It is somewhat philosophical(at least to me).
The question is as above. What is the 'substance' referred to as by the word set or class? Especially how is the thing called class defined? I cannot get a satisfactory answer from usual textbooks..So I ask here
On
See George Boolos, The Iterative Conception of Set (1971) :
A set, according to Cantor ["Beiträge zur Begründung der transfiniten Mengenlehre", Mathematische Annalen (1895-97), Engl.transl.1915, §1, page 85], is
"any collection into a whole (Zusammenfassung su einem Ganzen) $M$ of definite and separate objects $m$ of our intuition or our thought. These objects are called the 'elements' of $M$."
Cantor also defined a set as a "many, which can be thought of as one, i.e., a totality of definite elements that can be combined into a whole by a law."
One might object to the first definition on the grounds that it uses the concepts of collection and Whole, which are notions no better understood than that of set, [...] But it cannot be denied that Cantor's definitions could be used by a person to identify and gain some understanding of the sort of object of which Cantor wished to treat. Moreover, they do suggest -although, it must be conceded, only very faintly- two important characteristics of sets:
(i) that a set is "determined" by its elements in the sense that sets with exactly the same elements are identical, and
(ii) that, [...] the elements of a set are "prior to" it.
On
In most ontology textbooks ( by ontology I mean that branch of philosophy that deals with " being as being" ) sets are often given as examples of (1) abstract (2) particulars.
Reference : Lowe, A Survey Of Metaphysics.
They are abstract ( i.e. non-concrete) entities, since they do not belong to the physical universe, that is they do not belong to space-time. Although the Eiffel Tower is in Paris and the Statue of Liberty is in NewYork the set {Eiffel Tower, Statue of Liberty} is neither in Paris nor in NewYork, nor anywhere else.
You may say that this set has at least temporal being , that is, that it could not exist before both the tower and the statue came into being. I would agree; but this fact is a good reason to consider that our set is not a genuine one. The genuine sets mathematicians consider have as elements objects that do not belong to time, so that this question of time ( coming into being, duration, ceasing to exist, etc.) does not arise for the collection they compose.
Second, these abstract entities are also particulars. "Particular" is opposed to " universal". An example of universal would be : house, chair, father of. A universal is an entity that can have many instances. The chair I am sitting on is not "THE" chair, it is only "A" chair ( amongst others) , an instance of " chair-ness" as some would say. By contrast, a particular cannot be instantiated. This is true of concrete particulars ( Winston Churchill is not " a Winston Churchill" as if " WinstonChurchill-ness " could be instantiated at different places and times). This is also true of abstract particulars. Sets cannot have instances (although a particular set can be taken as standard in a class of equivalent sets to represent the whole class).
Remark. I think the category of substance is not appropriate for sets
(1) primary substances are concrete particulars (Aristotle)
(2) secondary substances are universals ( Aristotle)
(3) sets are neither concrete particulars nor universals
(4) therefore, sets are not substances
Remark. I do not know whether the way ontologists treat sets can be accepted by mathematicians. If I remember well, Jech/Hrbacek's Introduction to set theory begins ( quite amazingly ) with something like : " sets are creations of our minds..." . In that case sets would not have more than " mental being", or rather " intentional being" ( in the sense of Husserl).
There are many different approaches to how objects like a set might be constructed (and indeed thought of philosophically), and this also has a lot to do with your choice of axioms.
One perspective I particularly enjoy is the constructivist one. It doesn't allow for the full range of objects present in classical mathematics but it grounds a great many of them with a nice computational interpretation. We can construct sets (or, things that are isomorphic to everything else that we would ever call a set) in a number of ways, such as physical data on a computer using something like the calculus of inductive constructions, by building them out of types. If you're interested in foundations of mathematics, I can strongly recommend trying to pick up some type theory -- you could have a look at how an automated theorem proving language like Leanprover or Coq operate and I think that line of inquiry might clear some things up for you (or raise additional, but more precise questions!).
On the other hand, some take a view of platonic idealism and I do not honestly know enough about this perspective to give it the credit it deserves, but where mathematical objects are thought to sort of exist in an a priori way outside of the physical world.
Unfortunately, this question is perhaps a little too vague for anybody to give any really substantial answer to as is.