In the book "Aspects of Vagueness", the article "The alternative set theory and its approach to Cantor's set theory" by A. Sochor proposes the following definition:
We will say that a set universe is perfect if for each $X$ lying in it, each actualizable subset of $X$ also lies in it.
Later in the book, it is suggested that no ZFC set universe can be perfect:
With the assumption that the power set of an infinite existing set was actualizable, mathematics entered a peculiar world. It is a world only in the sense of a fairy tale, in which we meet a large number of wondrous creatures in the form of fantastic topological spaces, algebraic structures and the like. The immortality of Cantor's theory of sets lies not in its truth but just in the opposite - in its fantasticness and its suggestivity, by which it came to captivate the mathematics of an entire epoch.
If I replace "actualizable" by element of an inner model of ZFC, I get the following defintion:
We will say that an inner model $M$ of ZFC is perfect if for each $X\in M$, every subset $Y\subset X$ of $X$ also lies in $M$, i.e. $Y\in M$.
If A. Sochor had a valid point, then it should at least be possible to prove that "no inner model of ZFC can be perfect". This statement is not completely implausible, because a model of a first order theory is not allowed to use a proper class as its domain. So my question is on the one hand whether this statement is true, but I would also like to know whether this statement would have any relevance besides "pure" crankery against ZFC.