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On the relationship between the partition property and the weak partition property for normal ultrafilters on Pκλ1

Published online by Cambridge University Press:  12 March 2014

Julius B. Barbanel
Julius B. Barbanel*
Affiliation:
Department of Mathematics, Union College, Schenectady, New York 12308, E-mail: BARBANEJ@GAR.UNION.EDU
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Abstract

Suppose κ is a supercompact cardinal and λ > κ. We study the relationship between the partition properly and the weak partition properly for normal ultrafilters on Pκλ. On the one hand, we show that the following statement is consistent, given an appropriate large cardinal assumption: The partition property and the weak partition properly are equivalent, there are many normal ultrafilters that satisfy these properties, and there are many normal ultrafilters that do not satisfy these properties. On the other hand, we consider the assumption that, for some λ > κ, there exists a normal ultrafilter U on Pκλ such that U satisfies the weak partition property but does not satisfy the partition property. We show that this assumption is implied by the assertion that there exists a cardinal γ > κ such that γ is γ+-supercompact, and, assuming the GCH, it implies the assertion that there exists a cardinal γ > κ such that γ is a measurable cardinal with a normal ultrafilter concentrating on measurable cardinals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 1 , March 1993 , pp. 119 - 127
Copyright
Copyright © Association for Symbolic Logic 1993

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Footnotes

1

We would like to thank the referee for a number of useful suggestions on a previous version of this paper.

References

REFERENCES

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