Editing Von Neumann universe (section)

===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.