Using the theory of rudimentary recursion and provident sets expounded in [MB], we give a treatment of set forcing appropriate for working over models of a theory PROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen’s $J_{\omega }$. Much of the development is rudimentary or at worst given by rudimentary recursions with parameter the notion of forcing under consideration. Our development eschews the power set axiom. We...