I cannot agree more! Of course, and that's what I was saying. But you totally misread what I was writing. I think because of you "apparently" not having experience with the topic of Mereo-topology.
What I'm saying is a little bit complicated. Seeing your comments, I realize that you completely mis-understood me. But I do concede that what I wrote was too compact. — Zuhair
Sorry for the misunderstanding, I'll try to stay on track.
Lets come to what I meant by "UNIT", I mean by that an individual. For example an apple is a unit, while the collection of two separate apples is not a unit. Now I envision a unit as an object that is not the whole of two separate objects, that is at the same time separate form other objects. This has something to do with separateness and contact. So a single apple has any two parts of it connected by a part of the apple, so it is in continuity, there is no breach to its material. — Zuhair
I tend to think that this is not a very good representation of what an object, or "unit" really is. It isn't based on an accurate description of the relationship between parts and wholes. A "unit" for you is something with existence separate from other units, yet its parts do not have such separate existence. In reality though, there is vagueness in what constitutes "separate". Due to this vagueness, there may be discrepancy in judgements as to what is the unit, and what is a part of a unit, depending on one's perspective. For example, the apple is really a part of the tree. Its generation, existence and subsistence is dependent on the tree, such that as soon as it gains "separate" existence it starts to degenerate. Also, consider a "unit" like the earth. You might think of it as "separate" from the sun, but really it only exists as a part of the solar system. Then the solar system only exists as a part of the galaxy, and so on. And if we look the other way, we are faced with the question of why this composition of molecules which is "the apple" is properly "the unit", and not the molecules themselves. After the apple separates from the tree, the molecules of the apple separate from each other in the process of degeneration. That is why I think your determination of what constitutes an object or "unit" is rather arbitrary, and dependent on one's perspective.
So a single apple has any two parts of it connected by a part of the apple, so it is in continuity, there is no breach to its material. While the collection of some two separate apples is not like that, you have one apple being a part of that collection and the other apple also being a part of that collection but you have a breach of material between them, i.e. the two apples are separate, i.e. not in contact with each other and no part of that collection is in contact with these two parts, such collections are NOT units, they are collections of separate units. — Zuhair
Now your description of a "collection" doesn't seem to provide principles to distinguish between an artificial collection and a natural collection. So for example you do not distinguish between a collection of apples in a bag, placed there artificially, and a collection of apples hanging on a tree. The tree might be an object, a unit, and the collection of apples exist as parts of that unit, but they could also be rearranged as parts of an artificial collection.
So a single apple has any two parts of it connected by a part of the apple, so it is in continuity, there is no breach to its material. — Zuhair
So this appears to be the critical question, what constitutes a "breach to its material"? The apples hanging on the tree clearly have no breach of material and are therefore part of the tree, but that's a simple example. Is there a breach of material between the earth and the sun, when the two are connected by things like gravity and light?
On the other hand, an artificial collection might very well be connected by something. Apples in a plastic bag are "connected". Apples of the same variety are "connected". Furthermore, when we manufacture things like cars for example, we connect parts together to produce a unit. So the distinction between artificial and natural, though it serves as an example, is not even a good distinction itself.
The issue here seems to be what constitutes a "material" connection. You would say that having a material connection to something else negates the status of being a "unit", making the thing a "part" of a unit instead. If we switch to Aristotelian terms we'd replace "material" with "substance". In his "Categories", "substance" in the truest and primary sense, is defined as that which is neither predicable of, nor present in, a subject. Notice that this produces a more rigorous restriction than your "material" connection. Not only do we have "present in" as a restriction, but also "predicable of". So for example, if X is predicable of Y, X cannot be given the status of substance, and cannot therefore be a unit or object. This would extend your category of material connection to include predication as representing a material connection.
I believe that the goal here is not to produce the artificial/natural distinction mentioned above, but to distinguish between substantial and non-substantial, or material and non-material collections. Consider my criticism of your last post accusing you of a "random collection of elements" which you flatly denied, accusing me of misunderstanding. If a collection is truly random, the so-called "parts" of that collection are actually units, there is nothing substantial connecting them, and the collection itself cannot be an object or unit. Therefore the "parts" of that collection are not properly "parts". But if the parts are connected for any valid reason, this must qualify as a substantial, or a material connection. Then the whole of the collection is a valid object or unit, and the parts cannot be understood as independent objects.
