Of course, the interpretations are horrifically complicated and useless in actual formalization. In this particular posting, we are interested in interpretations that meet some additional criteria. We will endeavor to make our set theories conform to the usual axioms of ZFC, in order to facilitate comparison. Every nonempty set has an epsilon minimal and an epsilon maximal element. Separation for bounded formulas.
Every set is in one-one correspondence with an ordinal.
The set of all ordered pairs from any set is a set. This system also interprets FCA, where finite sets of nonnegative integers get interpreted as sets of ordinals. There is a least set containing all elements of any given set. More can be said. EFA which has axioms for finite sets of nonnegative integers.