Minimal model (set theory)
Encyclopedia
In set theory, a minimal model is a minimal standard model of ZFC.
Minimal models were introduced by .
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 Lκ is the class of constructible set
s of W. If there is a set which is a standard model of ZF, then the smallest such set is such a Lκ. 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).
Minimal models were introduced by .
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
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
of ZF, and the ordinal κ is the set of ordinals which occur in W, then Lκ is the class of constructible set
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
s of W. If there is a set which is a standard model of ZF, then the smallest such set is such a Lκ. This set is called the minimal model of ZFC, and also satisfies the axiom of constructibility
Axiom of constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neumann universe and the constructible universe, respectively.- Implications :The axiom of...
V=L. The downward Löwenheim–Skolem theorem
Löwenheim–Skolem theorem
In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ...
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).