Editing Von Neumann universe (section)

===Applications of ''V'' as models for set theories===
If ω is the set of [[natural number]]s, then ''V''<sub>ω</sub> is the set of [[hereditarily finite set]]s, which is a [[model (logic)|model]] of set theory without the [[axiom of infinity]].<ref>{{harvnb|Roitman|2011|p=136}}, proves that: "''V''<sub>ω</sub> is a model of all of the axioms of ZFC except infinity."</ref><ref>{{harvnb|Cohen|2008|p=54}}, states: "The first really interesting axiom [of ZF set theory] is the Axiom of Infinity. If we drop it, then we can take as a model for ZF the set ''M'' of all finite sets which can be built up from ∅. [...] It is clear that ''M'' will be a model for the other axioms, since none of these lead out of the class of finite sets."</ref>

''V''<sub>ω+ω</sub> is the [[universe (set theory)|universe]] of "ordinary mathematics", and is a model of [[Zermelo set theory]] (but not a model of [[Zermelo–Fraenkel set theory|ZF]]).<ref>{{harvnb|Smullyan|Fitting|2010}}.
See page 96 for proof that ''V''<sub>ω+ω</sub> is a Zermelo model.</ref>  A simple argument in favour of the adequacy of ''V''<sub>ω+ω</sub> is the observation that ''V''<sub>ω+1</sub> is adequate for the integers, while ''V''<sub>ω+2</sub> is adequate for the real numbers, and most other normal mathematics can be built as relations of various kinds from these sets without needing the [[axiom of replacement]] to go outside ''V''<sub>ω+ω</sub>.

If κ is an [[inaccessible cardinal]], then ''V''<sub>κ</sub> is a model of [[Zermelo–Fraenkel set theory]] (ZFC) itself, and ''V''<sub>κ+1</sub> is a model of [[Morse–Kelley set theory]].<ref>{{harvnb|Cohen|2008|p=80}}, states and justifies that if κ is strongly inaccessible, then ''V''<sub>κ</sub> is a model of ZF.
: "It is clear that if A is an inaccessible cardinal, then the set of all sets of rank less than A is a model for ZF, since the only two troublesome axioms, Power Set and Replacement, do not lead out of the set of cardinals less than A."</ref><ref>{{harvnb|Roitman|2011|pp=134–135}}, proves that if κ is strongly inaccessible, then ''V''<sub>κ</sub> is a model of ZFC.</ref> (Note that every ZFC model is also a ZF model, and every ZF model is also a Z model.)