As explained, this is absolutely false — Metaphysician Undercover
Here is a paper that questions the 'diagonal argument'. — sandman
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
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
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
You and sandman might as well complain about the rules of chess for not conforming to your metaphysical preferences. — Eee
To be a mathematician it suffices to prove things using 'the rules.' — Eee
One can think of it as a game with symbols. — Eee
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
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 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. — Zuhair
This object is the smallest object that has both of these apples as parts of. — Zuhair
It is simply this object that I've asked you to tell me whether it exists or not. — Zuhair
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. — Zuhair
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
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
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
According to this view a set is always a unit, and that unit act to represent a collection of units. — Zuhair
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
Now you claim that any random collection of elements is a "unit" — Metaphysician Undercover
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
to say that any random collection of elements is a unity is to utter nonsense. — Metaphysician Undercover
Here is a paper that questions the 'diagonal argument'.
https://app.box.com/s/vdop6iqhi8azgoc2upd76ifu8zacq8e4 — sandman
Cranky. — fishfry
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
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
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
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 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
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
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
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
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
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
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
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
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
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
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 seems from your accounts that you call a totality of unconnected parts as a random totality... — Zuhair
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
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
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
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
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. — Zuhair
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
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
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
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. — Metaphysician Undercover
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
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
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
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
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
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
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
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
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
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
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
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
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
In real life having eternal objects is itself faulty. — Zuhair
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
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
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 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
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
I find your idea that an object cannot have parts unless its subject to temporal separability as un-supported. Especially under imaginary grounds. — Zuhair
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
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