Indestructibility of the tree property

Journal of Symbolic Logic 85 (1):467-485 (2020)
  Copy   BIBTEX

Abstract

In the first part of the article, we show that if $\omega \le \kappa < \lambda$ are cardinals, ${\kappa ^{ < \kappa }} = \kappa$, and λ is weakly compact, then in $V\left[M {\left} \right]$ the tree property at $$\lambda = \left^{V\left[ {\left} \right]} $$ is indestructible under all ${\kappa ^ + }$-cc forcing notions which live in $V\left[ {{\rm{Add}}\left} \right]$, where ${\rm{Add}}\left$ is the Cohen forcing for adding λ-many subsets of κ and $\left$ is the standard Mitchell forcing for obtaining the tree property at $\lambda = \left^{V\left[ {\left} \right]} $. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that λ is supercompact and generalize the construction and obtain a model ${V^{\rm{*}}}$, a generic extension of V, in which the tree property at ${\left^{{V^{\rm{*}}}}}$ is indestructible under all ${\kappa ^ + }$-cc forcing notions living in $V\left[ {{\rm{Add}}\left} \right]$, and in addition under all forcing notions living in ${V^{\rm{*}}}$ which are ${\kappa ^ + }$-closed and “liftable” in a prescribed sense.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 93,296

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Fragility and indestructibility of the tree property.Spencer Unger - 2012 - Archive for Mathematical Logic 51 (5-6):635-645.
A Gitik iteration with nearly Easton factoring.William J. Mitchell - 2003 - Journal of Symbolic Logic 68 (2):481-502.
Laver Indestructibility and the Class of Compact Cardinals.Arthur W. Apter - 1998 - Journal of Symbolic Logic 63 (1):149-157.
Strong tree properties for small cardinals.Laura Fontanella - 2013 - Journal of Symbolic Logic 78 (1):317-333.
On Cohen and Prikry Forcing Notions.Tom Benhamou & Moti Gitik - 2024 - Journal of Symbolic Logic 89 (2):858-904.

Analytics

Added to PP
2019-09-18

Downloads
18 (#860,222)

6 months
11 (#272,000)

Historical graph of downloads
How can I increase my downloads?

References found in this work

No references found.

Add more references