Given any complete, consistent extension of , we showed that there is a* minimal* model of . This model is unique up to (unique) isomorphism, it is rigid (i.e., it has no automorphisms other than the identity), and has no proper elementary substructures.

Given a model , let , the *standard system* of , be the set of those coded by elements of , where codes iff . Thus is the class of finite sets. We showed that if is nonstandard, contains all recursive sets, and that for any non-recursive there is a nonstandard such that .