Minimal model (set theory)

In set theory, a minimal model is a minimal standard model of ZFC. Minimal models were introduced by (Shepherdson 1951, 1952, 1953).

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. If there is a set W in the von Neumann universe V which is a standard model of ZF, and the ordinal κ is the set of ordinals which occur in W, then L_κ is the class of constructible sets of W. If there is a set which is a standard model of ZF, then the smallest such set is such a L_κ. This set is called the minimal model of ZFC, and also satisfies the axiom of constructibility V=L. The downward Löwenheim–Skolem theorem implies that the minimal model (if it exists as a set) is a countable set. 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.

Cohen (1963) gave another construction of the minimal model as the strongly constructible sets, using a modified form of Godel's constructible universe.

Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets which are models of ZF (assuming ZF is consistent). However, those set models are non-standard. In particular, they do not use the normal element relation and they are not well founded.

If there is no standard model then the minimal model cannot exist as a set. 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).