  We lay the ground for an Isabelle/ZF formalization of Cohen's technique of
forcing. We formalize the definition of forcing notions as preorders with top,
dense subsets, and generic filters. We formalize the definition of forcing
notions as preorders with top, dense subsets, and generic filters. We formalize
a version of the principle of Dependent Choices and using it we prove the
Rasiowa-Sikorski lemma on the existence of generic filters.
  Given a transitive set $M$, we define its generic extension $M[G]$, the
canonical names for elements of $M$, and finally show that if $M$ satisfies the
axiom of pairing, then $M[G]$ also does. We also prove $M[G]$ is transitive.
