An axiomatic approach to forcing and generic extensions

This paper provides a conceptual analysis of forcing and generic extensions. Our goal is to give general axioms for the concept of standard forcing-generic extension and to show that the usual (poset) constructions are unified and explained as realizations of this concept. According to our approach, the basic idea behind forcing and generic extensions is that the latter are uniform adjunctions which are ground-controlled by forcing, and forcing is nothing more than that ground-control. As a result of our axiomatization of this idea, the usual definitions of forcing and genericity are derived.