Editing Universal set (section)

===Russell's paradox===
{{Main|Russell's paradox}}
[[Russell's paradox]] concerns the impossibility of a set of sets, whose members are all sets that do not contain themselves. If such a set could exist, it could neither contain itself (because its members all do not contain themselves) nor avoid containing itself (because if it did, it should be included as one of its members).{{sfnp|Irvine|Deutsch|2021}} This paradox prevents the existence of a universal set in set theories that include either [[Zermelo]]'s [[axiom of comprehension|axiom of restricted comprehension]], or the [[axiom of regularity]] and [[axiom of pairing]].

====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}}

====Comprehension====
Russell's paradox prevents the existence of a universal set in set theories that include [[Zermelo]]'s [[axiom of comprehension|axiom of restricted comprehension]].
This axiom states that, for any formula <math>\varphi(x)</math> and any set <math>A</math>, there exists a set 
<math display=block>\{x \in A \mid \varphi(x)\}</math>
that contains exactly those elements <math>x</math> of <math>A</math> that satisfy <math>\varphi</math>.{{sfnp|Irvine|Deutsch|2021}}

If this axiom could be applied to a universal set <math>A</math>, with <math>\varphi(x)</math> defined as the predicate <math>x\notin x</math>,
it would state the existence of Russell's paradoxical set, giving a contradiction.
It was this contradiction that led the axiom of comprehension to be stated in its restricted form, where it asserts the existence of a subset of a given set rather than the existence of a set of all sets that satisfy a given formula.{{sfnp|Irvine|Deutsch|2021}}

When the axiom of restricted comprehension is applied to an arbitrary set <math>A</math>, with the predicate <math>\varphi(x)\equiv x\notin x</math>, it produces the subset of elements of <math>A</math> that do not contain themselves. It cannot be a member of <math>A</math>, because if it were it would be included as a member of itself, by its definition, contradicting the fact that it cannot contain itself. In this way, it is possible to construct a witness to the non-universality of <math>A</math>, even in versions of set theory that allow sets to contain themselves. This indeed holds even with [[Axiom schema of predicative separation|predicative comprehension]] and over [[intuitionistic logic]].