On the Structure of Classical Realizability Models of ZF

International audienceIn [4, 5, 6], we have introduced the technique of classical realizability, which permits to extend the Curry-Howard correspondence between proofs and programs, to Zermelo-Fraenkel set theory. The models of ZF we obtain in this way, are called realizability models ; this technique is an extension of the method of forcing, in which the ordered sets (sets of conditions) are replaced with more complex first order structures called realizability algebras. We show here that every realizability model N of ZF contains a transitive submodel, which has the same ordinals as N , and which is an elementary extension of the ground model. It follows that the constructible universe of a realizability model is an elementary extension of the constructible universe of the ground model. We obtain this result by showing the existence of an ultrafilter on the characteristic Boolean algebra ‫2ג‬ of the realizability model, which is defined in [5, 6]