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Cardinalities of ultraproducts of finite sets

Published online by Cambridge University Press:  12 March 2014

Sabine Koppelberg
Sabine Koppelberg*
Affiliation:
II., Mathematisches Institut der Freien Universität, Berlin, Federal Republic of Germany
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Extract

Keisler, in [1], defined a set F(D) of infinite cardinals for every ultrafilter D on a set I, and, assuming GCH, gave a sufficient condition for a set C of infinite cardinals to have the form F(D) for suitable D and I. In this paper we prove a similar theorem (Theorem 1) under considerably weaker assumptions. Our main tool is a construction of elementary end extensions of Boolean ultrapowers of ω outlined by Shelah in [6, Exercise VI.3.35]. Hence this paper will mostly be concerned with Boolean ultrapowers of ω.

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Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 3 , September 1980 , pp. 574 - 584
Copyright
Copyright © Association for Symbolic Logic 1980

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References

REFERENCES

[1]Keisler, H. J., Ultraproducts offinite sets, this Journal, vol. 32 (1967), pp. 47–57.Google Scholar
[2]Koppelberg, B. and Koppelbero, S., A Boolean ultrapower which is not an ultrapower, this Journal, vol. 41 (1976), pp. 245–249.Google Scholar
[3]Koppelberg, S., Boolean algebras as unions of chains of subalgebras, Algebra Universalis, vol. 7 (1977), pp. 195–203.CrossRefGoogle Scholar
[4]Mansfield, R., The theory of Boolean ultrapowers, Annals of Mathematical Logic, vol. 2 (1971), pp. 297–323.CrossRefGoogle Scholar
[5]Shelah, S., On the cardinality of ultraproducts of finite sets, this Journal, vol. 35 (1970), pp. 83–84.Google Scholar
[6]Shelah, S., Classification theory and the number of nonisomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[7]Sikorski, R., Boolean algebras, 2nd edition, Springer-Verlag, Berlin and New York, 1964.Google Scholar
[8]Solovay, R. M. and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, vol. 94 (1970), pp. 201–245.CrossRefGoogle Scholar