  We introduce a formal theory called Flow where the intended interpretation of
its terms is that of function. We prove ZF, ZFC and ZFU (ZF with atoms) can be
immersed within Flow as natural consequences from our framework. Our first
important application is the introduction of a model of ZFU where the Partition
Principle holds but the Axiom of Choice fails, if Flow is consistent. So, our
framework allows us to address the oldest open problem in set theory: if the
Partition Principle entails the Axiom of Choice.
