Diamonds on trees

Abstract

We generalize the diamond principle and its variants using the notion of stationarity in trees introduced by Brodsky in [Brodsky, A. M., A theory of stationary trees and the balanced Baumgartner--Hajnal--Todorcevic theorem for trees. The Bulletin of Symbolic Logic]. In particular, we show that if $T$ is a nonspecial $\ddot{I}{‰}_1$-tree, then ${♢}_T⟹♢$, and if $T$ is a Suslin tree, then ${♢}_T⟺♢$. We also prove that ${♢}^{\ast }$ implies ${♢}_T$ (yielding the consistency of ${♢}_T$) and establish the consistency of $\neg {♢}^{\ast }+(\forall T\text{ nonspecial }\ddot{I}{‰}_1\text{-tree }({♢}_T))$. Finally, we demonstrate that it is consistent with $♢$ that there exists a nonspecial $\ddot{I}{‰}_1$-tree with $(\neg {♢}_T)$, introducing two forcing properties -- $\ddot{I}ƒ(S)$-closed and strategically closed in models -- which are preserved under countable support iterations.

41 pages