We mechanize, in the proof assistant Isabelle, a proof of the axiomscheme of Separation in generic extensions of models of set theory by using the fundamental theorems of forcing. We also formalize the satisfaction of the axioms of Extensionality, Foundation, Union, and Powerset. The axiom of Infinity is likewise treated, under additional assumptions on the ground model. In order to achieve these goals, we extended Paulson's library on constructibility with renaming of variables for internalized formulas, improved results on definitions by recursion on well-founded relations, and sharpened hypotheses in his development of relativization and absoluteness.

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 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.

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We formalize the theory of forcing in the set theory framework of Isabelle/ZF. Under the assumption of the existence of a countable transitive model of ZFC , we construct a proper generic extension and show that the latter also satisfies ZFC . In doing so, we remodularized Paulson's ZF-Constructibility library.

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