 Solution

A forcing name for $\mathbb{P}$ is a set of ordered pairs $(\sigma ,p)$ where $p\in P$ and $\sigma$ is itself a forcing name. This recursive definition is made well-founded by assigning the forcing name rank $\sup \{\mathrm{nrk}(\sigma )+1:(\sigma ,p)\in \tau \}$. Names over the ground model are those names belonging to $M$; conditions remain ground-model objects. A name describes which recursively interpreted elements are activated by the generic filter.