When JFK was assassinated, the general population could not accept that an ordinary individual could remove a popular public figure, so some thought it must be a conspiracy. I was never an advocate for that. Tragedies don't discriminate.
I said 'many', not 'all'. SR is a great theory, and has so much experimental support, why is it still questioned.

Here is a paper that questions the 'diagonal argument'.
https://app.box.com/s/vdop6iqhi8azgoc2upd76ifu8zacq8e4

As explained, this is absolutely false — Metaphysician Undercover

Well you seem to refuse tribe as an object, well this is a deep point. Anyhow to me a tribe, a herd, a bunch, etc.. all of those are objects, and they are well specified objects as long as each individual member of them is a well specified entity. Anyhow I don't think I can discuss refusal of such clear kinds of objects.

Here is a paper that questions the 'diagonal argument'.
https://app.box.com/s/vdop6iqhi8azgoc2upd76ifu8zacq8e4 — sandman

Cranky.

Wittgenstein also criticized the diagonal argument. He was wrong but his objections were at least coherent and interesting.

Here is a paper that questions the 'diagonal argument'. — sandman

It's not a good paper. And Cantor was definitely sophisticated enough to see what he is supposed to have missed. The author is instead failing to see.

When JFK was assassinated, the general population could not accept that an ordinary individual could remove a popular public figure, so some thought it must be a conspiracy. I was never an advocate for that. Tragedies don't discriminate. — sandman

What you still refuse to see is the proof is solid in the same way that a game of chess is legal. All of the moves are according to the rules. It does not prove something metaphysical about reality. Or at least the rules are agnostic about their 'real world' meaning.

Speaking more personally (indulging in a real-world interpretation), I experience its intuitive content this way. If someone claimed to have a way/algorithm to list all infinite sequences of bits, I'd know they were wrong. I would just flip bits along the diagonal and have a sequence they didn't include on their list.

Well you seem to refuse tribe as an object, well this is a deep point. Anyhow to me a tribe, a herd, a bunch, etc.. all of those are objects, and they are well specified objects as long as each individual member of them is a well specified entity. Anyhow I don't think I can discuss refusal of such clear kinds of objects. — Zuhair

Correct, under your definitions, I refuse a 'tribe" as an object. If we are to consider a "tribe" as an object, the relations between the members are essential to the existence of that object. Remove those relations and you have no object.

You seem to think that you can randomly point to a bunch of objects, say "abracadabra", and suddenly there is another object, which exists as the unity of those objects you have pointed to. Sorry, but that's not reality.

Yes, those metaphysical beliefs clearly do play a role in mathematical proofs because they are entrenched in the axioms, as foundational support for those axioms. And Cantor is a good example. What is at issue here is how we conceive of an "object". — Metaphysician Undercover

I agree that a community's embrace of a set of axioms manifests among other things something like metaphysical preferences or basic intuitions. But this is obvious. And where are the mathematicians that deny it? You and @sandman might as well complain about the rules of chess for not conforming to your metaphysical preferences.

Anti-Cantor cranks are fencing with their own shadows. To be a mathematician it suffices to prove things using 'the rules.' One can think of it as a game with symbols. One can also, to be sure, think that one is doing the True Metaphysics. One can, as I do, think of it as working within a system that strives imperfectly to articulate and accord with intuitions of space, quantity, and algorithm. Imperfectly! I like non-mainstream versions of mathematics. They are fascinating. No need for dogmatism or a fixed position. And that's also how I enjoy philosophy.

So to me the idea that mathematicians are true believers is in general ridiculous, and, in my experience, most online anti-Cantorism is purveyed by those who seemingly can't even play the game agnostically. That would take work, serious interest, and not just self-inflating online conspiracy theory.

You and sandman might as well complain about the rules of chess for not conforming to your metaphysical preferences. — Eee

I don't see how that's an adept analogy. Metaphysical principles are based in how one apprehends the nature of reality. Chess is a game which we can choose not to play if we don't like the rules, how does that relate to reality? Not even by killing oneself can one choose not to partake in reality.

To be a mathematician it suffices to prove things using 'the rules.' — Eee

If the rules do not conform to reality, then I'd have to ask you what are these proofs sufficient for? I'd say they're sufficient to produce unsound conclusions. What exactly would the mathemagician be proving, if one uses poorly formed rules? I'd say that the mathemagician would be proving that confusion follows from the use of poorly formed rules.

One can think of it as a game with symbols. — Eee

I don't like playing games, I'd rather be engaged in something meaningful.

So to me the idea that mathematicians are true believers is in general ridiculous, and, in my experience, most online anti-Cantorism is purveyed by those who seemingly can't even play the game agnostically. That would take work, serious interest, and not just self-inflating online conspiracy theory. — Eee

That's the point, why put serious effort, work, toward something which is just a game? I know that athletes do it. Sorry, but I'm not interested.

You seem to think that you can randomly point to a bunch of objects, say "abracadabra", and suddenly there is another object, which exists as the unity of those objects you have pointed to. Sorry, but that's not reality. — Metaphysician Undercover

So in your sense if I bought two applies today, then I only have two objects, that is the apples themselves, there is no other object that is the totality of these two applies, i.e. the sum material of these two apples, i.e. an object such that each of these two apples is a part of it, and that doesn't have a part of it that is disjoint of these two apples. To my naive understanding, I see it obvious that there is that object.

I see each apple as an individual object. The claim that an apple is an object is justified by sense perception of its existence separate from other things. If you show me two apples and assert that the two are one, then you need to supply a principle to justify this claim.

Let's say that the two apples are "the same", in the sense that both are apples. So we place them both in that category, the set of apples. Notice that "the same" here is not being used in a way which is consistent with the law of identity. The apples are not really "the same" in that sense, as they remain two distinct objects. Now, what is one is the category, or set we have created called "apples". The apples are not really unified to be one, they are judged as being members of one set, according to the principle (the Idea) whereby we class them both as apples. This is explained by Plato in his famous theory of participation. The distinct objects partake in the Idea. It is well explained in The Symposium.

But it is important to note the deficiency of the theory of participation which is developed in Plato's Parmenides, which some people argue leads to the refutation of Pythagorean/Platonic idealism by Aristotle. Let's say that the Idea, which is "the set of apples", provides the unity whereby the two apples are judged as one. It is the Idea itself, which is one object, not the two apples. The problem is that no matter how many apples partake in this Idea (the set of apples), the Idea as one object does not change. Apples may come and go from the set, as time passes, but despite this activity the set itself, as an object never changes. We see that the sense of "the same" here means that the Idea is unchanging over time, whereas "the same" in the law of identity allows that the same object may be changing as time passes.

This makes the "Idea", or "set", as an object, passive and unchanging, and therefore independent from, separate and distinct from, the objects identified through the law of identity, which are the members of the set. Through generation and corruption, identified objects which are members of a set, like apples, come into being and cease being, while the idea, or set itself is supposed to be unchanging.

So when you show me two apples and say that they are one object, you might as well show me three or four apples, a thousand, or a million apples, and each time you are showing me the exact same object, the set, or Idea of apples. That object is the Idea, the principle whereby you unify them as one. But that seems like nonsense, that showing me two apples, or four apples, or a million apples, is showing me the exact same object. Therefore it appears like we cannot properly refer to these "Ideas", or "sets", as objects, because we do not allow them to change when change is warranted by what is exhibited. Showing me two apples cannot be showing me the same object as showing me a hundred apples. Therefore either the two apples do not properly make an object, or we do not understand the way that this object, being the Idea or set, changes when members are added or subtracted from it.

So when you show me two apples and say that they are one object, you might as well show me three or four apples, a thousand, or a million apples, and each time you are showing me the exact same object — Metaphysician Undercover

No that is not correct. If you show me three or more apples, the totality object would be some OTHER object. I just showed you two particular apples (those that I've bought today), and I asked you simply if by today when I bought them, do I have an object that is the whole of both of them, and I explained this object in terms of Part-whole relationship, an object such that each apple (that I bought today) is a part of it, and such that it doesn't have a part of it that is disjoint (doesn't share a common part) of both these apples. This object is the smallest object that has both of these apples as parts of. It is simply this object that I've asked you to tell me whether it exists or not. I didn't speak about a Naming category (like the one you've spoken about) nor did I pose the problem of changing material of a set. I said at the moment when I bought both of these apples is there the object that I've defined or not? To me this 'whole' or 'totality' object of these two particular apples, to me I say, is as concrete as the existence of each apple, and it is an object as each apple is an object. Of course this object can be ruined with time, as each apple can be, and actually only when one of the apples constituting it would start to ruin. But this is another question. My simple question is whether such an object exists in the first place.

