CONICET | Buscador de Institutos y Recursos Humanos

We lay the ground for an Isabelle/ZF formalization of Cohen´s technique of emph{forcing}. 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 that $M[G]$ is transitive.