It is natural, given the usual iterative description of the universe of sets, to investigate set theories which in some way take account of the unfinished character of the universe. We do not here consider any arguments aimed at justifying one system over another, or at clarifying the basic philosophy. Rather, we look at an obvious candidate which is similar to a system discussed by L. Pozsgay in [1]. Pozsgay sketched the development of the ordinary theorems in such a system and attempted to show it equiconsistent with ZF. In this paper we show that the consistency of the system we call IZF can be proved in the usual ZF set theory.