No that is not correct. If you show me three or more apples, the totality object would be some OTHER object. I just showed you two particular apples (those that I've bought today), and I asked you simply if by today when I bought them, do I have an object that is the whole of both of them, and I explained this object in terms of Part-whole relationship, an object such that each apple (that I bought today) is a part of it, and such that it doesn't have a part of it that is disjoint (doesn't share a common part) of both these apples. — Zuhair

By showing me the apples you are not showing me the "totality object". This you would show to me in your explanation of "part-whole relationship", etc.. That is why number as an object, if it is an object, is something other than the objects which you use to demonstrate its rationality. The proper demonstration is an explanation, and the apples are just props.

This object is the smallest object that has both of these apples as parts of. — Zuhair

So by the time you are talking about a number of apples, "number" is completely abstract, and you are applying that abstract idea back, onto the apples. The claimed object which is signified by "2" does not have any apples as a part of it, it is abstract. And you cannot demonstrate that object to me by showing me two apples, because such a demonstration can only be done through an explanation of what it means to be 2.

It is simply this object that I've asked you to tell me whether it exists or not. — Zuhair

That claimed object, the "totality object", is what I see no reason to believe exists. You made some claim about "part-whole" relationship, but I see no reason why two apples, ten apples, or a million apples is a "whole" anything. Therefore there is no "whole" to this object which you are describing, and your descriptive terms are misleading. You are assuming that two apples is a whole, without any reasons why two apples may be a whole, and why it doesn't take three, four, five, or the proper totality of all apples, to make a whole. Surely there are more apples than two, so by what principle do you apprehend two as a whole?

This is what Plato's theory of participation demonstrates to us. The "whole" is an Idea, the parts participate in the Idea. Now the existence of the Idea, as the whole which unifies the participants producing the object, must be supported or else the whole theory of participation falls apart. "One" is fundamentally a whole, that is its essence, a unified entity, but "one" is fundamentally different from "two" by the difference between a singularity and a multiplicity. So what principle will you introduce to support your assertion that "two" is a whole, just like "one"?

You are assuming that two apples is a whole, without any reasons why two apples may be a whole, and why it doesn't take three, four, five, or the proper totality of all apples, to make a whole. Surely there are more apples than two, so by what principle do you apprehend two as a whole? — Metaphysician Undercover

All of those have their wholes, For any predicate that hold of apples there is a totality of all apples fulfilling that predicate. And those totalities would be different totalities if the apples constituting them are different. But I've just presented to you a particular case. There is nothing special about two here or three or any number.

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 units 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 units. So they are not the elements of themselves. Now for the sake of simplicity let's assume the ideal condition of all units 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 units 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 units, by stable units. So any two stable collections (i.e. their units are unchangeable over time) would have distinct representative units (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 units 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 ways, 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 represents a collection of units, now those units of the represented collection are the members 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 units, 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 (i.e. units) representing parts of that empty object, then to the collection of all sets (i.e. units) representing parts of the resulting objects, etc... According to this view a set is always a unit, and that unit act to represent a collection of units.

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.

Sets not only can represent finished collections, it can also represent 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 of naturals.

So set theory of mathematics like in ZFC are just a theory about representation of actually finished collections 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 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 terms of element-hood of collections, and this is a great confusion. Sets 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 a stronger concept than the collective concept. The former is representational and latter is mereological.

All of those have their wholes, For any predicate that hold of apples there is a totality of all apples fulfilling that predicate. And those totalities would be different totalities if the apples constituting them are different. But I've just presented to you a particular case. There is nothing special about two here or three or any number. — Zuhair

Right, that's why I say you're a magician. You claim that you can point to any random objects, and say those objects make another object, a whole. Your claim here amounts to an assertion that you can state "the predicate" which pertains to these two apples, or three apples, or any other number of apples, but no other apples. In reality there is no such predicate, it is a hollow claim. Therefore you have no principles for what constitutes an object, only "if I say it's a whole, then it's a whole". A principled "whole" has a defined completion, not the hollow (and sometimes impossible) claim that it's possible to define the completion.

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. — Zuhair

See, you are using "totality" as if any collection of things is a totality of things. Now "totality" is meaningless and redundant because any collection of things is, by your usage, automatically a totality of things. But to have a proper totality, which gives "totality" meaning, it is required that we name a type of thing, and sum the complete number of those named things. To say that any random number of things is a "totality" is nonsense, because it's not the total of anything, it's just a random collection.

Now you claim that any random collection of elements is a "unit", but you have no principle of unity to substantiate that assertion, only the hollow claim that there is such a predicate which unifies them while excluding others. In reality, the elements cannot be a unit unless they are united by something which produces a unity. To say that any random collection of elements is a unity is to utter nonsense. By what means are those elements united?

Do you not respect the fact, that "a collection" must be defined? And, to complete the specified collection, to produce a totality, or "whole", it is required that the entire collection be summed. To insist that my collection is 'definable', and demonstrably the totality or whole of that 'definable' collection, does not justify the claim that it is a totality or whole. The definition must be produced, and it must be demonstrated that there are no other elements which would fulfill the conditions of that definition in order to justify the claim that the collection is whole, a totality, complete.

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 ways, 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 represents a collection of unites, now those unites of the represented collection are the members of that set. — Zuhair

If I correctly understand what you are saying here, a "set" is a collection. As a "unit" the set is complete, a totality, or whole. The completion is determined, fixed, perhaps even "caused", by the definition, the predicate. The definition makes the unit definite, and that is complete.

Here is the issue which I see. In reality, it is only possible that the definition may produce a complete collection, in some cases it may be the case that a complete collection is impossible according to some definitions. Completion of the collection is really dependent on the quality of the definition. A good definition may produce a complete set, while a bad definition will produce a set which is impossible to complete. Notice your opening sentence in the paragraph above: "Set theory can be explained as an imaginary try to REPRESENT stable collections...". The operative word here being "try". So if we take a poorly defined set, and try to produce a completion, or whole, or if we assert that a poorly defined set has produced a complete whole, this is a mistake.

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. — Zuhair

I see this as a self-refuting, contradictory start. The empty set would be complete, whole, with no members. It requires a definition which nothing could fulfill, a definition of nothing. But it is the definition itself which produces the entity, the unit, so the empty set is at the same time a unit, and also nothing. In common terms, to try and produce the empty set is to try to produce something out of nothing. There is no such thing as the empty set, it is an impossibility, by way of contradiction.

Therefore I propose as the true start, the definition of a unit, what it means to be an object, complete, whole, a totality. We cannot start with an empty symbol because that creates infinite random possibilities for nonsense. We need to start with what the symbol is really supposed to represent, a set, which is an object, a unit, or whole. In this way we may restrict the use of the symbol, to eliminate vain attempts to produce an impossible whole.

Remember, the possibility of completion is directly dependent on the quality of the definition. Therefore we ought to restrict poor quality definitions which are not conducive to the possibility of completion. Such definitions may produce the illusion of completion when no such thing is possible, allowing for misleading, or deception. This can be done with a proper definition of what it means to be a unity, a whole, complete or totality.

So the empty set, as a starting point ought to be replaced with the set of one, a whole, an object, complete, whole and total in itself, by its very nature; so that the principal or primary set is consistent with itself, and not self-contradictory as the empty set is.

According to this view a set is always a unit, and that unit act to represent a collection of units. — Zuhair

One problem, the empty set cannot be a unit, as described above, that's self-contradictory. It's an object composed of nothing.

