I am interested in the foundations of mathematics. I know some of the ideas of set theory, especially ZFC. One way to define a function $f$ in such theory is to take $E$ and $F$ two sets, $G \in\mathcal{P}(E \times F)$ and set $f = (E,F,G)$ with some properties verified by $G$, mainly $f(x)=y \iff (x,y) \in G$.
I thought about the example of the following "function" $f = (\Omega,\overline{\mathbb{N}},G)$ where for all set $E$, $f(E) = \lvert E \rvert$. But I see two problems with this. First, we would need $\Omega$ to be the set of all sets, which does not exist. Second, $f(G) = +\infty$ and so $(G,+\infty) \in G$, which leads to self-reference of $G$.
However, the idea for defining this function seems pretty intuitive and "doable". Are there any theories where this function makes sense? I even imagined that some theory could have functions as its base objects, defined over all other possible object, and we would only need to focus on the objects we want for the domain. This would (I think) solve the "set of all sets" problem, but not the self-reference one, and I don't see how to build other objects (such as sets) from these base objects.
The solution to the conundrum is that if you take $\Omega$ to be a proper class (whose elements are sets), then $G$ is a subclass of the product class $\Omega \times \overline{\mathbb{N}}$ (however you wish to define the latter). Further, $G$ is a proper class, as, for instance, there is a surjective class function onto $\Omega$ (take the restriction to $G$ of the projection on the first component, $\Omega \times \overline{\mathbb{N}} \to \Omega$, and it's surjective as the function is claimed to be total). Then since elements of $\Omega$ are sets, elements of $G$ are sets. Thus $G$ cannot be in the domain of $G$, and so there is no "self-reference".