Editing Von Neumann universe (section)

==Applications and interpretations==

===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.)

===Interpretation of ''V'' as the "set of all sets"===
V is not "the [[set of all sets|set of all (naive) sets]]" for two reasons. First, it is not a set; although each individual stage ''V''<sub>α</sub> is a set, their union ''V'' is a [[proper class]]. Second, the sets in ''V'' are only the well-founded sets. The [[axiom of foundation]] (or regularity) demands that every set be well founded and hence in ''V'', and thus in ZFC every set is in ''V''. But other axiom systems may omit the axiom of foundation or replace it by a strong negation (an example is [[Aczel's anti-foundation axiom]]). These non-well-founded set theories are not commonly employed, but are still possible to study.

A third objection to the "set of all sets" interpretation is that not all sets are necessarily "pure sets", which are constructed from the empty set using power sets and unions. Zermelo proposed in 1908 the inclusion of [[urelement]]s, from which he constructed a transfinite recursive hierarchy in 1930.<ref name=Zermelo /> Such urelements are used extensively in [[model theory]], particularly in Fraenkel-Mostowski models.<ref name=Howard>{{harvnb|Howard|Rubin|1998|pp=175–221}}.</ref>

===Hilbert's paradox===
The von Neumann universe satisfies the following two properties:

* <math>\mathcal{P}(x) \in V</math> for every ''set'' <math>x \in V</math>.
* <math>\bigcup x \in V</math> for every ''subset'' <math>x \subseteq V</math>.

Indeed, if <math>x \in V</math>, then <math>x \in V_\alpha</math> for some ordinal <math>\alpha</math>.  Any stage is a [[transitive set]], hence every <math>y \in x</math> is already <math>y \in V_\alpha</math>, and so every subset of <math>x</math> is a subset of <math>V_\alpha</math>.  Therefore, <math>\mathcal{P}(x) \subseteq V_{\alpha+1}</math> and <math>\mathcal{P}(x) \in V_{\alpha+2} \subseteq V</math>. For unions of subsets, if <math>x \subseteq V</math>, then for every <math>y \in x</math>, let <math>\beta_y</math> be the smallest ordinal for which <math>y \in V_{\beta_y}</math>.  Because by assumption <math>x</math> is a set, we can form the limit <math>\alpha = \sup \{ \beta_y : y \in x \}</math>.  The stages are cumulative, and therefore again every <math>y \in x</math> is <math>y \in V_\alpha</math>.  Then every <math>z \in y</math> is also <math>z \in V_\alpha</math>, and so <math>\cup x \subseteq V_\alpha</math> and <math>\cup x \in V_{\alpha+1}</math>.

Hilbert's paradox implies that no set with the above properties exists .<ref>A. Kanamori, "[https://math.bu.edu/people/aki/10.pdf Zermelo and Set Theory]", p.490. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.</ref>  For suppose <math>V</math> was a set.  Then <math>V</math> would be a subset of itself, and <math>U = \cup V</math> would belong to <math>V</math>, and so would <math>\mathcal{P}(U)</math>.  But more generally, if <math>A \in B</math>, then <math>A \subseteq \cup B</math>.  Hence, <math>\mathcal{P}(U) \subseteq \cup V = U</math>, which is impossible in models of ZFC such as <math>V</math> itself.

Interestingly, <math>x</math> is a subset of <math>V</math> if, and only if, <math>x</math> is a member of <math>V</math>.  Therefore, we can consider what happens if the union condition is replaced with <math>x \in V \implies \cup x \in V</math>.  In this case, there are no known contradictions, and any [[Grothendieck universe]] satisfies the new pair of properties.  However, whether Grothendieck universes exist is a question beyond ZFC.

===''V'' and the axiom of regularity===
The formula ''V'' = ⋃<sub>α</sub>''V''<sub>α</sub> is often considered to be a theorem, not a definition.<ref name=Bernays /><ref name=Mendelson /> Roitman states (without references) that the realization that the [[axiom of regularity]] is equivalent to the equality of the universe of ZF sets to the cumulative hierarchy is due to von Neumann.<ref name=Roitman>{{harvnb|Roitman|2011}}. See page 79.</ref>

===The existential status of ''V''===

Since the class ''V'' may be considered to be the arena for most of mathematics, it is important to establish that it "exists" in some sense. Since existence is a difficult concept, one typically replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed by [[Gödel's incompleteness theorems]], which effectively imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consistent.<ref>See article [[On Formally Undecidable Propositions of Principia Mathematica and Related Systems]] and {{harvnb|Gödel|1931}}.</ref>

The integrity of the von Neumann universe depends fundamentally on the integrity of the [[ordinal number]]s, which act as the rank parameter in the construction, and the integrity of [[transfinite induction]], by which both the ordinal numbers and the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to rest upon von Neumann's 1923 and 1928 papers.<ref>{{harvnb|von Neumann|1923}}, {{harvnb|von Neumann|1928b}}. See also the English-language presentation of von Neumann's "general recursion theorem" by {{harvnb|Bernays|1991|pp=100–109}}.</ref> The integrity of the construction of ''V'' by transfinite induction may be said to have then been established in Zermelo's 1930 paper.<ref name=Zermelo />