I just wanted to put you in the picture, that sets (as used in mathematics) are different from the 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 terms of element-hood of collections, and this is a great confusion. Sets 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 a stronger concept than the collective concept. The former is representational and latter is mereological. — Zuhair

I would say that this confusion is actually produced by the convention of allowing for empty sets. By allowing for a set with no members we produce a separation between the set, and the members of the set. But no such separation is warranted. The set itself is the object, the elements comprise that object, and the object is created by the definition. The quality of the definition determines the definiteness or definitiveness of the object. A vague definition produces a vague object. So the set, and the elements or members, must be one and the same, in order that the set be the object composed of those elements, while the definition is what has separate existence. Therefore judgement lies in conformity between the definition and the elements, such that a poorly defined set makes a vague object, or in some (impossible or contradictory) cases not an object at all.

Now you claim that any random collection of elements is a "unit" — Metaphysician Undercover

I never said that, nor did I claim it. Actually what I said refutes that!

If I correctly understand what you are saying here, a "set" is a collection. As a "unit" the set is complete, a totality, or whole — Metaphysician Undercover

You didn't correctly understand what I was saying!

to say that any random collection of elements is a unity is to utter nonsense. — Metaphysician Undercover

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.

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. 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. The only collection that is a unit is a collection that have one individual, like the collection of one apple, like the collection of one bird, etc.. those are unit collections. You need some experience with Mereology (Part-whole formal study) and connectedness (Separate-contact) formal study, joining both fields you have what is known as Mereo-topology. You need to be familiar with the axiomatization of Mereo-topology, in order to get the grasp of what I'm writing here. These are particular concepts, they are not that philosophical, but of course they can be realized on philosophical grounds.

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. 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.

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.

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 non-representing 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....

So one need to discriminate sets (which are unit collections that act as representatives of collections) from collections (which are totalities of unit objects). If one manage to fathom that discrimination, then one can of course understand the difference between being an element of a collection, which is being an individual (i.e. a unit) part of that collection, and between being a "member" of a set, here a set is a representative of a collection, and with well founded sets, they are always external representatives of collections (like in the lawyer, accused example), now being a member of a set is actually to be an element (i.e. a unit part) of the collection represented by that set. Membership of sets is a representational act, it is a kind of a singular representational act. Discrimination between the concepts of Collections and their elements, from Sets and their members, is vital for a proper understanding of the subject of sets and classes, and it is something often misunderstood, and misrepresented even at official text-books unfortunately.

Actually If I was to rename matters, I'd call collections as sets, and what is termed as "sets" in set theory I'll call as representatives, and epsilon membership, I'll re-name as "representation step". Anyhow

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".

I hope you can re-read my prior posting with this clarification.

As about the question of random collections and defined ones, this is another matter, I didn't allude to those yet. I want to define the basic terms, and then if we have some agreement over those, we can go to those issues. But basically I do agree with the sentiment that ALL collections are aught to be definable!

Here is a paper that questions the 'diagonal argument'.
https://app.box.com/s/vdop6iqhi8azgoc2upd76ifu8zacq8e4 — sandman

Cranky. — fishfry

The author giving just one reference and that being the Wikipedia page of the diagonal argument is telling by itself.

And seems like the author is simply confused about infinite sets. And one really has to understand how different the reals are.

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.

See, under this system, the collection, as an object, can only be apprehended as an object, to the extent provided by the representative. — Metaphysician Undercover

No this is wrong. A collection can exist and be apprehended without having any representative, or even if it has a representative, the apprehension of the collection need not depend on it. Having representatives is and ADDITIONAL feature. It enables that collection to be a member of higher collections through the representation relation.In other words, representative singulars are only essential for having a hierarchical development of collections of collections of collections... etc.. It is not essential for our apprehension of the collection itself, which could be described in fairly specific manner without reliance on having a representative whatsoever.

What I'm trying to achieve is a hierarchical buildup like bringing separate bricks, define a collection of them, assign some brick (external to them) to act as a representative of them, actually just a label of the collection of those bricks, now there are other representative bricks representing other collections of bricks, now put those representative bricks into collections and also assign other bricks as representative of those collections, and so on... going up. Each brick is a unit, but a collection of separate bricks is not a unit. It is something like this envisioning that I want to construct.

OK, let me simplify this method. Lets use the concept of Mereological atom. An atom is an object that doesn't have parts other than themselves. Now a totality of atoms, is a collection. and an element of a collection is being an atom part of that collection.

Now the buildup I want to speak about is to have a Representation relation that assigns in a unique manner to each collection of atoms, some atom that act as a label (name) for that collection, I call this uniquely distinguishing label as a representative. Of course a collection of atoms might have a representative or might not have any one.

The buildup is to have collections of atoms, each of these collections is represented uniquely by a unique atom, now the next tier is to have collections of those representing atoms, and those collections would in turn also have representative atoms, and so on....

Set theory is about such a hierarchical build up.

Is this artificial. Yes it is! Not only that even the representation relation can be a fixed one. Much as naming symbols are arbitrary in nature.

You can call this hierarchy of names, or Naming Hierarchy. Each name can be understood as a mereological atom. Now we have collections of names, those collections themselves have names that names them, then we have collection of names of collections of names, then we have names of those, then collection of those, then names of those..etc... I'm claiming that Set theory of mathematics thrives in such a naming hierarchy which I happen to call the representative hierarchy.

One need to completely disentangle the concept of representation (unique naming) from that of collection, that's the point that I'm trying to insist on here. A collection of mereological atoms satisfying a predicate \phi, lets denote it as C^phi, is the totality of all of those atoms, i.e. C^phi is an object that has each of those atoms (satisfying phi) as a part of, and such that any object that has each of those atoms as a part of, would have C^phi as a part of! This is substantially different from the *SET* of all \phi atoms, lets denote it by S^phi, here S^phi is the atom that names the collection C^\phi. Now being an *element* of the collection C^phi means being an atom that satisfy phi, that is a part of C^phi, while being a *member* of set S^phi means being an atom that is part of the collection named by S^phi, so it doesn't necessarily mean being an atom that is a part of S^phi. In some sense a set is one step higher than its members, while a collection is not higher than its elements.

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. — Metaphysician Undercover

My outcast is descriptive, while your's is largely etiologic!

Collections can be fairly described and recognized up to identity without resorting to any representative of them, yes I do agree with that. Representatives are neither essential for the existence nor for the characterization of a collection. However representatives of collections are essential for developing a hierarchical account about collections, i.e. when we want to speak about collections of collections of collections, etc...

I don't believe in random collections, yet I don't refute them. However the concept of "random" seem to be different to me than to what you mean by it. It seems from your accounts that you call a totality of unconnected parts as a random totality, because there is NO etiologic like connection between its parts in your sense, so you call such collections as arbitrary, random, etc.. While to me the concept of random only raise versus definable. To me a definable collection of separate unit objects, is itself an object, and it is not a random object because there is a strict "descriptive" rule that joins its separate unit parts. However that descriptive joining of its unit parts should NOT be understood as a kind of "connection" between its unit parts that renders them inseparable, otherwise those would seize to be units, the unit parts still remain "separated" since there is no material (or if you like call it substance) that joins them together, so they remain separate apart, even though they are descriptively linked in some manner. Now as long as there is a descriptive characterization of the collection in a unique manner, i.e. the collection is a definable entity in terms of its unit parts, i.e. like in saying it is the totality of all unit objects satisfying predicate \phi for example, then this definability is (to me) against saying that this collection is a random collection. To me randomness only arise if there is no such a definition, so we have an object that is the totality of unit objects and yet there is no description of those unit objects in our language. Those so called as indefinable collections are really what can be said to be random collections. So for example there is no clear etiologic connection between some particular tear drop and some particular orange, but we can descriptively define a collection of both of those individual objects. That collection is an artificial hybrid, a chimeras, still it is not random, since it has a unique description. Now if a collection is definable in terms of being a totality of unit objects satisfying some particular predicate, then we can assign a representative unit object that can serve to label it (i.e., represent it). However can we assign a label to an undefinable collection? My guess is NO, we CAN'T. Because intuitively we can only label what we can describe. That's why in my philosophic line of thought all sets in set theory are ought to be definable! That is they are names for definable collections!

