I was reading about older definitions of the natural numbers on Wikipedia here (in retrospect, not the best place to learn mathematics) and came across the following definition for the natural numbers: (paraphrased)
Let $\sigma$ be a function such that for every set $A$, $\sigma(A) := \{ x \cup \{ y \} \mid x \in A \wedge y \notin x \} $. Then $\sigma(A)$ is the set obtained by adding any new element to all elements of $A$. Then define $0 := \{ \emptyset \}$, $1 := \sigma(0)$, $2 := \sigma(1)$ et cetera.
The way I understood this definition is that the natural number $n$ is "defined" as the set of all sets with exactly $n$ elements. This sounded straightforward to me, until I read the next paragraph:
This definition works in naive set theory, type theory, and in set theories that grew out of type theory, such as New Foundations and related systems. But it does not work in the axiomatic set theory ZFC and related systems, because in such systems the equivalence classes under equinumerosity are "too large" to be sets. For that matter, there is no universal set V in ZFC, under pain of the Russell paradox.
Why exactly doesn't this definition work in ZFC? I don't fully understand how the sets in this definition are "too large". Is part of the problem just that there is no "universal set" to pick the element $y$ from?
I tried to do some more reading to find my answer, but the material was way out of my depth. (I am only familiar with the basics of set theory, Russell Paradox, Cantor diagonal argument, and not much more. ) So I apologize in advance if this is a really simple question...