Editing Inner model (section)

==Definition==

Let ''L''&nbsp;=&nbsp;⟨∈⟩ be the language of set theory. Let ''S'' be a particular set theory, for example the [[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 {{nowrap|''L''-structure}} such that

# ''N'' is a substructure of ''M,'' i.e. the [[interpretation (model theory)|interpretation]] ∈<sub>''N''</sub> of ∈ in ''N'' is ∈<sub>''M''</sub>&nbsp;∩&nbsp;''N''<sup>2</sup>
# ''N'' is a model of ''T''
# the domain of ''N'' is a [[transitive class]] of ''M''
# ''N'' contains all [[ordinal number|ordinals]] in ''M''
then we say that ''N'' is an '''inner model''' of ''T'' (in ''M'').<ref>{{cite book | last = Jech | first = Thomas |authorlink = Thomas Jech| title = Set Theory | publisher = [[Springer-Verlag]] | location = Berlin | year = 2002 | isbn = 3-540-44085-2 }}</ref> 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 (set theory)|standard model]] of ''T'' (in ''M''), a ''standard submodel'' of ''T'' (if ''S''&nbsp;=&nbsp;''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]] is not assumed (that is, is not in ''S'') all three of these concepts are given the additional condition that ''N'' be [[well-founded set|well-founded]]. 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]] (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 (set theory)|minimal model]]'' contained in all standard submodels. The minimal submodel contains no standard submodel (as it is minimal) but (assuming the [[consistency]] of ZFC) it contains
some model of ZFC by the [[Gödel completeness theorem]]. This model is necessarily not well-founded otherwise its [[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]] so is "internally" well-founded. Being well-founded is not an absolute property.<ref>{{cite book | last = Kunen | first = Kenneth |authorlink = Kenneth Kunen| title = Set Theory | publisher = North-Holland Pub. Co | location = Amsterdam | year = 1980 | isbn = 0-444-86839-9 }}, Page 117</ref>)
In particular in the minimal submodel there is a model of ZFC but there is no standard submodel of ZFC.