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
.