The definable tree property for successors of cardinals

Archive for Mathematical Logic 55 (5-6):785-798 (2016)

Abstract
Strengthening a result of Leshem :1204–1214, 2000), we prove that the consistency strength of GCH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{GCH}$$\end{document} together with the definable tree property for all successors of regular cardinals is precisely equal to the consistency strength of existence of proper class many Π11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPi ^{1}_1$$\end{document}-reflecting cardinals. Moreover it is proved that if κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} is a supercompact cardinal and λ>κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda > \kappa $$\end{document} is measurable, then there is a generic extension of the universe in which κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document} is a strong limit singular cardinal of cofinality ω,λ=κ+,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega, ~ \lambda =\kappa ^+,$$\end{document} and the definable tree property holds at κ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa ^+$$\end{document}. Additionally we can have 2κ>κ+,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^\kappa > \kappa ^+,$$\end{document} so that SCH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{SCH}$$\end{document} fails at κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}.
Keywords No keywords specified (fix it)
Categories No categories specified
(categorize this paper)
ISBN(s)
DOI 10.1007/s00153-016-0494-7
Options
Edit this record
Mark as duplicate
Export citation
Find it on Scholar
Request removal from index
Revision history

Download options

Our Archive


Upload a copy of this paper     Check publisher's policy     Papers currently archived: 47,350
External links

Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library

References found in this work BETA

No references found.

Add more references

Citations of this work BETA

No citations found.

Add more citations

Similar books and articles

On the Consistency of the Definable Tree Property on ℵ.Amir Leshem - 2000 - Journal of Symbolic Logic 65 (3):1204 - 1214.
On the Consistency of the Definable Tree Property on $\Aleph_1$.Amir Leshem - 2000 - Journal of Symbolic Logic 65 (3):1204-1214.
Aronszajn Trees and the Successors of a Singular Cardinal.Spencer Unger - 2013 - Archive for Mathematical Logic 52 (5-6):483-496.
The Tree Property at Successors of Singular Cardinals.Menachem Magidor & Saharon Shelah - 1996 - Archive for Mathematical Logic 35 (5-6):385-404.
Fragility and Indestructibility of the Tree Property.Spencer Unger - 2012 - Archive for Mathematical Logic 51 (5-6):635-645.
The Tree Property and the Failure of SCH at Uncountable Cofinality.Dima Sinapova - 2012 - Archive for Mathematical Logic 51 (5-6):553-562.
Successors of Singular Cardinals and Coloring Theorems I.Todd Eisworth & Saharon Shelah - 2005 - Archive for Mathematical Logic 44 (5):597-618.
Strong Tree Properties for Two Successive Cardinals.Laura Fontanella - 2012 - Archive for Mathematical Logic 51 (5-6):601-620.
Strong Tree Properties for Small Cardinals.Laura Fontanella - 2013 - Journal of Symbolic Logic 78 (1):317-333.
A Remark on the Tree Property in a Choiceless Context.Arthur W. Apter - 2011 - Archive for Mathematical Logic 50 (5-6):585-590.

Analytics

Added to PP index
2017-11-06

Total views
0

Recent downloads (6 months)
0

How can I increase my downloads?

Downloads

Sorry, there are not enough data points to plot this chart.

My notes

Sign in to use this feature