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THE KETONEN ORDER

Part of:Set theory

Published online by Cambridge University Press:  18 June 2020

GABRIEL GOLDBERG
GABRIEL GOLDBERG*
Affiliation:
DEPARTMENT OF MATHEMATICS UC BERKELEY BERKELEY, CA 94720, USA E-mail: ggoldberg@berkeley.edu
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Abstract

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We study a partial order on countably complete ultrafilters introduced by Ketonen [2] as a generalization of the Mitchell order. The following are our main results: the order is wellfounded; its linearity is equivalent to the Ultrapower Axiom, a principle introduced in the author’s dissertation [1]; finally, assuming the Ultrapower Axiom, the Ketonen order coincides with Lipschitz reducibility in the sense of generalized descriptive set theory.

Keywords

ultrafilterLipschitz orderUltrapower Axiom

MSC classification

Primary:03E55: Large cardinals

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 2 , June 2020, pp. 585 - 604
Copyright
© The Association for Symbolic Logic 2020

References

REFERENCES

Goldberg, G., The ultrapower axiom , Ph.D. thesis, Harvard University, 2019. Available at https://dash.harvard.edu/bitstream/handle/1/42029483/GOLDBERG-DISSERTATION-2019.pdf.Google Scholar
Ketonen, J., Strong compactness and other cardinal sins . Annals of Mathematical Logic, vol. 5 (1972/1973), pp. 47–76.CrossRefGoogle Scholar
Kunen, K. and Paris, J. B., Boolean extensions and measurable cardinals . Annals of Mathematical Logic, vol. 2 (1970/1971), no. 4, pp. 359–377.CrossRefGoogle Scholar
Mitchell, W. J. and Steel, J. R., Fine Structure and Iteration Trees, Lecture Notes in Logic, vol. 3, Springer-Verlag, Berlin, 1994.CrossRefGoogle Scholar