Indestructibility of the tree property

Radek Honzik, Sarka Stejskalova

Published 2019-07-06Version 1

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