Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-17T15:01:20.216Z Has data issue: false hasContentIssue false
  • English
  • Français

THE TREE PROPERTY UP TO אω+1

Published online by Cambridge University Press:  25 June 2014

ITAY NEEMAN
ITAY NEEMAN*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES LOS ANGELES, CA 90095-1555E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Assuming ω supercompact cardinals we force to obtain a model where the tree property holds both at אω+1, and at אn for all 2 ≤ n < ω. A model with the former was obtained by Magidor–Shelah from a large cardinal assumption above a huge cardinal, and recently by Sinapova from ω supercompact cardinals. A model with the latter was obtained by Cummings–Foreman from ω supercompact cardinals. Our model, where the two hold simultaneously, is another step toward the goal of obtaining the tree property on increasingly large intervals of successor cardinals.

Keywords

03E3503E0503E55Aronszajn treestree propertysupercompact cardinals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 79 , Issue 2 , June 2014 , pp. 429 - 459
Copyright
Copyright © The 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

Abraham, Uri, Aronszajn trees on א2 and א2. Annals of Pure and Applied Logic, vol. 24 (1983), no. 3, pp. 213–230.Google Scholar
Cummings, James and Foreman, Matthew, The tree property. Advances in Mathematics, vol. 133 (1998), no. 1, pp. 1–32.CrossRefGoogle Scholar
Foreman, Matthew, Magidor, Menachem, and Schindler, Ralf-Dieter, The consistency strength of successive cardinals with the tree property, this JOURNAL, vol. 66 (2001), no. 4, pp. 1837–1847.Google Scholar
Magidor, Menachem and Shelah, Saharon, The tree property at successors of singular cardinals. Archive for Mathematical Logic, vol. 35 (1996), no. 5–6, pp. 385–404.Google Scholar
Mitchell, William, Aronszajn trees and the independence of the transfer property. Annals of Mathematical Logic, vol. 5 (1972/73), pp. 21–46.CrossRefGoogle Scholar
Neeman, Itay, Aronszajn trees and failure of the singular cardinal hypothesis. Journal of Mathematical Logic, vol. 9 (2009), no. 1, pp. 139–157.Google Scholar
Sinapova, Dima, The tree property and the failure of the singular cardinal hypothesis at $\aleph _{\omega ^2 } $, this journal, vol. 77 (2012), no. 3, pp. 934–946.Google Scholar
Sinapova, Dima, The tree property atאω+1, this journal, vol. 77 (2012), no. 1, pp. 279–290.Google Scholar
Specker, E., Sur un problème de Sikorski. Colloquium Mathematicum, vol. 2 (1949), pp. 9–12.CrossRefGoogle Scholar
Unger, Spencer, A model of Cummings and Foreman revisited. To appear.Google Scholar
Unger, Spencer, Fragility and indestructibility of the tree property. Archive for Mathematical Logic, vol. 51 (2012), no. 5–6, pp. 635–645.CrossRefGoogle Scholar