Dynamic Mereo-topology had been introduced in this website at page titled "Dynamic Mereotopology" at:
https://sites.google.com/site/zuhairaljohar/dynamic-mereotopolgy
I'll use exactly the same axiom system presented there, however, the definition of "set" and "set membership" would follow a rather simpler approach.
One of the most important concepts of Dynamic Mereotopology is the definition of a "unit", now informally a unit is a solid object that has no external contact to any other object, where a "solid" object means an object that is not the disjoint union of two non contacting parts, of course "unit-hood" is time dependent, so if an object at moment t is a unit, it might fail to be so at another moment. Of course "unit-hood" is a sort of packaging of the material of an object, so if the material of an object is packaged in unit, at other times it might be packaged in two units, or even infinitely many units, or even at other times, it might not be packaged into any units.
Now we will work in a kind of Mereo-topology that is not atomic, this can be called as Gunk Dynamic Mereo-topology, so here every object is infinitely divisible into smaller and smaller parts without ending in indivisible parts, usually called "atoms", so here it is the packaging form "unit" that plays the central role in this method.
Now an object that at a moment of time can be packaged into a unit, is an object that in some sense or another can be regarded as forming a kind of a ONE WHOLE, while objects whose material cannot be gathered together as a unit, are discrete objects that functionally speaking cannot be regarded as constituting a single whole, they rather constitute a SHATTERED object. It is objects that can be units at a moment of time that are the most important objects and all of our concentration would be made on those, and especially how their parts can be packaged over times, it is this packaging of the object and its parts over time that we would seek to interpret "set-hood" and "set membership" in. This kind of interpretation seems in some sense to be a reasonable and not a made up concept, i.e. not ad hoc.
So lets define the above concepts before moving forwards:
We say Unit(x,t) to mean x is a unit at moment t, similarly Solid(x,t) to
mean x is solid at moment t.
Define: Solid(x,t) <=df=> ~Exist y,z (y P x ^ z P x ^ y disjoint z ^ x= y U z ^ ~ Contact (y,z,t))
Define: Unit(x,t) <=df=> Solid(x,t) ^ ~Exist k ( Contact (x,k,t))
We'll call an object as a "unit" to mean that it can packaged into a unit at some moment of time
Define: Unit(x) <-> Exists t ( Unit(x,t) )
Similarly Contact(x,y) and Solid(x) can be defined in a timeless manner as:
Define: Contact (x,y) <=df=> Exist t (Contact(x,y,t))
Define: Solid(x) <=df=> Exist t (Solid(x,t))
Now we come into an important definition of parts of objects, the most important definition after the definition a proper part (a part that is distinct from the whole) is the definition of "integral" part. Now an integral part of an object is a proper part of it that is in contact with its complementary part of that object. Formally this is:
Define:
x PP y -> [Complementary (x,y)= k <=df=> for all z (z O k <-> Exists u (u P y ^ u disjoint x ^ z O u))]
Define: x is an integral part of y <=d=> x PP y ^ Contact (x, complementary(x,y))
The opposite of an integral part is a "displaced" part at t, which is a proper part that is not in contact with its complementary part at moment t. And an object that is always a displaced part is an object that remains displaced at all moments, i.e. never an integral part.
Now we come to the definition of "Splash" at moment t as an object in which all proper parts are not in contact with each other at that moment. We need to remember that we are working in atom-less Mereo-topology. In other words a splash object at moment t is an object that do not have a solid part at moment t. Also a splash object at moment t is an object having every proper part of it being non-integral part at t, i.e. all proper parts of it are displaced parts at t.
Define: Splash(x,t) <=d=> for all yz (y PP x ^ z PP x -> ~ Contact(y,z,t))
Now we come to define "integral unit" of an object, this is somewhat tricky, since it involves some difference from the definition of "unit" given above. Here we say that x is an integral unit of y iff at some moment t, x is a solid integral part of y that only comes in contact with a splash part of y at that moment.
Define: x is an integral unit of y <=df=> Exist t (x PP y ^ Solid(x,t) ^ x is an integral part of y at t ^ for all z (Contact(x,z,t) -> z P y ^ Splash(z,t)))
Now we come to the definition of "set": an object is a set if and only if at some moment t it can be packaged in a 'set status', where a set status is defined as the fusion of splash material and otherwise integral units connected to that splash material.