However my account is different totality from your account. You refuse to admit a collection of "unconnected parts" being an object, to you there should be a kind of necessary relationship between the parts of an entity for it to be an object. That's why you call any try to describe a collection of unrelated objects, as an object, as being magical, since it brings to existence something out of nothing, to you it is some kind of fuzzy entity that doesn't qualify of being an object. While to me it qualifies as being an object no matter how much fuzzy it is, actually even if it is indefinable, still it is an object, so in principle I admit the possibility of the existence of fuzzy collections as well as indefinable collections. Yet I don't see such collections as useful, and I would be sympathetic with a line of argument rejecting their existence. But I don't admit fuzzy sets, since sets are labels, representatives, and it would be against the nature of naming to have them name fuzzy collections, this would be very confusional. And to say that we can name indefinable collections is even contradictory, it is like naming the unnameable. You see the difference of how I use the words "collection" and "set" above.

Cranky.
— fishfry

The author giving just one reference and that being the Wikipedia page of the diagonal argument is telling by itself.

And seems like the author is simply confused about infinite sets. And one really has to understand how different the reals are. — ssu

Are you saying I was excessively judgmental? Perhaps.

No this is wrong. A collection can exist and be apprehended without having any representative, or even if it has a representative, the apprehension of the collection need not depend on it. Having representatives is and ADDITIONAL feature. — Zuhair

I don't see how this could be true. There is nothing to validate the collection as a "collection" without a representative. In both your examples, the lawyer and the tribe, the collection cannot be apprehended as a true collection without the representative. The representative is the thing which tells you things about the collection such that you might apprehend it as a collection. Without the representative, you might apprehend any group of objects as a collection, or not as a collection, with nothing to tell you whether it truly is a collection of not. If you move to predicates to validate the existence of the collection, rather than a representative, then you move to my system. But then we must work top down instead of bottom up. And, what would be the point to having a representative if you could apprehend the collection without a representative?

What I'm trying to achieve is a hierarchical buildup like bringing separate bricks, define a collection of them, assign some brick (external to them) to act as a representative of them, actually just a label of the collection of those bricks, now there are other representative bricks representing other collections of bricks, now put those representative bricks into collections and also assign other bricks as representative of those collections, and so on... going up. Each brick is a unit, but a collection of separate bricks is not a unit. It is something like this envisioning that I want to construct. — Zuhair

But this is not how we understand collections ontologically. We proceed toward understanding them by determining what they have in common, not by looking at a representative. So it seems to me like this idea of representatives is a step in the wrong direction.

However representatives of collections are essential for developing a hierarchical account about collections, i.e. when we want to speak about collections of collections of collections, etc... — Zuhair

It appears to me, like creating a hierarchical account through representatives would be prone to arbitrariness and error. Any hierarchy needs to be created through reference to real relationships between objects, not reference to representatives.

It seems from your accounts that you call a totality of unconnected parts as a random totality... — Zuhair

I would say that this is not even a totality. A totality must be the completion of a defined collection. if the parts are "unconnected' then they cannot be a defined collection.

To me a definable collection of separate unit objects, is itself an object, and it is not a random object because there is a strict "descriptive" rule that joins its separate unit parts. — Zuhair

So how could "unconnected parts" make a collection, or a totality?

and it is not a random object because there is a strict "descriptive" rule that joins its separate unit parts. However that descriptive joining of its unit parts should NOT be understood as a kind of "connection" between its unit parts that renders them inseparable, otherwise those would seize to be units, the unit parts still remain "separated" since there is no material (or if you like call it substance) that joins them together, so they remain separate apart, even though they are descriptively linked in some manner. — Zuhair

Here's the issue which you do not quite seem to grasp. If there is a "descriptive rule" which joins parts, then those parts have a real commonality, which joins them and renders them inseparable in reality. For example, two apples have a single descriptive rule, and they also have a common background, they came from a tree, and they have a common origin. This makes them part of that whole, which is understood through the common origin. We can also say the same thing about two oxygen atoms they have a common source. Being descriptively linked implies a material connection. So I don't think it's correct to make the separation which you want to.

However my account is different totality from your account. You refuse to admit a collection of "unconnected parts" being an object, to you there should be a kind of necessary relationship between the parts of an entity for it to be an object. — Zuhair

Actually, I'm looking for compromise, by allowing that a descriptive rule implies a material connection.

That's why you call any try to describe a collection of unrelated objects, as an object, as being magical, since it brings to existence something out of nothing, to you it is some kind of fuzzy entity that doesn't qualify of being an object. — Zuhair

The fuzziness is now due to the inability to determine the material connections through the descriptive rules. The descriptive rule indicates that there is a material connection. But we cannot get to the material connection through the descriptive rule, and that's why the representative idea is faulty. We cannot get to the real connections which constitute the real existence of the object, through the description. We need to determine the real relationships.

Actually, I'm looking for compromise, by allowing that a descriptive rule implies a material connection. — Metaphysician Undercover

Well I do agree that having a common description imply some material connection, but that connection is not the connection that imply inseparability. You can call these connections "loose" connections, as opposed to "tight" connection which is what causes continuity (inseparability), so if object K has tight connection to object L then they are in continuity, i.e. they are not separate, ie. they are in contact; while if object K has loose connection to object L then they are separate. Now what I call as an "individual" or a "unit" or sometimes I call as a "singular", this is an object that posses tight connections between parts of it in such a manner that it is not the totality of two parts that loosely connected to each other, and at the same manner the object itself must not possess tight connections to external objects. Now if we have many individuals such that there is a description that isolates them from others, i.e. there is a description common to all of them but not to other objects, then those individuals would be said to be LOOSELY connected by this descriptive joining, so they are still separate form each other. Now this would be an object! I call it a collection of objects satisfying this property, so it is a collection and its elements are the individuals that are loosely connected in it. So far for collections.

But this is not enough. You need representatives, or actually NAMEs, you can also call them tokens, or labels, those would be singular objects (units) that we arbitrarily assign to each collection, but provided that the assignment works along unique lines, I mean each collection is assigned only one name, and each name only names one collection. So although the choice of which object would name a collection is arbitrary, but once done naming of other collections cannot use that name, so the naming function is not totally arbitrary. Of course this is not Ontologically innocent, it involves adding unrelated material into the picture!

But why names? why should we assign an external object that is singular to act as a name to a collection that may have multiple elements, so why represent a multiplicity by a singular object? With external naming, there is no clear intimacy between the name and what is named, the assignment is arbitrary for that particular aspect. And this is what actually happens with naming generally, its artificial, for example the names used in language are all arbitrary, there is no special connection between the string of letters "horse" and the animal group it is used to represent. So that's the question: why we should bring an external object that doesn't bear a necessary relationship to a collection and make it act as a name, actually a "representative" for that collection?

The answer is to develop a hierarchical account about collections! This cannot be done in an efficient manner without the use of singular names. The idea is that through this artificially made unique naming process, we can define a new relation, called "membership", that act to copy the relation of element-hood in collections but raises this relation to the name of the collection, and since names are singular objects so they can be elements of collections (while collections when they are non-singular objects cannot be elements of collections, so we can't have a hierarchy of collections in collections using directly the "element-hood" relation!!!), so all elements of a collection wold be "members" of the name of that collection. The "name" of a collection, is what we call as "set" in set theory. So for example the set of objects k,l, denoted by {k,l}, is actually the name given to the collection whose only elements are k,l. so k,l would be "members" of that set, i.e. they bear the membership relation to the NAME of that collection, which is the set itself. Through this copying process of elements to members, one can speak of a hierarchy of sets that are members of sets and so on.... And so indirectly speak of collections of collection of...This would give the powerful mainframe needed to interpret almost all of mathematics.

Now you might be suspicious, and actually object, to such a buildup. Since its pivotal rule is built up through an intermediary that involves some arbitrariness, which is the choice of a name per particular collection of course. So its like building a big building that involves multiple big junks of tightly connected material put on top each other using light joining material, so the the whole buildup is bound to fail!

