Editing Universal set (section)

====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]].