The minimal model was introduced by Shepherdson (1951, 1952, 1953) and rediscovered by Cohen (1963).
The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is consistent, but follows from the existence of a standard model as follows.
This set is called the minimal model of ZFC, and also satisfies the axiom of constructibility V=L.
More precisely, every element s of the minimal model can be named; in other words there is a first-order sentence φ(x) such that s is the unique element of the minimal model for which φ(s) is true.
However in this case the class of all constructible sets plays the same role as the minimal model and has similar properties (though it is now a proper class rather than a countable set).