I'd say it would be extremely difficult to make that hierarchical buildup without using names, I tried myself to figure out these possibilities. For example you can use various grades of tightness of connections, like in saying we have: degree 0 loose connection (which are the tightest connections), degree 1 loose, degree 2 loose, etc.... where for each i the i+1 loose connection is looser than the i loose connection. We can do that and define elements of a collection by those bearing the loosest kind of connection between each other and internally of course they use harder degree of connections, and so on...

This can be done but largely on disjoint collections. When there are overlaps, for example like with the case of power-sets, then here it would become very bleak. And even worst in trying to capture non-well founded sets, that it becomes even impossible to use this method for that sake.

So we needs NAMES, to do the intermediary role in developing a hierarchy of sets of sets of..,etc.. It is the simplest way to do it! And this proves to be very powerful logically speaking, that almost all of mathematics can be encoded in it.

This build-up is not due to me. It is largely David Lewis's idea. But here I used 'sets' as names, while in his approach he use them as collection of names. The approaches are equivalent, but mine is more extensional than his. Also related work can be found in point-set topology although taking different technical paths.

According to this line of thought of mine, to me, sets being names, then they out to be assigned only to "definable" collections. Because with naming procedure, you need name something that you can describe first. And so to me all sets must be parameter free definable, i.e. they must name collections that are parameter free definable collections. And naming must proceed in a hierarchical build up from the simplest to the most complicated in a step-wise manner. So definable collections that are not the result of that buildup cannot have names, and this include the collection of all singular labeling objects (names) , and of all singular names that do are not part of the collections they label, etc.. those collections usually called as "big classes", are not reach-able by a hierarchical naming build-up from below, so they cannot be named, even though they are definable!

The nice corollary to this line of thought is that it proves the axiom of choice! and actually of much stronger form of choice, of a definable global choice!

So having a pertinent line of thought about what sets really are, can solve some technical problems, like with the famous problem of axiom of choice here.

Well I do agree that having a common description imply some material connection, but that connection is not the connection that imply inseparability. You can call these connections "loose" connections, as opposed to "tight" connection which is what causes continuity (inseparability), so if object K has tight connection to object L then they are in continuity, i.e. they are not separate, ie. they are in contact; while if object K has loose connection to object L then they are separate. — Zuhair

We agree to an extent, but you seem to have some ambiguity. Material connection is what makes parts into an object, and when its an object, you say that the parts cannot exist as separate objects. Yet now you say that material connection does not imply "inseparability". I would agree with this if we clarify by distinguishing between actually separate, and separable.

I find your distinction between loose and tight connections to be very fuzzy, and I think we can replace it with the more substantial distinction between temporal continuity and spatial continuity. Suppose for example, that the apple exists as a collection of molecules. There is a tight connection between the molecules, and an even tighter connection between the atoms, and this spatial continuity indicates that these parts are not actually separate. Yet the molecules, and atoms are in principle separable, and this is a function of the temporal continuity of the object. The parts are not separate spatially, yet they are separable temporally.

So we need to distinguish temporal continuity from spatial continuity. If the parts are inseparable in the sense of temporal continuity, then the object is eternal, and it would appear contradictory to even talk about the object as being composed of parts. Such an object is the fundamental, or base "unit". It can have no parts because that would imply that the unit is separable in time. The spatial extension of such an object is dubious.

From experience, it appears like spatial extension implies parts which are separable. It's difficult if not impossible to conceive of a object that occupies space, which is not divisible into parts. If you think about this, you will probably be able to conclude that the actual unity, which is an object with spatial continuity, is really an illusory "unity" due to temporal separability. The spatial continuity which we observe really hides the underlying temporal separability, The parts of the spatial object which are connected through a spatial continuity, may appear to have a loose or tight spatial connection, but the true strength of that continuity will only be understood through an understanding of the temporal separability. Since the temporal aspect is what hands the parts of the object separability, the nature of time is better understood through the terms of discontinuity. Therefore we end up with spatial continuity which is an illusion, and temporal discontinuity which describes the reality of the object in terms of separability. The reality of the eternal object (true temporal continuity) is dubious, as synonymous with the object that has no spatial extension.

But this is not enough. You need representatives, or actually NAMEs, you can also call them tokens, or labels, those would be singular objects (units) that we arbitrarily assign to each collection, but provided that the assignment works along unique lines, I mean each collection is assigned only one name, and each name only names one collection. So although the choice of which object would name a collection is arbitrary, but once done naming of other collections cannot use that name, so the naming function is not totally arbitrary. Of course this is not Ontologically innocent, it involves adding unrelated material into the picture!

But why names? why should we assign an external object that is singular to act as a name to a collection that may have multiple elements, so why represent a multiplicity by a singular object? With external naming, there is no clear intimacy between the name and what is named, the assignment is arbitrary for that particular aspect. And this is what actually happens with naming generally, its artificial, for example the names used in language are all arbitrary, there is no special connection between the string of letters "horse" and the animal group it is used to represent. So that's the question: why we should bring an external object that doesn't bear a necessary relationship to a collection and make it act as a name, actually a "representative" for that collection? — Zuhair

OK, I'm with you here. I apprehend a name as an object, which is a representative of something else, as described. And I wonder why there is a need to bring names into the picture. The type of object which a name is, I think is a tool, and this tool has the purpose of understanding. It's the tool we use for understanding.

The answer is to develop a hierarchical account about collections! This cannot be done in an efficient manner without the use of singular names. The idea is that through this artificially made unique naming process, we can define a new relation, called "membership", that act to copy the relation of element-hood in collections but raises this relation to the name of the collection, and since names are singular objects so they can be elements of collections (while collections when they are non-singular objects cannot be elements of collections, so we can't have a hierarchy of collections in collections using directly the "element-hood" relation!!!), so all elements of a collection wold be "members" of the name of that collection. The "name" of a collection, is what we call as "set" in set theory. So for example the set of objects k,l, denoted by {k,l}, is actually the name given to the collection whose only elements are k,l. so k,l would be "members" of that set, i.e. they bear the membership relation to the NAME of that collection, which is the set itself. Through this copying process of elements to members, one can speak of a hierarchy of sets that are members of sets and so on.... And so indirectly speak of collections of collection of...This would give the powerful mainframe needed to interpret almost all of mathematics. — Zuhair

This I find to be very confused and I am unable to follow. I agree that we may use names to develop a hierarchical account, that is part of understanding, but where I find confusion is in your failure to recognize the distinction between naming an object, and naming a relation. These are two distinct uses of a name, like the difference between noun and verb.

So if we name "membership", what that name actually refers to is the relationship implied by "membership". Now we must guard against deception. We can name membership when no reasonable relationship has been identified. Therefore there is no point to naming "membership", unless to deceive. The relationship ought to be named directly, without the medium "membership". Furthermore, what follows from this, is that this "name", which is the name of the collection, but actually represents membership in the collection, which in itself represents a relation, is an even further layer of representation. So we have three levels of representation now, the name represents the collection, the collection represents membership, and membership represents a relationship. Plato warned us against such multi levels of representation, calling them "narrative". Any hierarchy produced in this manner would be extremely unreliable, as we ought to refer directly to "the good" to produce a hierarchy. Names are tools used for understanding, so the good here is understanding. Multi levels of representation are conducive to confusion rather than understanding.

Now you might be suspicious, and actually object, to such a buildup. Since its pivotal rule is built up through an intermediary that involves some arbitrariness, which is the choice of a name per particular collection of course. So its like building a big building that involves multiple big junks of tightly connected material put on top each other using light joining material, so the the whole buildup is bound to fail! — Zuhair

Exactly! A name is an object with extremely unreliable temporal continuity so we ought not construct structures using names. And, since it is required to produce a multi level representation to make the name into a building material, this makes the structure even more unreliable.

So we needs NAMES, to do the intermediary role in developing a hierarchy of sets of sets of..,etc.. It is the simplest way to do it! And this proves to be very powerful logically speaking, that almost all of mathematics can be encoded in it. — Zuhair

But there is no need for a hierarchy of sets, that is a faulty premise. What is needed is to understand the relationships between things, that is the good which we might put the tools (names) to use toward. The hierarch is produced naturally by understanding the nature of continuity, and the strength of relationships. So we begin with some fundamental determinations concerning the nature of continuity; for example what is prescribed above concerning spatial and temporal existence, and we proceed to name relationships according to their strength, producing a hierarchy of relations. There is no need for a hierarchy of sets.

