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INDESTRUCTIBILITY OF THE TREE PROPERTY

Published online by Cambridge University Press:  16 September 2019

RADEK HONZIK  and
ŠÁRKA STEJSKALOVÁ
RADEK HONZIK
Affiliation:
DEPARTMENT OF LOGIC CHARLES UNIVERSITY CELETNÁ 20, PRAHA 1 116 42, CZECH REPUBLIC E-mail: radek.honzik@ff.cuni.czURL: logika.ff.cuni.cz/radekE-mail: sarka.stejskalova@ff.cuni.czURL: logika.ff.cuni.cz/sarka
ŠÁRKA STEJSKALOVÁ
Affiliation:
DEPARTMENT OF LOGIC CHARLES UNIVERSITY CELETNÁ 20, PRAHA 1 116 42, CZECH REPUBLIC E-mail: radek.honzik@ff.cuni.czURL: logika.ff.cuni.cz/radekE-mail: sarka.stejskalova@ff.cuni.czURL: logika.ff.cuni.cz/sarka
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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 $V\left[M {\left( {\kappa ,\lambda } \right)} \right]$ the tree property at $$\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $$ is indestructible under all ${\kappa ^ + }$-cc forcing notions which live in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$, where ${\rm{Add}}\left( {\kappa ,\lambda } \right)$ is the Cohen forcing for adding λ-many subsets of κ and $\left( {\kappa ,\lambda } \right)$ is the standard Mitchell forcing for obtaining the tree property at $\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \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 ${\left( {{\kappa ^{ + + }}} \right)^{{V^{\rm{*}}}}}$ is indestructible under all ${\kappa ^ + }$-cc forcing notions living in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$, and in addition under all forcing notions living in ${V^{\rm{*}}}$ which are ${\kappa ^ + }$-closed and “liftable” in a prescribed sense (such as ${\kappa ^{ + + }}$-directed closed forcings or well-met forcings which are ${\kappa ^{ + + }}$-closed with the greatest lower bounds).

Keywords

03E5503E35tree propertylifting of embeddingsindestructibility
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 467 - 485
Copyright
Copyright © The Association for Symbolic Logic 2019 

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