set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

, a branch of mathematics, the minimal model is the minimal

standard model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...

of ZFC. The minimal model was introduced by and rediscovered by . The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is

consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...

, but follows from the existence of a standard model as follows. If there is a ''set'' ''W'' in the

von Neumann universe In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (Z ...

V that is a

of ZF, and the ordinal ''κ'' is the set of ordinals that occur in ''W'', then L_''κ'' is the class of constructible sets of ''W''. If there is a set that 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 The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible universe, constructible. The axiom is usually written as ''V'' = ''L'', where ''V'' and ''L'' denote the von Neumann unive ...

V=L. The downward

Löwenheim–Skolem theorem In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-order t ...

implies that the minimal model (if it exists as a set) is a

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

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. gave another construction of the minimal model as the strongly constructible sets, using a modified form of Gödel's constructible universe. Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets that are models of ZFC (assuming ZFC is consistent). However, these set models are non-standard. In particular, they do not use the normal membership 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). The minimal model of set theory has no inner models other than itself. In particular it is not possible to use the method of inner models to prove that any given statement true in the minimal model (such as the

continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

) is not provable in ZFC.

References

* * * *{{Citation , last1=Shepherdson , first1=J. C. , title=Inner models for set theory. III , mr=0057828 , year=1953 , journal=The Journal of Symbolic Logic , volume=18 , pages=145–167 , doi=10.2307/2268947 , jstor=2268947 , issue=2 , publisher=Association for Symbolic Logic Constructible universe