But there is no need for a hierarchy of sets, that is a faulty premise. — Metaphysician Undercover

Actually from experience with mathematics. We do need a hierarchy of sets. The other alternatives are not so promising.

I find temporal versus spatial separability not easy to fathom. I'll have a better try to fathom your notes about it.

I do admit that there is some ambiguity with my characterization of "loose" versus "tight" connection.

I view the connection between adjacent parts of an apple as being tight connections, while I view the descriptive joining of many apples satisfying some predicate as being a kind of loose connection. "Separability" in my sense means absence of tight connections, so loosely connected objects are separate, while tightly connected objects are not separate. I define a "unit" or a "singular" as an object K such that for any part x of K, the part of K that is the complementary part of x, denoted by x'^K, is in contact with x, i.e. x'^K is in contact with x. So when an object is a unit, then it is not the totality of non-tightly connected objects, and the other condition is that a unit must be not be in tight connection with an external object, i.e. it must be separate from external objects, so it can bear loose connections to external objects, but it cannot possess tight connections with them.

So a collection is a totality of one singular object, or a totality of many singular objects loosely connected to each other through descriptive joining of its singular parts through having a common description that isolates them from other singulars not fulfilling that description.

A name is a singular (a unit) that as you called it a "tool" that helps us understanding, here it helps us direct our attention to a specific collection, in such a manner that we can speak of multiplicity of collections in a hierarchical manner. Each collection has only one name, and each name only names one collection. Sets are singular names of collections. They are not names of relations. Naming of relations is a different subject, and I've never attempted to speak about it in any of my prior comments. I've been always speaking about naming collections, and so speaking about naming objects, and not relations.

When a set say set x names some collection C, then we call each "element" of C (i.e. each singular part of C) as a "member" of x. In some sense membership would copy element-hood but transfer it to an object external to the collection, that is to the name of the collection. But you need not confuse "membership" as a name for "element-hood", No! That is not the case. Membership is not a name, it is a relation, so it is not an object.

Now through membership relation and sets (i.e., names of collections), one can easily define a hierarchy of sets. And that build-up proves to be an extremely useful tool in our understanding of many mathematical entities. And the witness to that is SET THEORY. In particular ZFC set theory (Zermelo-Frankel set theory with Choice), which proves to be very powerful in understanding mathematical entities and rules, through the iterative buildup of a hierarchy of sets.

Of course for the development of set theory, all of our units are un-breakable over time, and they don't change their tight connections with time, so they are remotely different from natural objects which rut over time or combine with other objects to build bigger units, etc... Here in the platonic mathematical imaginary world, all individuals (units) have non-changeable tight connections over time. So they are as you said "eternal". Then we can freely form collections of them using the descriptive tool, and with the help of the naming relation, we can speak of a hierarchy of them, which helps us encode almost all of mathematical entities in it. Thus serving as a FOUNDATION for MATHEMATICs.

I'll try to keep track with your notions of temporal versus spatial continuity, and come back with comments about it.

Yet now you say that material connection does not imply "inseparability". I would agree with this if we clarify by distinguishing between actually separate, and separable. — Metaphysician Undercover

I'm speaking within the confines of a mathematical realm, some platonic realm in which time doesn't cause any change to connection relations. So what is actually separate is always separate, so separable is separate, and so temporal x spatial connection is immaterial in this realm. We only have spatial connection and separation. That said we need to revert again to loose versus tight connections.

My account entails that the existence of connections between parts of an entity is what qualifies that entity to be an object. So having loose connection is fairly enough for that quest. You don't need tight connections between parts of an entity to qualify it for being an object! NO! loose connection can do the job, so an entity in which loose connections between its singular parts exist, is perfectly qualified of being an "object. However, you need tight connections to form units (singulars) but units are just special kinds of "objects", so an entity that has tight connections over its parts and it itself doesn't have that kind of connection to external objects, that would qualify it to be a unit object. But objects need not be units. They can be totalities of loosely connected units, or what I call as "collections". So as such collections qualifies for being "objects". I hope this resolves the confusion.

Now the loosest kind of a connection is the one made through descriptions. And so an entity of separate singular parts that are commonly describable in a manner that isolates them from objects not fulfilling that description, is the least needed condition to qualify it as an object. And we have this situation with collections as I defined.

The above coupled with a naming function that name collections by singular names, is enough to build up the required hierarchy in which almost all of mathematical objects can be carved!

The hierarch is produced naturally by understanding the nature of continuity, and the strength of relationships. So we begin with some fundamental determinations concerning the nature of continuity; for example what is prescribed above concerning spatial and temporal existence, and we proceed to name relationships according to their strength, producing a hierarchy of relations. There is no need for a hierarchy of sets. — Metaphysician Undercover

As attractive as it sounds, this proves to be extremely difficult. Experience along such lines are moot. Its hopeless. Without a hierarchy of sets, or similar structure, there is almost no hope to encode most of mathematics. You will only have the sketchy picture of prior to the twentieth century mathematics. But again this is one of the most useful kinds of mathematics.

So we need to distinguish temporal continuity from spatial continuity. If the parts are inseparable in the sense of temporal continuity, then the object is eternal, and it would appear contradictory to even talk about the object as being composed of parts. Such an object is the fundamental, or base "unit". It can have no parts because that would imply that the unit is separable in time. The spatial extension of such an object is dubious. — Metaphysician Undercover

Ok, I agree it would be eternal since its not actually breakable. But why it can have no parts? Any object is itself a part of itself. Perhaps you mean it doesn't have "proper parts" [parts of an object other than itself]. I'm under the impression that you think that an object must be breakable to into parts in time in order for us to say that it has parts. But this is not correct. Even if we have an object that is eternally not breakable, still it can have many parts connected by tight connection in a manner that renders it a unit, it doesn't mean it doesn't have proper parts, it only means its no breakable to them, but it can have them always as parts of it. In real life having eternal objects is itself faulty. So the idea of eternal objects is hypothetical or actually imaginary, it suits framing mathematical objects, because usually we work with mathematical objects in some Platonic realm, and that realm appears to be time free, sometimes even disrupt spatial reasoning as well, anyhow.

Now if we work in an imaginary space in which time has no effect, i.e. doesn't change connection relations of objects to each other, still it is imaginable for those objects to have parts, so having parts is not a function of temporal separability as you hold. Not only that, still without time we can fathom of having objects that are composed of units that are loosely connected to each other. So we can have collections having many elements.

However, we may reach into a definition of "loose" and "tight" connections using this spatial x temporal distinction. For example you can say that:

x tightly connected to y if and only if at all times x is connected to y;

while

x is loosely connected to y if and only if sometimes x is connected to y and sometimes x is not connected to y.

According to this definition a "true unit" would be an object that is never breakable nor is continually in connection with an external object.

Of course in a mathematical realm in which time is not operable, like "most" of mathematical contexts, then all unit objects in that realm are true units.

Of course in an imaginary context one can incorporate time into the mathematical world and actually use that time-dependent definition of "true unit" and of loose and tight connection. So a collection would be either a true unit or a pseudo unit (a temporally appearing continuous object that is temporally separable from external objects), and being an element of a collection would be being a true unit part of that collection. Then we introduce naming of these collections with true units, and everything would run as I intended.

The problem is that this would add additional features to the picture, namely temporarily, which is not all that desirable in a mathematical realm. For the purpose of defining sets, we can simply hold the dichotomy of loose and tight connection as primitive concepts without relation to time. Our aim is largely descriptive. Since set theory serves as a foundation for mathematics then the particularities of what decides the "units" of a certain mathematical discipline is stuff related to the particularities of that discipline itself, so in Geometry units would be "points", in arithmetic units would be "numbers", in set theory units would be "sets", etc.... Here we are only concerned in introduced a general descriptive framework that can be applicable to diverse mathematical disciplines, and possibility even non-mathematical spheres of knowledge as well! For example the idea of having a "true unit" in time, might be useful in understanding the ontology of time and space?

