Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-08T07:32:25.827Z Has data issue: false hasContentIssue false
  • English
  • Français

THE STRONG TREE PROPERTY AT SUCCESSORS OF SINGULARCARDINALS

Published online by Cambridge University Press:  17 April 2014

LAURA FONTANELLA
LAURA FONTANELLA*
Affiliation:
EQUIPE DE LOGIQUE MATHÉMATIQUE, UNIVERSITÉ PARIS DIDEROT PARIS 7, UFR DE MATHÉMATIQUES CASE 7012, SITE CHEVALERET, 75205 PARIS CEDEX 13, FRANCE, KURT GÖDEL RESEARCH CENTER FOR MATHEMATICAL LOGIC, UNIVERSITY OF VIENNA, DEPARTMENT OF MATHEMATICS, WÄHRINGER STRASSE 25, VIENNA 1090, AUSTRIA E-mail: [email protected], URL:http://www.logique.jussieu.fr/∼fontanella
Get access Rights & Permissions [Opens in a new window]

Abstract

An inaccessible cardinal is strongly compact if, and only if, it satisfies thestrong tree property. We prove that if there is a model of ZFC with infinitelymany supercompact cardinals, then there is a model of ZFC where ${\aleph _{\omega + 1}}$ has the strong tree property. Moreover, we prove that everysuccessor of a singular limit of strongly compact cardinals has the strong treeproperty.

Keywords

Tree propertylarge cardinalsforcing03E55.
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 1 , March 2014 , pp. 193 - 207
Copyright
Copyright © Association for Symbolic Logic 2014 

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.)

References

REFERENCES

Cummings, J. and Foreman, M., The tree property. Advances in Mathematics, vol. 133 (1998),pp. 1–32.Google Scholar
Di Prisco, C. A. and Zwicker, S., Flipping properties and supercompact cardinals. Fundamenta Mathematicae, vol. CIX (1980), pp. 31–36.Google Scholar
Erdös, P. and Tarski, A., On some problems involving inaccessible cardinals, Essays on the Foundations of Mathematics (Bar-Hillel, Y., Poznanski, E. I. J., Rabin, M. O., and Robinson, A., editor), Magnes Press, Jerusalem, 1961.Google Scholar
Fontanella, L., Strong tree properties for two successive cardinals. Archive for Mathematical Logic, vol. 51 (2012), no. 5–6, pp. 601–620.Google Scholar
Fontanella, L., Strong tree properties at small cardinals, this Journal, vol. 78 (2012), no. 1, pp. 317–333.Google Scholar
Jech, T., Some combinatorial problems concerning uncountable cardinals. Annals of Mathematical Logic, vol. 5 (1973), pp. 165–198.Google Scholar
Jech, T., Set theory: The third millennium edition revised and expanded, Springer, Berlin, 2006.Google Scholar
Kanamori, A., The higher infinite. Large cardinals in set theory from their beginnings, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994.Google Scholar
Kunen, K., Set theory. An introduction to independence proofs, North-Holland, Amsterdam, 1980.Google Scholar
Magidor, M., Combinatorial characterization of supercompact cardinals. Proceedings of the American Mathematical Society, vol. 42 (1974), pp. 279–285.Google Scholar
Magidor, M. and Shelah, S., The tree property at successors of singular cardinals. Archive for Mathematical Logic, vol. 35 (1996), no. 5–6, pp. 385–404.Google Scholar
Mitchell, W. J., Aronszajn trees and the independence of the transfer property.Annals of Mathematical Logic, vol. 5 (1972), pp 21–46.Google Scholar
Neeman, I., The tree property up to ${\aleph _{\omega + 1}}$ To appear, (2012).Google Scholar
Sinapova, D., The tree property at ${\aleph _{\omega + 1}}$ this Journal, vol. 77 (2012), no. 1, pp. 279–290.Google Scholar
Unger, S., The ineffable tree property, unpublished (2012).Google Scholar
Weiss, C., The combinatorial essence of supercompactness.Annals of Pure and Applied Logic, vol. 163 (2012), pp. 1710–1717.Google Scholar