Minimal model (set theory)

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 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.

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).

References

Shepherdson, J. C. (1951), "Inner models for set theory. I", The Journal of Symbolic Logic (Association for Symbolic Logic) 16 (3): 161–190, doi:10.2307/2266389, JSTOR 2266389, MR 0045073

Shepherdson, J. C. (1952), "Inner models for set theory. II", The Journal of Symbolic Logic (Association for Symbolic Logic) 17 (4): 225–237, doi:10.2307/2266609, JSTOR 2266609, MR 0053885

Shepherdson, J. C. (1953), "Inner models for set theory. III", The Journal of Symbolic Logic (Association for Symbolic Logic) 18 (2): 145–167, doi:10.2307/2268947, JSTOR 2268947, MR 0057828

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