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On the cofinality of ultrapowers

Published online by Cambridge University Press:  12 March 2014

Andreas Blass  and
Heike Mildenberger
Andreas Blass
Affiliation:
Mathematics Department, University of Michigan, Ann Arbor, MI 48109, U.S.A. E-mail: [email protected]
Heike Mildenberger
Affiliation:
Mathematisches Institut der Universität Bonn, Beringstrasse 1, 53115 Bonn, Germany E-mail: [email protected]
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Abstract

We prove some restrictions on the possible cofinalities of ultrapowers of the natural numbers with respect to ultrafilters on the natural numbers. The restrictions involve three cardinal characteristics of the continuum, the splitting number s, the unsplitting number r, and the groupwise density number g. We also prove some related results for reduced powers with respect to filters other than ultrafilters.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 2 , June 1999 , pp. 727 - 736
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1] Blass, A., Near Coherence of filters, I: Cofinal equivalence of models of arithmetic, Notre Dame Journal of Formal Logic, vol. 27 (1986), pp. 579–591.Google Scholar
[2] Blass, A., Applications of superperfect forcing and its relatives, Set theory and its applications (Steprāns, J. and Watson, S., editors), Lecture Notes in Mathematics, no. 1401, Springer-Verlag, 1989, pp. 18–40.Google Scholar
[3] Blass, A. and Shelah, S., There may be simple - and -points and the Rudin-Keisler ordering may be downward directed, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 213–243.Google Scholar
[4] Canjar, R. M., Model-Theoretic Properties of Countable Ultraproducts Without the Continuum Hypothesis, Ph.D. thesis , University of Michigan, 1982.Google Scholar
[5] Canjar, R. M., Countable ultraproducts without CH, Annals of Pure and Applied Logic, vol. 37 (1988), pp. 1–79.Google Scholar
[6] Canjar, R. M., Cofinalities of countable ultraproducts: the existence theorem, Notre Dame Journal of Formal Logic, vol. 30 (1989), pp. 539–542.Google Scholar
[7] Nyikos, P., Special ultrafilters and cofinal subsets ofωω, (to appear).Google Scholar
[8] Roitman, J., Non-isomorphic H-fields from non-isomorphic ultrapowers, Mathematische Zeitschrift, vol. 181 (1982), pp. 93–96.Google Scholar
[9] Solomon, R. C., Families of sets and functions, Czechoslovak Mathematical Journal, vol. 27 (1977), pp. 556–559.Google Scholar
[10] van Douwen, E., The integers and topology, Handbook of Set Theoretic Topology (Kunen, K. and Vaughan, J., editors), North-Holland, 1984, pp. 111–167.Google Scholar
[11] Vaughan, J., Small uncountable cardinals and topology, Open Problems in Topology (van Mill, J. and Reed, G., editors), North-Holland, 1990, pp. 195–218.Google Scholar