In set theory, a standard model for a theory (in the language of set theory) is a model for where the membership relation is the same as the membership relation of a set theoretical universe (restricted to the domain of ). In other words, is a substructure of . A standard model that satisfies the additional transitivity condition that implies is a standard transitive model (or simply a 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 .
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 } (essentially, we add to every set and to its elements recursively). 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.
A standard transitive model will in many aspects behave "exactly like ". For example, the element that satisfies the axiom of empty set in will be , the empty set of . Similarly, all sets that can be built up from the empty set, the axiom of pairing and the axiom of union (i.e., all hereditarily finite sets) are all the same as their counterparts in . In other words, sentences such as " is the empty set" or " (the von Neumann ordinal 3)" have the truth value for the same , whether they are interpreted in or any standard transitive model . Such sentences are known as absolute for standard transitive models.
An example of a sentence that is not absolute for standard transitive models is " is the power set of ", which by definition means "For all , if and only if ", or more formally:
The qualifier means the same thing whether interpreted in or : As long as , transitivity ensures that all elements of must also be in . Therefore, the right hand side of the biconditional does mean " is a subset of ". However, the qualifier means when interpreted in , but when interpreted in . In the latter case, only subsets of that is in is required to be in (in fact, only those subsets could be in due to transitivity). Since an arbitrary subset of is not necessarily in (the axiom of separation only works for definable sets), the that satisfies this sentence in may be a proper subset of the "real" power set of (i.e., the that satisfies this sentence in ).
In general, a sentence is absolute as long as it is equivalent to a formula with only bounded quantifiers like . For example, assuming the axiom of regularity:
On the other hand:
In fact, the downward LöwenheimâÂÂSkolem theorem together with Mostowski collapse lemma can convert any standard (set) model of ZFC into a standard transitive model that is itself countable. Every set in must be countable in , but at the same time there must exist sets in that are uncountable in , such as the sets playing the role of or (the power set of ). This does not lead to any contradiction because countability is not absolute.