I am having some trouble interpreting axiom of (naive) comprehension in a graph, $G(V,E)$ Now suppose I define $x\in y \leftrightarrow xEy$, $x, y \in V$. So we now have a structure for set theory.
So how do I interpret the (naive/unbounded/unrestricted) axiom of comprehension in $G$. Originally, based on Jech's book, it reads "If $P$ is a property, then there exist a set $Y$ = {$x$ | $P(x)$}". I tried to interpret this in $G$, and then it reads:
"If $P$ is a property, then there exist a vertex $Y$ = {$x$ | $P(x)$}"
This is really confusing, on the left hand side it is a vertex, on the right hand side, it is a collection/set. But there are no "sets" in $G$, only vertices and edges
How do I resolve this ?
In your graph, a vertex $y$ is viewed as the set of its $E$-elements: $\{x\in V:xEy\}$.