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On P-points over a measurable cardinal

Published online by Cambridge University Press:  12 March 2014

A. Kanamori*
Affiliation:
Harvard University, Cambridge, Massachusetts 02138

Extract

This paper continues the study of κ-ultrafilters over a measurable cardinal κ, following the sequence of papers Ketonen [2], Kanamori [1] and Menas [4]. Much of the concern will be with p-point κ-ultrafilters, which have become a focus of attention because they epitomize situations of further complexity beyond the better understood cases, normal and product κ-ultrafilters.

For any κ-ultrafilter D, let iD: VMDVκ/D be the elementary embedding of the universe into the transitization of the ultrapower by D. Situations of U < RKD will be exhibited when iU(κ) < iD(κ), and when iU(κ) = iD(κ). The main result will then be that if the latter case obtains, then there is an inner model with two measurable cardinals. (As will be pointed out, this formulation is due to Kunen, and improves on an earlier version of the author.) Incidentally, a similar conclusion will also follow from the assertion that there is an ascending Rudin-Keisler chain of κ-ultrafilters of length ω + 1. The interest in these results lies in the derivability of a substantial large cardinal assertion from plausible hypotheses on κ-ultrafilters.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1981

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References

REFERENCES

[1]Kanamori, A., Ultrafilters over a measurable cardinal, Annals of Mathematical Logic, vol. 11 (1977), pp. 315356.Google Scholar
[2]Ketonen, J., Ultrafilters over measurable cardinals, Fundamenta Mathematicae, vol. 77 (1973), pp. 257269.CrossRefGoogle Scholar
[3]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.CrossRefGoogle Scholar
[4]Menas, T., Some results on κ-ultrafilters, Fundamenta Mathematicae (to appear).Google Scholar
[5]Miller, A., Ph.D. Thesis, University of California at Berkeley, 1977.Google Scholar