I define "collection" as a totality of units, of course that totality itself may be a unit (in the case the collection has only one unit part of it), or might not be a unit (like a collection of multiple units: like of two apples, 10 cats, etc...). I need to stress here that "being a unit" or not, has nothing to do with the collection being definable or not, even if it is definable after some predicate still the collection if it contains many units, still it is NOT a unit. Being a unit depends on the continuity of the material in the collection, and not on definability issues or the alike. — Zuhair
According to what I've explained above, I dismiss your criteria of "the continuity of the material", as being too vague, and replace it with the Aristotelian concept of substance. Therefore any valid "collection" is itself an object or unit, the parts having a substantial relation to one another, demonstrating the existence of a "whole". The parts are therefore not independent units. Having something in common for example, cannot be taken as merely coincidental, and must be understood as indicating that the parts are not independent objects, but parts of a whole.
The only collection that is at the same time a unit, is the singular collection, i.e. the collection having one element, i.e. has one unit part. Otherwise collections having multiple elements whether definable or not, are always not units. — Zuhair
I foresee an issue with this concept of a "singular collection". I'm afraid it might be somewhat contradictory like the empty set, or simply purposeless. Let's say that every object is unique, as per the law of identity. Any individual thing which we come across could therefore be a singular collection. However, it is pointless to make such a collection, because the reason for making a collection is to acknowledge relationships between things. So we could only place a thing in the category of "singular collection" if and only if there could be no relations between that thing and anything else. Having a relation would make it a part of a collection negating the status of "singular collection". Perhaps we could keep the category of "singular collection", but it would most likely remain an empty category. It's not an empty set though, but an empty category, because under my categorization a valid set (reasonable relations) constitutes an object.
A set (as that term is used in set theory) is a unit object that represent a collection of units, like in how a lawyer represent a collection of many accused persons. Each accused person is a unit object (because its material is in continuity, and it itself is separate form other material) and the lawyer is also a unit object, so here you have an example of some Representation relation where a collection of unit objects (that is itself (i.e. the collection) not a unit since there are many accused person in that collection of our example) that is represented by a unit object (the lawyer). That was an example of EXTERNAL REPRESENTATION. On the other hand there is INTERNAL REPRESENTATION where a single unit in the collection can stand to represent the total collection, like for example when the HEAD of some tribe represents the whole of its tribe in some meeting of head of tribes. The head of a tribe is a unit part of that tribe, and yet it can represent the whole tribe. Any group (collection) of people can always chose one among them that can stand to represent the whole group. This is internal representation. — Zuhair
So this is where your system gets very confused, and mine becomes much more practical. In your example, why not simply say that the collection of people represented by the lawyer is itself an object? They all have X in common, so they have that valid relationship to one another, and therefore exist together as that mentioned object. There is no need to assign to a member of the group the task of representation, such as the person who represents the group, and hand the group real existence through that representation, the group already has real existence through the real experience which they share, which constitutes a real relationship. Such real relations make real objects. And, in the other example, the tribe has real existence as an object, due to the relations between members, it does not require a "head" of the tribe, or representative of the tribe, to give it real existence as a collection or object. Requiring that the collection has a representative creates all sorts of problems, beginning with the representative's real capacity to adequately represent the collection. See, under this system, the collection, as an object, can only be apprehended as an object, to the extent provided by the representative. But the representative is not the true object, and we are better off to look directly at the object to understand its true existence.
The usual set theory with well founded sets is a theory of external representation of collections of representatives of collections of representatives of..... It is about tiers of representation of collections.
The empty set can be ANY individual object. For example take any particular apple. This can serve as the empty set, since apples are not representatives of collections of representatives..
Now take some unit object that serves to represent the chosen apple above (the one we called the empty set). This must be different form that apple, because the apple is not a representative of anything, while that object is representing that apple itself. This latter object would act as the singleton set of the empty set, denoted by {{}}. Now you can take a third object that act as a representative of the collection of the apple (the empty set) and the object that represents that apple (the singleton of the empty set), now this representative object would be the set of the empty set and the singleton of the empty set, denoted by { {}, {{}} }. And so on.... — Zuhair
Yes, this complexity is exactly why your system is bound to failure. As I described above, any individual object may be used to create a "singular collection". But that is to assume that the individual has no relations to anything else, and this produces an empty set. To make the set meaningful, another object must represent the singular collection, but that negates the status of singular collection. So your whole set system is based in something meaningless, or even contradictory, the empty set, which is represented by the singular collection. It's like you're building your sets bottom up, when they need to be produced top down to have any substance. The set must be principled on the real existence of parts to a whole, as an object, and not based on a part which is meant to represent a whole.
One needs to be careful! Not every collection has a representative! Even some well definable collections might not have representatives. Although this largely depends on what is meant by "well definable". — Zuhair
This is evidence that your system is faulty. We need to recognize a collection as an entity itself, and not rely on a representative. A representative is often incapable of representing to us, the "thing" which is responsible for the real and valid existence of the collection. And this is proven by the fact that some valid collections have no representative.