Indestructibility of the tree property

Abstract

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 $V[\M(κ,λ)]$ the tree property at $λ= κ^{++V[\M(κ,λ)]}$ is indestructible under all ${\kappa }^{+}$-cc forcing notions which live in $V[\Add(κ,λ)]$, where $\Add(κ,λ)$ is the Cohen forcing for adding $\lambda$-many subsets of $\kappa$ and $\M(κ,λ)$ is the standard Mitchell forcing for obtaining the tree property at $λ= (κ^{++})^{V[\M(κ,λ)]}$. 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 $V^{\ast }$, a generic extension of $V$, in which the tree property at $({\kappa }^{++}{)}^{V^{\ast }}$ is indestructible under all ${\kappa }^{+}$-cc forcing notions living in $V[\Add(κ,λ)]$, and in addition by all forcing notions living in $V^{\ast }$ 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).

22 pages, submitted