Different cofinalities of tree ideals

Abstract

We introduce a general framework of generalized tree forcings, GTF for short, that includes the classical tree forcings like Sacks, Silver, Laver or Miller forcing. Using this concept we study the cofinality of the ideal $I\left(Q\right)$ associated with a GTF Q. We show that if for two GTF's $Q_0$ and $Q_1$ the consistency of $\mathrm{add}\left(I\right(Q_0\left)\right)<\mathrm{add}\left(I\right(Q_1\left)\right)$ holds, then we can obtain the consistency of $\mathrm{cof}\left(I\right(Q_1\left)\right)<\mathrm{cof}\left(I\right(Q_0\left)\right)$. We also show that $\mathrm{cof}\left(I\right(Q\left)\right)$ can consistently be any cardinal of cofinality larger than the continuum.