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

Flow: the Axiom of Choice is independent from the Partition Principle in ZFU