Define: Set status(x,t) <=df=> Exist z (z P x ^ z is splash at t ^ for all y (y O x -> y O z or Exists u (u is an integral unit of x at t ^ contact (u,z,t) ^ y O u))
Define: x is Set <=df=> Exist t ( Set status(x,t))
Now the idea is to define "element" as being an integral unit of a set at the moment when the set is packed in a set status.
Define: x E y <=df=> Exist t (x is integral unit of y at t ^ Set status (y,t))
The above definition can be strengthened by adding the condition that for x to be a set, then x must be a unit. However 'set' as defined above really can stand for any "class" including proper classes. Still I prefer adding it to the above definitions, because it cast a sense of complete whole functionally so to speak to sets and classes, which is plausible, since they are seen as such.
Of course the interpretation of the common sets in the cumulative hierarchy of ZFC is quite direct, unit objects that can be splash objects at some moment of time and never having an integral unit part of them at any moment, those play the role of the empty set, while the union of each of those with a splash object that can unify at some moment of time and that has the empty set being the only integral part of it, plays the role of the singleton set of the empty set, and so we can go up building the hierarchy, we can even impose axioms like saying that the union of sets is the sum of their material! and also that subsets are parts of their set, without endangering the set membership relation. Transitivity of part-hood relation is not at play here, since being an integral unit of an object is not necessarily transitive! However this method can only interpret Acyclic sets, i.e. sets that are not elements of their transitive closures, this limitation is imposed because elements are defined after being proper parts so this forbids cyclic membership.
One of the disadvantages of this method is that it doesn't grant Extensionality, since there is no clear justification for the uniqueness of the splash material in a set being determined after its integral parts. We can have two unit objects that have the same integral parts and yet its splash material being distinct. There is no clear reason why this cannot be, in other words there is nothing in this definition that would forbid that. So this method indeed can allow Non-extensional sets. However since Extensionality is a restriction, then there is also nothing to forbid having that restriction in this method. This might supply some flexibility to this method, however the lack to enforce Extensionality is in some sense a great defect conceptually speaking, since Extensionality is clearly an inherent feature of sets. However we can still artificially define a set world as a totality of unit objects that satisfies Extensionality.
Bearing in mind the minimalistic approach to sets to make them as nearly as possible near to the sum of all their members, this can be done for multipletons easily, we can leave only the singletons being involved with topping splash material on top of their sole members, otherwise multipletons can be just unions of their singletons, perhaps at special cases they can even be the unions of their elements. Of course Extensionality suits a minimalistic apprach as well.
One of the possible ways to motivate Extensionality is to add identity axioms after contact relationships, like no two distinct objects can have the same contacts, i.e.
for all x,y (for all z (Contact(z,x) <-> Contact(z,y)) -> x=y)
Even a stronger form of Extensionality of contacts is that no two distinct objects can have the same graphical relations of contacts, for example there cannot exist two distinct objects that can only have contact with each other, here the graphical contact of those wold be identical since b would be only in contact with a that is in contact with it, and a would be only in contact with b that is in contact with it, both have the same graphical membership relation of only being in contact with a single other object. Such rules might provide some background for Extensionality. On the other hand it is difficult see how two distinct objects would exactly have the same pattern of topping of splash objects all up the hierarchy of sets, so there seems to be some justification for Extensionality.
Also other ways of providing a background for Extensionality is to really specify which splash material a set can have, most obviously it would be those splash material that is in contact with the integral units of it at the moments of set status and of course those that are parts of its integral units at other moments. I thik this can manage to prove Extensionality for non empty sets. However, what remains is the possibility of having multiple empty sets (Ur-elements), this in some sense can be done if the fusion of all empty sets is an empty set, then we take this fusion to be the empty set, this is in some sense plausible.
Also this method doesn't provide any motivation for any of the known set theoretic principles, with the exception of sets having acyclic structure, non of the axioms of ZF have clear motivation in that method. The main problem is that there is no clear control over the momentary Splash material of a set, especially a kind of a function determining the splash material after the identity of membership of sets. I'm not sure if we this method can raise some thought about that aspect, that can provide a natural explanation of the axioms of set theory, perhaps it can. The whole merit of this method is that it has a natural genre, elements of sets are proper parts of it, and really the most important kind of proper parts, sets on the other hand are units which are functionally complete objects, this is a nice way to look at sets and their members. This method no doubt can lead to definition of different parts of sets which would be Ur-elements, as well as to definition of proper classes, and possibly can lead to deeper scenarios about material inside sets which might prove to be important?
Zuhair