What are Sets?
in [Logic]
|
Prev: concept of Consistency as a variable, not a constant for math and physics #263: Correcting Math
Next: ~~~~~~~~~~~~~~ CABBAGE SOUP ~~~~~~~~~~~~~~
From: John Jones on 25 Dec 2009 19:40 zuhair wrote: > Hi all, > > One of the facts of set theory is that the primitive relation > "e" is totally vague relation, there is no way of understanding this > relation at all, even at informal level! Which is actually a great > drawback, since in set theory we are building hierarchy of objects > using this relation, a relation that nobody understand what it stands > for? > > Here I will present a simple trial to understand this relation "e" > and to give it some informal intuitive background. > > First I shall speak about the concept of set as used in ordinary > language. > > When we say we have a set of objects x,y,z , then we usually mean that > there is an entity that represent the *Whole* of > x,y,z, so although x,y,z are distinct objects, but yet > the *whole* that is constituted from these objects is considered the > set of these objects, or the *collection* of these objects. > > However this *Whole* of x,y,z is in a sense these objects themselves. So far so good, but that last statement is incorrect. There's no relationship between parts and their whole. > > So a set is a form of describing multiplicity by unity, that unity is > taken to be the totality of the Whole of these multiple objects. > > However if we collect ONE object, then it is pretty intuitive to say, > that the set of all objects collected IS that object. The set "the set of all objects collected" doesn't look like a set to me. It begs the question of what a set is, and if its a collection then it looks tautologous. It just says "set"? > > Also it is pretty intuitive to say that if there are no objects > collected, then there is NO set of objects collected, in other words > there do not exist an empty collection. > > Of course the traditional membership relation "e" doesn't capture all > these pretty intuitive concepts of the customary use of the word > membership and of the word "set" or the word "collection". > > To try to put all of the above informal account into rigorous > mathematical context, we can define a theory in FOL with identity > and the primitive binary relation symbol e' , which stands for > what I call "intuitive membership", or in reality "membership", since > epsilon doesn't capture the intuitive concept of membership. > > So we can have the following axioms that characterize this binary > relation e'. I don't think you will find the resources you need from what is currently available to you. I would look at the fringes.
From: Virgil on 25 Dec 2009 20:01 In article <hh3m28$8ii$1(a)news.eternal-september.org>, John Jones <jonescardiff(a)btinternet.com> wrote: > zuhair wrote: > > Hi all, > > > > One of the facts of set theory is that the primitive relation > > "e" is totally vague relation, there is no way of understanding this > > relation at all, even at informal level! Which is actually a great > > drawback, since in set theory we are building hierarchy of objects > > using this relation, a relation that nobody understand what it stands > > for? > > > > Here I will present a simple trial to understand this relation "e" > > and to give it some informal intuitive background. > > > > First I shall speak about the concept of set as used in ordinary > > language. > > > > When we say we have a set of objects x,y,z , then we usually mean that > > there is an entity that represent the *Whole* of > > x,y,z, so although x,y,z are distinct objects, but yet > > the *whole* that is constituted from these objects is considered the > > set of these objects, or the *collection* of these objects. > > > > However this *Whole* of x,y,z is in a sense these objects themselves. > > So far so good, but that last statement is incorrect. There's no > relationship between parts and their whole. > > > > > So a set is a form of describing multiplicity by unity, that unity is > > taken to be the totality of the Whole of these multiple objects. > > > > However if we collect ONE object, then it is pretty intuitive to say, > > that the set of all objects collected IS that object. > > The set "the set of all objects collected" doesn't look like a set to > me. It begs the question of what a set is, and if its a collection then > it looks tautologous. It just says "set"? > > > > > Also it is pretty intuitive to say that if there are no objects > > collected, then there is NO set of objects collected, in other words > > there do not exist an empty collection. > > > > Of course the traditional membership relation "e" doesn't capture all > > these pretty intuitive concepts of the customary use of the word > > membership and of the word "set" or the word "collection". > > > > To try to put all of the above informal account into rigorous > > mathematical context, we can define a theory in FOL with identity > > and the primitive binary relation symbol e' , which stands for > > what I call "intuitive membership", or in reality "membership", since > > epsilon doesn't capture the intuitive concept of membership. > > > > So we can have the following axioms that characterize this binary > > relation e'. > > I don't think you will find the resources you need from what is > currently available to you. I would look at the fringes. A necessary and sufficient realization of the idea of a "set" is the ability to distinguish between the objects one wishes to include in the set from tho objects one wishes to exclude. I.e., if one can deal successfully with the issue of membership, then one has grasped the idea of "set".
