3]Lavrov, I. A., The effective non-separability of the set of identically true formulae and the set of finitly refutable formulae for certain elementary theories, Algebra i Logika, vol. 2 (1963). no. 1, pp. 518, (Russian).Google Scholar
[4]Shelah, Saharon, There are just four second-order quantifiers, Israel Journal of Mathematics, vol. 15 (1973), pp. 282300.CrossRefGoogle Scholar
[5]Shelah, Saharon, Classifying of generalized quantifiers, Around classification theory of models, Lecture Notes in Mathematics, no. 1182, Springer-Verlag, 1986. pp. 146.CrossRefGoogle Scholar
[6]Shelah, Saharon, On quantification with a finite universe, this Journal, vol. 65 (2000), pp. 10551075.Google Scholar