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Descendingly Incomplete Ultrafilters and the Cardinality of Ultrapowers

Published online by Cambridge University Press:  20 November 2018

Andrew Adler  and
Murray Jorgensen
Andrew Adler
Affiliation:
University of British Columbia, Vancouver, British Columbia
Murray Jorgensen
Affiliation:
Waterloo Lutheran University, Waterloo, Ontario
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Let D be an ultrafilter on I, and let k be a cardinal. D is said to be k-descendingly incomplete (k-d.i.) if there exists a chain Xα : α < k of elements of D such that α < β → Xα ⊆ Xβ and Xα = ϕ. Such a chain will be called a k-chain for D. The notion of k-descending incompleteness is due to Chang [3].

In this paper we explore the relationship between the cardinality of the ultrapower kI/D and the existence of certain chains on D. Since we deal so much with questions of size, we do not ordinarily make a notational distinction between a set and its cardinality. Where such a distinction is useful, the cardinality of a set A will be denoted by |A|.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 5 , October 1972 , pp. 830 - 834
Copyright
Copyright © Canadian Mathematical Society 1972

References

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3. Chang, C. C., Descendingly incomplete ultrafilters, Trans. Amer. Math. Soc. 126 (1967), 108–111.Google Scholar
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