The theory of classical realizability is a framework for the Curry-Howard correspondence which enables to associate program with each proof in Zermelo-Fraenkel set theory. But, almost all applications mathematics physics, probability, statistics, etc. use Analysis i.e. axiom dependent choice (DC) or even (full) (AC). It therefore important find explicit programs these axioms. Various solutions ...

The following notion of forcing was introduced by Grigorieff [2]: Let I ⊂ ω be an ideal, then P is the set of all functions p : ω → 2 such that dom(p) ∈ I. The usual Cohen forcing corresponds to the case when I is the ideal of finite subsets of ω. In [2] Grigorieff proves that if I is the dual of a p-point ultrafilter, then ω1 is preserved in the generic extension. Later, when Shelah introduced...