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On the ultrafilters and ultrapowers of strong partition cardinals

Published online by Cambridge University Press:  12 March 2014

J.M. Henle ,
E.M. Kleinberg  and
R.J. Watro
J.M. Henle
Affiliation:
Smith College, Northampton, Massachusetts 01063
E.M. Kleinberg
Affiliation:
State University of New York at Buffalo, Buffalo, New York 14214
R.J. Watro
Affiliation:
Villanova University, Villanova, Pennsylvania 19085
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Extract

A strong partition cardinal is an uncountable well-ordered cardinal κ such that every partition of [κ]κ (the size κ subsets of κ) into less than κ many pieces has a homogeneous set of size κ. The existence of such cardinals is inconsistent with the axiom of choice, and our work concerning them is carried out in ZF set theory with just dependent choice (DC). The consistency of strong partition cardinals with this weaker theory remains an open question. The axiom of determinacy (AD) implies that a large number of cardinals including ℵ1 have the strong partition property. The hypothesis that AD holds in the inner model of constructible sets built over the real numbers as urelements has important consequences for descriptive set theory, and results concerning strong partition cardinals are often applied in this context. Kechris [4] and Kechris et al. [5] contain further information concerning the relationship between AD and strong partition cardinals.

We assume familiarity with the basic results on strong partition cardinals as developed in Kleinberg [6], [7], [8] and Henle [2]. Recall that a strong partition cardinal κ is measurable; in fact every stationary subset of κ is measure one under some normal measure on κ. If μ is a countably additive ultrafilter extending the closed unbounded filter on κ, then the length of the ultrapower [κ]κ under the less than almost everywhere μ ordering is again a measurable cardinal. In §1 below we establish a polarized partition property on these measurable cardinals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 4 , December 1984 , pp. 1268 - 1272
Copyright
Copyright © Association for Symbolic Logic 1984

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References

REFERENCES

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