Collections and Sets
I like to present matters in a Mereo-topological manner. Now a unit is an object that is not a whole of two separate (not in contact) parts, and at the same time it is separate from any other object. A totality of unites is a collection. The smallest collection is a unit. An element of a collection is a unit part of that collection. So the unit collection is the sole element of itself. Multipleton collections are those that are constituted of many unites. So they are not the elements of themselves. Now for the sake of simplicity let's assume the ideal condition of all unites being unbreakable and actually in-changeable over time, and they won't be in contact with other objects at other moments of time. So no unit object can be split into two separate objects at some other moment of time, nor it would be a part of another unit object at other moment of time. Of course this is an ideal condition. Under that assumption we can have stable totalities and thus I can extend any predicate in the object world as far as that predicate only hold of unchangeable unit objects. If the unites are breakable (as it is the case with the real object world) or can come in contact with other units to form bigger units (as it is the case with the real object world) then this method fail, or at least becomes very extremely complex.
Set theory can be explained as an imaginary try to REPRESENT stable collections of unites, by stable unites. So any two stable collections (i.e. their unites are unchangeable over time) would have distinct representative unites (whether those representative units are part of those collections or external to them) as long as they are not the same, and each collection is only represented by one unit. This theory of representation of collections by units, is the essence of Set theory. Of course the representative unites are ideal, i.e. unchangeable over time. Now while element-hood of collections are being unit parts of those collections, yet "membership" in a set is another matter. Membership in sets can be defined in two says, l personally like the definition of them being elements of collections represented by the unit, i.e. every set is actually a unit object that represented collection of unites, now those unites of the represented collection are the elements of that set. Let me put it formally:
x is member of y if and only if there exists a collection z such that y is the representative of z and x is an element of z.
We start with the non representative unit, i.e. a unit object that do not represent any collection of unites, this would stand for the empty set. then go upwards in a hierarchy. What I call as the representational hierarchy, where collections are represented by sets (units) and sets themselves are collected into collections, which are represented by sets, etc..... This step-wise hierarchical approach enables a gradual transition from the less complex to the next more complex to the next, and so on... So a nice way would be to start with the empty object (the non representing unit), then to the collection of all sets representing parts of that empty object, then to the collection of all sets representing parts of the resulting objects, etc...
We can extend the representational hierarchy as long as we don't have a clear inconsistency with it. This way we can encode almost all of mathematical objects in that hierarchy.
Set not only can represent finished collections, it can also represented unfinished collection, as long as the process of producing the elements of that collection is well defined, like the process of making the naturals by succession from prior naturals and so on.. we can have a set that would represent the process of that natural production. And those are the infinite sets.
So set theory of mathematics like in ZFC are just a theory about representation of actually finished collection and of potentially non-finishing collections.
I just wanted to put you in the picture, that sets (as used in mathematics) are different from the Mereological collections I've spoken about. While the genre of collections is the same as the genre of their elements, sets on the other hand can be totally external to the collections they represent and can indeed be of a different nature. There is a lot of confusion between collections and sets, even in standard text-books of mathematics, and especially there is the confusion between element-hood of collections and membership in sets, that many mathematical textbooks on set theory introduce sets in terms of collections and set membership in term of element-hood of collections, and this is a great confusion. Set do not function as collections, no they function as unit representatives of collections, thereby enabling us to speak of a hierarchy of multiplicities within multiplicities and so on... So the set concept is stronger concept than the collective concept. The former is representational and latter is mereological.