Suppose that we have three bricks, A,B,C, now one can understand the Whole of those bricks to be an object that have every part of it overlapping with brick A or B or C, lets denote that whole by W. Of course clearly W is not a brick, W is the totality of all the three above mentioned bricks. However here I want to capture the idea of 'membership' of that whole, more specifically what do we mean when we say that brick A is a 'member' of W. We know that A is a part of W, but being a part of W is not enough by itself to qualify A as being a member of W, one can observe that brick A itself can have many proper parts of it and those would be parts of W of course (since part-hood is transitive) and yet non of those is a member of W. So for a part of W to be a member of W there must be some property that it must satisfy. I'll add another primitive binary relation in addition to part-hood, and that binary relation I'll call as "contact". So we'll be working within a kind of Mereotopology. However axioms to characterize contact relation would be different from those of Mereotopology. Here only disjoint (non overlapping) objects are allowed to be in contact. When we say A is in contact with B then we mean that for some x,y where x is a part of A and y is a part of B there do not exist a gap between x and y, of course as said above provided that A and B are disjoint objects.
Now we come to stipulate the sufficient condition for membership of a whole, this is:
x is a member of y iff x part of y And (there do not exist a part of y that is in contact with x) And every proper part of x is in contact with some part of x.
This supplies us with the impression that x is a 'solid' entity and yet x is separate (not in contact) from some other parts of y should those exist.
However this interpretation of membership using this kind of mereotopology has its shortcomings, the greatest is that it is limiting in the sense that only one level of membership is possible, that is between solid blocks and collections of them, any collection of several blocks would not be able to be a member of any object since it does have separate proper parts. So this would only be enough to interpret flat sets.
If we desire having a hierarchy of sets being members of other sets and if we want also to keep the above background of thinking of matters in terms of parts and wholes and contact etc.., then we need a more complex approach, one of those would be to invoke TIME in the picture!
This without doubt would complicate the whole picture but yet it does supply us with some interpretation of sets and their membership.
Now instead of having a binary relation C to represent contact, we upgrade that to a triple relation symbol C(x,y,t) to signify x in contact with y at moment t.
This would revolutionize how matters are tackled here. So for example if at moment t1 we have a whole apple P being a solid block i.e. it is not in contact with any object and all its proper parts in contact with some proper part of it, then P would be said to be a block at t1. However this doesn't mean that apple P would also remain in this block status, possibly at moment t2 the same apple P had been cut into two separate halfs, so at moment t2 P is a whole of two separate blocks P1 and P2 and no longer being as a solid block, so at moment t2 P cannot be an element of any object, while at moment t1 P could have been.
This development would introduce us to the concept of MEMORY traces! and of course the introduction of a new binary primitive 'memorized in' or 'carved in'.
A memory trace is supposed to be a record of objects in block status. So for example suppose that an rock was immersed in some mud at moment t1 and thus left its print on that piece of mud, then after a while that rock was broken into two smaller parts, and one part also fell down on another part of the mud and made another print on that piece of mud, so suppose that mud remained like that for years, now this piece of mud have the prints of the whole rock at solid status and also of a part of that rock at solid status, this piece of mud would be considered as a 'memory trace'.
Now we would coin another interpretation of 'set' as a 'memory trace'. Membership would be of objects in solid status carved in the memory trace.
so x carved in y or x memorized in y is taken to mean that x is a solid block at some moment t and memorized as such in y.
So we have the axiom.
x M y -> Exist t. x is solid at t.
So membership can be interpreted as this memorizing relation and sets can be interpreted as 'memory traces'
It is natural to assume identity of memory traces after what is memorized in them.
It needs to be stressed that memory traces are NOT the wholes of what is memorized in them! since the whole of an apple and a half of it is the apple itself and it is not different from the whole of three thirds of it, but the memory traces of those are different!
However if an object do not change its solid status over time, i.e. if we have the following property:
for all t. x is solid at t
then wholes (i.e. totalities) of such objects can be taken to be memory traces of them since time is not having any differential effect on those kinds of objects.
So all in all, sets here can be interpreted as memory traces and set membership as memorizing objects in solid block status.