Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-18T12:33:52.313Z Has data issue: false hasContentIssue false
  • English
  • Français

Supercompact cardinals, trees of normal ultrafilters, and the partition property1

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
Get access Rights & Permissions [Opens in a new window]

Abstract

Suppose κ is a supercompact cardinal. It is known that for every λ ≥ κ, many normal ultrafilters on Pκ(λ) have the partition property. It is also known that certain large cardinal assumptions imply the existence of normal ultrafilters without the partition property. In [1], we introduced the tree T of normal ultrafilters associated with κ. We investigate the distribution throughout T of normal ultrafilters with and normal ultrafilters without the partition property.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 3 , September 1986 , pp. 701 - 708
Copyright
Copyright © Association for Symbolic Logic 1986

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

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

References

REFERENCES

[1]Barbanel, J. B., Supercompact cardinals and trees of normal ultrafilters, this Journal, vol. 47 (1982), pp. 89–109.Google Scholar
[2]Baumgartner, J. E., Ineffability properties of cardinals. I, Finite and infinite sets ( P. Erdös sixtieth birthday colloquium), Colloquia Mathematica Societatis János Bolyai, vol. 10, part I, North-Holland, Amsterdam, 1975, pp. 109–130.Google Scholar
[3]Kunen, K., Some remarks on theorems of Ketonen, Menas, and Solovay, unpublished handwritten manuscript (4 pages), 1971.Google Scholar
[4]Kunen, K. and Pelletier, D. H., On a combinatorial property of Menas related to the partition property for measures on supercompact cardinals, this Journal, vol. 48 (1983), pp. 475–481.Google Scholar
[5]Levy, A., The sizes of indescribable cardinals, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 205–218.CrossRefGoogle Scholar
[6]Magidor, M., On the role of supercompact and extendible cardinals in logic, Israel Journal of Mathematics, vol. 10 (1971), pp. 147–157.CrossRefGoogle Scholar
[7]Menas, T. K., A combinatorial property of P κ(λ), this Journal, vol. 41 (1976), pp. 225–234.Google Scholar
[8]Pelletier, D. H., The partition property for certain extendible measures on supercompact cardinals, Proceedings of the American Mathematical Society, vol. 81 (1981), pp. 607–612.CrossRefGoogle Scholar
[9]Solovay, R., Reinhardt, W. and Kanamori, A., Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73–116.CrossRefGoogle Scholar