Standard model (set theory)
In set theory, a standard model for a theory is a model for where the membership relation is the same as the membership relation of a set theoretical universe . In other words, is a substructure of. A standard model that satisfies the additional transitivity condition that implies is a standard transitive model.
Often, when one talks about a model of set theory, it is assumed that is a set model, i.e. the domain of is a set in. If the domain of is a proper class, then is a class model. An inner model is necessarily a class model, because inner models are required to contain all the ordinals of.
Examples
It is difficult to exhibit an explicit set model of ZFC, because the very existence of a set model implies the consistency of ZFC, which is unprovable within ZFC. However, the universe itself, when equipped with the ordinary set membership relation, is an intuitive example of a class model that is standard transitive.To better illustrate the concepts of "standard" and "transitive", we compare the model with other models isomorphic to it. An arbitrary isomorphism such as will usually yield a non-standard class model, since does not imply in general. To construct a class model that is standard but not transitive, consider a function defined by -recursion as . Denote the image of as. Since itself is not in, we have iff, and thus is indeed a standard model, but it is not transitive because but is not in. Essentially, non-standard models have a membership relation different from the universe, and standard non-transitive models have elements with "superfluous" members.