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
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.