What is the interpretation of the set $W$ in a Kripke model $\mathcal{K}=\langle W,R,\phi\rangle$ of provability logic, where the $\Box$-operator stands for provability in a given arithmetic theory. What are the elements of this set?
Further what is the interpretation of the relation $R$ on the set $W$?
Neither Boolos or Smorynksi seems to give an answer to this question in their texts on the subject.