Given an arbitrary set $X$, can one always find an element outside $X$?

Question

Given an arbitrary set $X$, can one always find an element outside $X$? I think this question boils down to whether there exists a universal set that contains everything, including itself. But wiki says that assuming the existence of a universal set leads to Russell's paradox. So given a set $X$, can one explicitly construct an element that does not belong to $X$? For example, is $\{X\}$ such an element?

Answer 1

It depends what set theory we're using. All set theories have to find a way around Russell's paradox, but there are different possible responses.

The usual foundational system is $\mathsf{ZF(C)}$. Here we have the axiom of regularity (or foundation), which implies that we don't have any "$\in$-loops" - e.g. we don't have a pair of sets $a,b$ with $a\in b$ and $b\in a$. Since $X\in\{X\}$, this rules out $\{X\}\in X$.

In other systems things play out differently: systems like $\mathsf{NF}$ and $\mathsf{GPK_\infty^+}$ actually have a universal set, and systems like $\mathsf{ZFC-Foundation+Antifoundation}$ lack a universal set but do allow $\{X\}\in X$. These latter theories however do still, to the best of my knowledge, all allow an explicit construction of a $Y\not\in X$ given a set $X$: specifically, by the Burali-Forti paradox we argue that no set contains every ordinal, so "the least ordinal not in $X$" provides a non-element as desired.

I'm not aware of any "natural" set theory in which there is no universal set but also no way of explicitly building non-elements of given sets. However, my background in alternative set theories isn't too strong, so I could be missing something.

Answer 2

Russell's paradox itself gives a way to find such an element. Namely, let $$Y=\{x\in X:x\not\in x\}.$$ If $Y$ were an element of $X$, then Russell's paradox would give a contradiction: we would have $Y\in Y$ iff $Y\not\in Y$. Thus $Y$ is not an element of $X$.

(This construction works in any axiomatization of set theory that includes the axiom schema of separation, so that we can be sure there really does exist a set $Y$ whose elements are those $x\in X$ such that $x\not\in x$. As mentioned in Noah's answer, the usual ZFC axioms for set theory imply that actually no set is an element of itself, so $Y$ would just be $X$.)