I think that "collections" and "sets" are pervasive, so they can be used in any field of knowledge. That's why I'm dealing with that matter in a merely descriptive manner. So consider relations of "connection", "tight connection" to be primitive relations. Then using logic, we can define the terms "singular" (i.e. unit), collection, element, set, member. And that's all what we need. Then build the hierarchy in a gradual manner, and you are allowed to go as high as consistency permits. That set you get all the extensions of ZFC set theory, and thus encode any area of knowledge really, but specially mathematics.

So if we name "membership", what that name actually refers to is the relationship implied by "membership". Now we must guard against deception. We can name membership when no reasonable relationship has been identified. Therefore there is no point to naming "membership", unless to deceive. The relationship ought to be named directly, without the medium "membership". Furthermore, what follows from this, is that this "name", which is the name of the collection, but actually represents membership in the collection, which in itself represents a relation, is an even further layer of representation. So we have three levels of representation now, the name represents the collection, the collection represents membership, and membership represents a relationship. Plato warned us against such multi levels of representation, calling them "narrative". Any hierarchy produced in this manner would be extremely unreliable, as we ought to refer directly to "the good" to produce a hierarchy. Names are tools used for understanding, so the good here is understanding. Multi levels of representation are conducive to confusion rather than understanding. — Metaphysician Undercover

I couldn't manage to follow that really. But in my usage when I used names, I used them to name "collections" which are "objects" and not relations. When I write "member" or "membership" this is a symbol to denote the membership relation, I don't mean by those symbols to be "names" those are not names, those are definable relation symbols. Names only are assigned to name collections which are objects. I don't think that a hierarchical build-up would be confusional, why? The definitions I gave were very strict. So I'm using "names" in a particular context, that is to name collections. symbols used to "name" relations are not called as "names" here.

ut where I find confusion is in your failure to recognize the distinction between naming an object, and naming a relation. These are two distinct uses of a name, like the difference between noun and verb. — Metaphysician Undercover

You said I'm not discriminating between naming of objects and naming of relations. I need to see where exactly I made this confusion. I introduced names for the specific context of naming collections, and I defined collections as totalities of loosely connected singulars (units) and I justified my claim that such entities are totalities and such totalities are objects. So I was all the way speaking about naming objects. Where do you see me introducing names to name relations and confusing those relations as objects? I never said we can name for example the relation 'membership' and I've never introduced a name for it (in the particular context of name that I've used) where do you see me speaking about introducing a name for the relation "element-hood" and speaking about such naming? The relations that I've used are "connection" tight and loose, part, membership, element-hood, where do you see me speaking of attaching names to those? I never said that! I only spoke specifically about introducing singular names for collections, and I specifically defined what "collection" mean, and I justified that being an "object".

They are not names of relations. Naming of relations is a different subject, and I've never attempted to speak about it in any of my prior comments. I've been always speaking about naming collections, and so speaking about naming objects, and not relations. — Zuhair

The point is that your description of the distinction between "loose" and "tight" does not provide us with an indication as to what these terms really mean. The reality is that the parts of a collection are either loose or tight depending on what type of relation they have with each other. Therefore there are all sorts of different types of sets, which may be named dependent on the relations between the parts. We cannot just classify loose and tight sets, just like we cannot just class soft and hard physical objects, there are all sorts of different type of objects which we name, like 'cars' and "houses". The naming of a set, as it is supposed to be the naming of an object ought not be any different. The name ought to indicate to us what type of an object that particular set is, by indicating something about the relations of its parts, or its use, or something like that. W#hen someone says "house", or "car", it brings to mind a specified type of object, the same ought to be the case when sets are named.

When a set say set x names some collection C, then we call each "element" of C (i.e. each singular part of C) as a "member" of x. In some sense membership would copy element-hood but transfer it to an object external to the collection, that is to the name of the collection. But you need not confuse "membership" as a name for "element-hood", No! That is not the case. Membership is not a name, it is a relation, so it is not an object. — Zuhair

So this is a basic type of set. The set "C" has "members", "elements", that is what is specified of this type of set. Notice that there is no specification as to the relationship between members, the relations might be either loose or tight. Would you agree that the so-called "empty set", and the set with only one member, are different types of sets from the set which has members. We could have a set "A", which is empty, and a set "B" which has one member, and then the names A, B, C, would each specify a different type of set, therefore the name would be useful. We could then proceed to look at the different types of relations between members and have all sorts of subcategories of type C, but A and B would be distinguished as completely different.

Now through membership relation and sets (i.e., names of collections), one can easily define a hierarchy of sets. And that build-up proves to be an extremely useful tool in our understanding of many mathematical entities. And the witness to that is SET THEORY. In particular ZFC set theory (Zermelo-Frankel set theory with Choice), which proves to be very powerful in understanding mathematical entities and rules, through the iterative buildup of a hierarchy of sets. — Zuhair

Unless your hierarchy is built on real principles of distinguishing real difference between types of sets, that hierarchy will be arbitrary and misleading. Therefore the hierarchy is not built on the names themselves, but the type of set which the name identifies. We cannot make up relations between named sets, without first having a complete grasp of the relations between the members of the sets themselves, which gives us an understanding of the type of set, because such relations would be completely imaginary. Understanding the relations between the members, gives an understanding of the type of set, and understanding the type of set will allow us to proceed toward establishing relations between the types of sets. ZFC proceeds without a proper understanding of the types of sets, to produce imaginary relations between imaginary types.

Of course for the development of set theory, all of our units are un-breakable over time, and they don't change their tight connections with time, so they are remotely different from natural objects which rut over time or combine with other objects to build bigger units, etc... Here in the platonic mathematical imaginary world, all individuals (units) have non-changeable tight connections over time. So they are as you said "eternal". Then we can freely form collections of them using the descriptive tool, and with the help of the naming relation, we can speak of a hierarchy of them, which helps us encode almost all of mathematical entities in it. Thus serving as a FOUNDATION for MATHEMATICs. — Zuhair

As I said, the Platonic eternal, unchangeable object must be considered as imaginary, impossible, and must be excluded as not real, until its reality may be demonstrated. Therefore to base a set theory in this assumption is to start from a false, unjustified premise.

I'm speaking within the confines of a mathematical realm, some platonic realm in which time doesn't cause any change to connection relations. So what is actually separate is always separate, so separable is separate, and so temporal x spatial connection is immaterial in this realm. We only have spatial connection and separation. That said we need to revert again to loose versus tight connections. — Zuhair

But this is unreal, impossible, therefore I deny an such "speaking" as irrelevant, because you are speaking illogically, of the impossible as if it were possible.

My account entails that the existence of connections between parts of an entity is what qualifies that entity to be an object. So having loose connection is fairly enough for that quest. You don't need tight connections between parts of an entity to qualify it for being an object! NO! loose connection can do the job, so an entity in which loose connections between its singular parts exist, is perfectly qualified of being an "object. However, you need tight connections to form units (singulars) but units are just special kinds of "objects", so an entity that has tight connections over its parts and it itself doesn't have that kind of connection to external objects, that would qualify it to be a unit object. But objects need not be units. They can be totalities of loosely connected units, or what I call as "collections". So as such collections qualifies for being "objects". I hope this resolves the confusion. — Zuhair

But this is total nonsense. In the "platonic realm", all objects are eternal, so the distinction of loose and tight means nothing because one cannot be more susceptible to corruption than the other. And "loose and "tight" cannot refer to spatial relations because eternal objects are non-spatial. Therefore your specified relations are completely illogical, there is nothing to justify "tight" or "loose".

Ok, I agree it would be eternal since its not actually breakable. But why it can have no parts? Any object is itself a part of itself. — Zuhair

Again, this is illogical nonsense which has been accepted into set theory, the idea that a thing can be a part of itself. The adoption of that idea into set theory is representative of the breakdown in logic of set theory. "Part" means 'some but not all of', and "whole" means 'all of'. If we redefine "part" such that 'all of' may be 'part of', then we loose the essence of the distinction between part and whole, rendering a corrupted ontology.

