Can we characterize properties that don't make a set? Following to the ZFC system, there is no set that satisfies some kind of properties. For example, as in this question. Can we characterize what that property is? Or at least we can make useful sufficient conditions? Or do we have a list of these kinds of properties?
 A: In $\sf ZF(C)$ we have the von Neumann hierarchy, $V_\alpha$, which satisfies:


*

*$V_\alpha$ is a set for every ordinal $\alpha$, and $V_\alpha\subseteq V_\beta$ for $\alpha\leq\beta$.

*Every set $x$ lies in some $V_\alpha$.


Therefore, $\varphi(u)$ does not define a set if and only if for every $\alpha$, there is $x\notin V_\alpha$ such that $\varphi(x)$ holds. In other words, $\varphi$ defines a set if and only if all sets satisfying $\varphi$ are in some $V_\alpha$.
Of course, we can easily engineer $\varphi$'s that in different models of set theory, or under different extensions of $\sf ZFC$, will define different sets, or even a proper class. For example $\varphi(u)$ is defined as $\lnot\sf CH$ will either be satisfied by none of the sets assuming $\sf CH$, in which case it defines the empty set, or all of them if $\sf CH$ fails, in which case it does not define a set.
But $\varphi$ provably defines a set if and only if we can prove that there is some bound (although we don't necessary need to be able to produce that bound). An example to that is $\varphi(u)$ states that $u$ is an ordinal, and $u$ can be injected into $\mathcal P(\Bbb N)$, this provably defines a set, but we cannot prove what is the least $\alpha$ which bounds all the members of that set.
