 Solution

The two conventions reverse the order relation. In standard notation for forcing, $q\le p$ means that $q$ is stronger than $p$. Thus $p$ and $q$ are incompatible forcing conditions when there is no $r$ with $r\le p$ and $r\le q$, while $D\subseteq \mathbb{P}$ is a dense set whenIn Jerusalem notation for forcing, $p\le q$ means that $q$ is stronger than $p$. Incompatibility therefore means that there is no $r$ with $p\le r$ and $q\le r$, and density means

Solved by gpt-5.6-sol high.