The eternal thing cannot change, that's what defines its being as eternal. But if a thing is composed of parts, then the parts must exist in relations to each other. Sense observations demonstrate to us that all relations are changing (relativity). Therefore it is impossible that an eternal thing is composed of parts.

Even if we have an object that is eternally not breakable, still it can have many parts connected by tight connection in a manner that renders it a unit, it doesn't mean it doesn't have proper parts, it only means its no breakable to them, but it can have them always as parts of it. — Zuhair

As I've shown, your distinction of "tight" and "loose" amounts to meaningless nonsense. If the relations between an object's parts are so tight that these relations cannot change with an eternity of passing time, the claim that there are parts in relation to each other, rather than one changeless entity without parts, is unjustified. If the relationship between two so-called "parts" of an object cannot change, then the claim that they are "parts" is not justified. That is one object, a whole without parts.

In real life having eternal objects is itself faulty. — Zuhair

Exactly, therefore we must establish proper principles to distinguish between imaginary eternal objects, and real objects. An eternal object, by nature of what it means to be eternal, cannot consist of parts. This presents a problem to set theory. A set, by its nature consists of parts. Therefore those who work with "set theory" must resist the contradictory notion of treating sets as if they are eternal objects.

Now if we work in an imaginary space in which time has no effect, i.e. doesn't change connection relations of objects to each other, still it is imaginable for those objects to have parts, so having parts is not a function of temporal separability as you hold. Not only that, still without time we can fathom of having objects that are composed of units that are loosely connected to each other. So we can have collections having many elements. — Zuhair

Now, don't you see this as illogical? There is no such thing as a space in which time has no effect. that's like assuming absolute rest, which relativity theory has ruled out as impossible. So if we adopt this premise as a basis for "set theory", we are working from a premise which inherently contradicts relativity theory. If one were to apply this "set theory" in physics, all sorts of confused conclusions would follow from that application of contradictory premises.

Of course in a mathematical realm in which time is not operable, like "most" of mathematical contexts, then all unit objects in that realm are true units. — Zuhair

However, as I've shown, it is illogical to speak of "relations" in a realm where time is not operable. Relations must be justified. If the relation is spatial, it cannot be free from time. If you propose a type of relation which is not spatial, such a proposition needs to be substantiated, justified.

The problem is that this would add additional features to the picture, namely temporarily, which is not all that desirable in a mathematical realm. For the purpose of defining sets, we can simply hold the dichotomy of loose and tight connection as primitive concepts without relation to time. Our aim is largely descriptive. Since set theory serves as a foundation for mathematics then the particularities of what decides the "units" of a certain mathematical discipline is stuff related to the particularities of that discipline itself, so in Geometry units would be "points", in arithmetic units would be "numbers", in set theory units would be "sets", etc.... Here we are only concerned in introduced a general descriptive framework that can be applicable to diverse mathematical disciplines, and possibility even non-mathematical spheres of knowledge as well! For example the idea of having a "true unit" in time, might be useful in understanding the ontology of time and space? — Zuhair

The glaring problem here is that mathematics is fundamentally a tool which is used to understand the relations between objects, from the most fundamental, "order". In the mathematical realm, change to relations is as you say, not "desirable". However, change to relations is reality. So if mathematics assumes changeless relations, because this is desirable, but changeless relations are not real, then there will be a problem in the application of mathematics.

So we need to look directly at this fundamental idea of changeless relations, and determine whether we can make sense of it. I suggest that there is a type of relation, which is demonstrated by the existence of time itself, which may be changeless. This is "order". We can assume an "order" which is given by time itself, a changeless order which is the passing of time. However, we need to make this consistent with relativity theory.

The point is that your description of the distinction between "loose" and "tight" does not provide us with an indication as to what these terms really mean. The reality is that the parts of a collection are either loose or tight depending on what type of relation they have with each other. Therefore there are all sorts of different types of sets, which may be named dependent on the relations between the parts. We cannot just classify loose and tight sets, just like we cannot just class soft and hard physical objects, there are all sorts of different type of objects which we name, like 'cars' and "houses" — Metaphysician Undercover

You Can! if your aim was to FOUND (i.e., lay the basis for) matters with. We leave "tight" and "loose" like blanks to be filled with the relevant application. So tight an loose are left as "primitive" concepts, those would take different meanings according to the working application. Of course collections would have different meaning across all applications, but they will have consistent meaning within the same application. Like how number 1 can have different meaning across applications.

I also want to note, actually an apology, that what I'm saying here about Mereological, well actually Mereo-topological, understanding of set theory is not the conventional line. It has been loosely suggested by David Lewis, but not with all such detail. So set theory is not dependent on those views I'm posting here. But to a great extend those views can make one understand what's going on with set theory as far as applying the rules and the logical flow within these set theories is concerned.

I find your idea that an object cannot have parts unless its subject to temporal separability as un-supported. Especially under imaginary grounds. We can fathom the imaginative line of having an eternal unicorn. Now this entity (which do not exist in the real world) does have parts, for instance the corn is a part of the unicorn, its head, tail, legs, etc.. all of these are parts of a unicorn, so the unicorn is not a mereological atom, even if it was an eternal being. So descriptive imaginative wise we can fathom what it does mean to have eternal objects with parts. But in the real world if one thinks that an object if composed of parts then those parts must have been in existence "before" the whole, i.e. every object must be "formed" after parts by some force connecting those parts that occurs in some moment of time of course after the existence of the parts, then according to this synthetic hypothesis (which holds mostly in the real physical world), then of course one would be bond to reject eternal objects being synthesized from parts, since there should have been a moment where those parts were separate and then after that another moment came where they'll possess a relationship to each other that caused the unit of that object.

In the mathematical realm, we don't adhere to such observations. The mathematical realm is changeless with time (unless time itself is adopted in some mathematical models where change is studied) and objects in that realm can be dealt with as having parts (proper parts I mean) without having to have a formalization (synthetic) moment. That applies to classes (i.e. collections) with many elements, where their singular parts can be understood as their elements, now those multipleton collections do have parts and they are supposed to be eternal in the platonic realm. Now that is obviously false in the physical realm. But this doesn't mean it is false of every realm! Platonists holds that the platonic realm exists, so it is a true realm, while fictionists think its false.

I'd say even if that platonic realm is FALSE (i.e.doesn't exist), still, the logical-mathematical rules displayed in them are not necessarily false. And they can hold of some real scenarios, and so can possibly find applications, and that what really matters!

Of course collections would have different meaning across all applications, but they will have consistent meaning within the same application. Like how number 1 can have different meaning across applications — Zuhair

Then it would be impossible to create a reasonable hierarchy like you were talking about, if the meaning of tight and loose could vary.

I find your idea that an object cannot have parts unless its subject to temporal separability as un-supported. Especially under imaginary grounds. — Zuhair

Of course we can imagine things which would violate that proposed law, but the whole point is to exclude from our principles, things which are physically impossible. If we allow mathematical principles to include imaginary things which are physically impossible, and we apply mathematics within physics which employs inductive conclusions that exclude such things as impossible, then we will be employing contradictory premises in the very same application, as I described.

Relativity theory denies the possibility of eternal unchanging relations between parts (absolute rest). But if mathematical principles allow for eternal unchanging relations, then we have contradictory premises. To resolve this problem we cannot change our description of the physical world without loosing accuracy. So we must change these mathematical principles to provide consistency. That it would be difficult to make such changes, or that the existing principles are supported by simplicity, is no excuse.

I'd say even if that platonic realm is FALSE (i.e.doesn't exist), still, the logical-mathematical rules displayed in them are not necessarily false. And they can hold of some real scenarios, and so can possibly find applications, and that what really matters! — Zuhair

Sure, such mathematical rules would be applicable, and in the vast majority of cases they would give us very accurate conclusions. That is because the vast majority of cases don't deal with things like eternal objects, and infinity is never approached. And, when eternal objects are approached (fundamental particles for example), we can be fully aware of the faults within the principles, and take the conclusions with a grain of salt. However, as these faulty principles become more accepted, and work their way deeper and deeper into the hierarchical structure of the mathematical axioms, their application becomes more commonplace. At this time, they are subsumed by other principles, and we could loose track of when they are actually being applied, and not notice the mistakes which they produce.