On sequents of $\Sigma$ formulas

We investigate the position that foundational theories should be modelled on ordinary computability. In this context, we investigate the metamathematics of $\Sigma$ formulas. We consider theories whose axioms are implications between $\Sigma$ formulas, and we show that arbitrarily strong such theories prove their own correctness. We also show that a natural extension of such a theory proves the validity of intuitionistic reasoning for that theory. Finally, we show the equivalence of two completeness principles appropriate to a potentialist conception of the universe of sets.