Paper detail

Indestructibility of the tree property

In the first part of the paper, we show that if $ω\le κ< λ$ are cardinals, $κ^{<κ} = κ$, and $λ$ is weakly compact, then in $V[\M(κ,λ)]$ the tree property at $λ= κ^{++V[\M(κ,λ)]}$ is indestructible under all $κ^+$-cc forcing notions which live in $V[\Add(κ,λ)]$, where $\Add(κ,λ)$ is the Cohen forcing for adding $λ$-many subsets of $κ$ 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 $λ$ is supercompact and generalize the construction and obtain a model $V^*$, a generic extension of $V$, in which the tree property at $(κ^{++})^{V^*}$ is indestructible under all $κ^+$-cc forcing notions living in $V[\Add(κ,λ)]$, and in addition by all forcing notions living in $V^*$ which are $κ^+$-closed and ``liftable&#39;&#39; in a prescribed sense (such as $κ^{++}$-directed closed forcings or well-met forcings which are $κ^{++}$-closed with the greatest lower bounds).