Suppose I have a bag with a pouch of few items : pen, pencil, eraser etc, and I want to mathematical model the idea of things existing in my pouch using ZFC, is that actually possible?
The doubt raised when I read this answer, in it, it is said that every other set in ZFC are generated out of the axioms which confirm existence of some sets ( power set axiom, empty set axiom etc etc).
Hence, except of the sets generated out of the axioms, we can't insert sets of our liking to be described by the theory. So, what additional structure/ theory do we need to describe the procedure of lifting up what we see in reality into something which can be described by set theory?)
In ZFC, the only set which doesn't have only other sets as its members is the empty set. There are no sets which have as their members widgets, wombats or other non-sets.
Now you might suppose that it is more natural to think of set theory as a superstructure we can add to theories about other things. So then, starting perhaps from some given urelements -- i.e. elements which don't themselves have members -- we can form sets of them, and then sets of sets, sets of sets of sets, and so on and on. This indeed is how early set theorists originally thought of sets. (Likewise, more informally, think of schoolroom use of set talk in "new math").
However, for purely mathematical purposes such as reconstructing analysis, it seems that we only need a single non-membered base-level entity, and it is tidy to think of this as the empty set. So for internal mathematical purposes, we can take the whole universe of sets to contain only 'pure' sets (when we dig down and look at the members of members of ... members of sets, we find nothing other than more sets). Hence, for most mathematical purposes, we concentrate on 'pure' set theories like ZFC.
But what if we want to be able to apply our set-theoretic apparatus in talking about e.g. sets of widgets or wombats or (more seriously!) space-time points? Then it might seem that we will want the base level of non-membered elements to be populated with those widgets, wombats or space-time points as the case might be. However, we can always code for widgets, wombats or space-time points using some kind of numbers, and we can treat those numbers as sets. So our set-theory-for-applications can still involve only pure sets. That's why typical introductions to set theory either explicitly restrict themselves to talking about pure sets, or -- after officially allowing the possibility of urelements -- promptly ignore them.
[Excerpted in part from Beginning Mathematical Logic: A Study Guide]