Editing Universal set (section)

====Regularity and pairing====
In [[Zermelo–Fraenkel set theory]], the [[axiom of regularity]] and [[axiom of pairing]] prevent any set from containing itself. For any set <math>A</math>, the set <math>\{A\}</math> (constructed using pairing) necessarily contains an element disjoint from <math>\{A\}</math>, by regularity. Because its only element is <math>A</math>, it must be the case that <math>A</math> is disjoint from <math>\{A\}</math>, and therefore that <math>A</math> does not contain itself. Because a universal set would necessarily contain itself, it cannot exist under these axioms.{{sfnp|Cenzer|Larson|Porter|Zapletal|2020}}