I found the following statement in a book after the usual definition of preorder and preordered set and I am not sure about how I should interpret it.
"While a preordered set $(X, \succsim)$ is not a set, it is convenient to talk as if it is a set when referring to properties that apply only to $X$. For instance, by a 'finite preordered set', we understand a preordered set $(X, \succsim)$ with $|X|< \infty$."
So the question (probably trivial) is, in the end is the preordered set a set or not?
My interpretation is that here the only set is $X$, and even if we put a preorder on it, and we call the result a preordered set, still the only set around is $X$, even if we had some characteristics (i.e. the predorder) that describe the behavior of the elements of $X$ with more details.
Is this the right interpretation of this paragraph?
PS I was not sure about the title of this question. I hope it is ok.