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Quantifier elimination for neocompact sets

Published online by Cambridge University Press:  12 March 2014

H. Jerome Keisler
H. Jerome Keisler*
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA. E-mail: keisler@math.wisc.edu
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Abstract

We shall prove quantifier elimination theorems for neocompact formulas, which define neocompact sets and are built from atomic formulas using finite disjunctions, infinite conjunctions, existential quantifiers, and bounded universal quantifiers. The neocompact sets were first introduced to provide an easy alternative to nonstandard methods of proving existence theorems in probability theory, where they behave like compact sets. The quantifier elimination theorems in this paper can be applied in a general setting to show that the family of neocompact sets is countably compact. To provide the necessary setting we introduce the notion of a law structure. This notion was motivated by the probability law of a random variable. However, in this paper we discuss a variety of model theoretic examples of the notion in the light of our quantifier elimination results.

Keywords

eliminationquantifierneocompact

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 4 , December 1998, pp. 1442 - 1472
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1] Baratella, S. and Ng, S.-A., Quantifier elimination in Banach spaces, to appear.Google Scholar
[2] Billingsley, P., Convergence of probability measures, Wiley, 1968.Google Scholar
[3] Cutland, N. and Keisler, H. J., Neocompact sets and stochastic Navier-Stokes equations, Stochastic partial differential equations (Etheridge, A., editor), London Mathematical Society Lecture Notes Series, no. 216, Cambridge, 1995, pp. 31–54.Google Scholar
[4] Fajardo, S. and Keisler, H. J., Existence theorems in probability theory, Advances in Mathematics, vol. 120 (1996), pp. 191–257.Google Scholar
[5] Fajardo, S. and Keisler, H. J., Neometric spaces, Advances in Mathematics, vol. 118 (1996), pp. 134–175.CrossRefGoogle Scholar
[6] Henson, C. W., Nonstandard hulls of Banach spaces, Israel Journal of Mathematics, vol. 25 (1976), pp. 108–144.CrossRefGoogle Scholar
[7] Henson, C. W. and Iovino, J., Banach space model theory, to appear.Google Scholar
[8] Hoover, D. N. and Keisler, H. J., Adapted probability distributions, Transactions of the American Mathematical Society, vol. 286 (1984), pp. 159–201.Google Scholar
[9] Keisler, H. J., Rich and saturated adapted spaces, to appear.Google Scholar
[10] Keisler, H. J., Probability quantifiers. Model-theoretic logics (Barwise, K. J. and Feferman, S., editors), Springer-Verlag, 1985, pp. 506–556.Google Scholar
[11] Keisler, H. J., From discrete to continuous time, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 99–141.Google Scholar
[12] Keisler, H. J., A neometric survey, Developments in nonstandard analysis (Cutland, N. et al., editors), Longman, 1995, pp. 233–250.Google Scholar
[13] Kelley, J., General topology, D. Van Nostrand, 1955.Google Scholar
[14] Macintyre, A. and Wilkie, A., On the decidability of the real exponential field, to appear.Google Scholar
[15] Tarski, A., A decision method for elementary algebra and geometry, second ed., University of California Press, 1951.Google Scholar
[16] van den Dries, L., Macintyre, A., and Marker, D., The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics, vol. 140 (1994), pp. 183–205.CrossRefGoogle Scholar