# Inner model > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Inner_model > Markdown URL: https://mediated.wiki/source/Inner_model.md > Source: https://en.wikipedia.org/wiki/Inner_model > Source revision: 1350304458 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Transitive model containing all ordinals This article is about models of mathematical set theory. For the standard effective temperature (SET) model, see [Thermal comfort § Standard effective temperature](/source/Thermal_comfort#Standard_effective_temperature). In [set theory](/source/Set_theory), a branch of [mathematical logic](/source/Mathematical_logic), an **inner model**[1] for a [theory](/source/Theory_(mathematical_logic)) *T* is a [substructure](/source/Substructure_(mathematics)) of a [model](/source/Model_(mathematical_logic)) *M* of a [set theory](/source/Set_theory) that is both a model for *T* and contains all the ordinals of *M*. ## Definition Let *L* = ⟨∈⟩ be the language of set theory. Let *S* be a particular set theory, for example the [ZFC](/source/ZFC) axioms and let *T* (possibly the same as *S*) also be a theory in *L.* If *M* is a model for *S,* and *N* is an *L*-structure such that 1. *N* is a substructure of *M,* i.e. the [interpretation](/source/Interpretation_(model_theory)) ∈*N* of ∈ in *N* is ∈*M* ∩ *N*2 1. *N* is a model of *T* 1. the domain of *N* is a [transitive class](/source/Transitive_class) of *M* 1. *N* contains all [ordinals](/source/Ordinal_number) in *M* then we say that *N* is an **inner model** of *T* (in *M*).[2] Usually *T* will equal (or subsume) *S*, so that *N* is a model for *S* 'inside' the model *M* of *S*. If only conditions 1 and 2 hold, *N* is called a [standard model](/source/Standard_model_(set_theory)) of *T* (in *M*), a *standard submodel* of *T* (if *S* = *T* and) *N* is a *set* in *M*. A model *N* of *T* in *M* is called *transitive* when it is standard and condition 3 holds. If the [axiom of foundation](/source/Axiom_of_foundation) is not assumed (that is, is not in *S*) all three of these concepts are given the additional condition that *N* be [well-founded](/source/Well-founded_set). Hence inner models are transitive, transitive models are standard, and standard models are well-founded. The assumption that there exists a standard submodel of [ZFC](/source/ZFC) (in a given universe) is stronger than the assumption that there exists a model. In fact, if there is a standard submodel, then there is a smallest standard submodel called the *[minimal model](/source/Minimal_model_(set_theory))* contained in all standard submodels. The minimal submodel contains no standard submodel (as it is minimal) but (assuming the [consistency](/source/Consistency) of ZFC) it contains some model of ZFC by the [Gödel completeness theorem](/source/G%C3%B6del_completeness_theorem). This model is necessarily not well-founded otherwise its [Mostowski collapse](/source/Mostowski_collapse) would be a standard submodel. (It is not well-founded as a relation in the universe, though it satisfies the [axiom of foundation](/source/Axiom_of_foundation) so is "internally" well-founded. Being well-founded is not an absolute property.[3]) In particular in the minimal submodel there is a model of ZFC but there is no standard submodel of ZFC. ## Use Usually when one talks about inner models of a theory, the theory one is discussing is [ZFC](/source/ZFC) or some extension of ZFC (like ZFC + "a [measurable cardinal](/source/Measurable_cardinal) exists"). When no theory is mentioned, it is usually assumed that the model under discussion is an inner model of ZFC. However, it is not uncommon to talk about inner models of [subtheories](/source/Subtheories) of ZFC (like [ZF](/source/Zermelo%E2%80%93Frankel_set_theory) or [KP](/source/Kripke%E2%80%93Platek_set_theory)) as well. ## Related ideas [Kurt Gödel](/source/G%C3%B6del) proved that any model of ZF has a least inner model of ZF, the [constructible universe](/source/Constructible_universe), which is also an inner model of ZFC + [GCH](/source/Generalized_Continuum_Hypothesis). There is a branch of set theory called [inner model theory](/source/Inner_model_theory) that studies ways of constructing least inner models of theories extending ZF. Inner model theory has led to the discovery of the exact [consistency strength](/source/Consistency_strength) of many important set theoretical properties. ## References 1. **[^](#cite_ref-1)** [Shepherdson, J.C.](/source/John_Shepherdson) (1951–53). "Inner Models for Set Theory" (Document). [Journal of Symbolic Logic](/source/Journal_of_Symbolic_Logic). 1. **[^](#cite_ref-2)** [Jech, Thomas](/source/Thomas_Jech) (2002). *Set Theory*. Berlin: [Springer-Verlag](/source/Springer-Verlag). [ISBN](/source/ISBN_(identifier)) [3-540-44085-2](https://en.wikipedia.org/wiki/Special:BookSources/3-540-44085-2). 1. **[^](#cite_ref-3)** [Kunen, Kenneth](/source/Kenneth_Kunen) (1980). *Set Theory*. Amsterdam: North-Holland Pub. Co. [ISBN](/source/ISBN_(identifier)) [0-444-86839-9](https://en.wikipedia.org/wiki/Special:BookSources/0-444-86839-9)., Page 117 --- Adapted from the Wikipedia article [Inner model](https://en.wikipedia.org/wiki/Inner_model) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Inner_model?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.