From: zuhair on 26 Dec 2009 02:31 On Dec 25, 7:40 pm, John Jones <jonescard...(a)btinternet.com> wrote: > zuhair wrote: > > Hi all, > > > One of the facts of set theory is that the primitive relation > > "e" is totally vague relation, there is no way of understanding this > > relation at all, even at informal level! Which is actually a great > > drawback, since in set theory we are building hierarchy of objects > > using this relation, a relation that nobody understand what it stands > > for? > > > Here I will present a simple trial to understand this relation "e" > > and to give it some informal intuitive background. > > > First I shall speak about the concept of set as used in ordinary > > language. > > > When we say we have a set of objects x,y,z , then we usually mean that > > there is an entity that represent the *Whole* of > > x,y,z, so although x,y,z are distinct objects, but yet > > the *whole* that is constituted from these objects is considered the > > set of these objects, or the *collection* of these objects. > > > However this *Whole* of x,y,z is in a sense these objects themselves. > > So far so good, but that last statement is incorrect. There's no > relationship between parts and their whole. Well the wrong thing is that I used the word "whole", if you examine carefully what I wrote, you will see that I am not advocating Mereology here. So what I am speaking is not about "whole" and "part". Zuhair > > > > > So a set is a form of describing multiplicity by unity, that unity is > > taken to be the totality of the Whole of these multiple objects. > > > However if we collect ONE object, then it is pretty intuitive to say, > > that the set of all objects collected IS that object. > > The set "the set of all objects collected" doesn't look like a set to > me. It begs the question of what a set is, and if its a collection then > it looks tautologous. It just says "set"? > > > > > > > Also it is pretty intuitive to say that if there are no objects > > collected, then there is NO set of objects collected, in other words > > there do not exist an empty collection. > > > Of course the traditional membership relation "e" doesn't capture all > > these pretty intuitive concepts of the customary use of the word > > membership and of the word "set" or the word "collection". > > > To try to put all of the above informal account into rigorous > > mathematical context, we can define a theory in FOL with identity > > and the primitive binary relation symbol e' , which stands for > > what I call "intuitive membership", or in reality "membership", since > > epsilon doesn't capture the intuitive concept of membership. > > > So we can have the following axioms that characterize this binary > > relation e'. > > I don't think you will find the resources you need from what is > currently available to you. I would look at the fringes.
From: Marshall on 26 Dec 2009 13:38 On Dec 24, 7:41 pm, zuhair <zaljo...(a)gmail.com> wrote: > Hi all, > > One of the facts of set theory is that the primitive relation > "e" is totally vague relation, there is no way of understanding this > relation at all, even at informal level! Which is actually a great > drawback, since in set theory we are building hierarchy of objects > using this relation, a relation that nobody understand what it stands > for? This is completely untrue. A simple, "intuitive" understanding of of the is-an-element-of relation is trivial, even for middle school children; they are taught about sets and it is quite easy stuff. At the formal level, we have the axioms. I can see no basis whatsoever for these claims. > However if we collect ONE object, then it is pretty intuitive to say, > that the set of all objects collected IS that object. It is a matter of choice. It we think of sets as just the collections of elements, maybe. If we think of sets as containers, then no. It is not difficult to distinguish between a social club with one member and the person himself. Even intuitively. > Also it is pretty intuitive to say that if there are no objects > collected, then there is NO set of objects collected, in other words > there do not exist an empty collection. Whether it is intuitive or not is debatable and unlikely to be of much importance except from a pedagogical view. I subscribe to the oft-quoted view that the only intuitive concept is to suck on the nipple, and anything else is learned. > Of course the traditional membership relation "e" doesn't capture all > these pretty intuitive concepts of the customary use of the word > membership and of the word "set" or the word "collection". > > To try to put all of the above informal account into rigorous > mathematical context, we can define a theory in FOL with identity > and the primitive binary relation symbol e' , which stands for > what I call "intuitive membership", or in reality "membership", since > epsilon doesn't capture the intuitive concept of membership. Trying to build systems by extrapolating from natural language has been attempted many times, and has never yielded anything fruitful. It is much more successful to consider things like expressive power, simplicity, elegance, etc. I think your choice of design criteria is fundamentally poor; even if you succeed in meeting your criteria, you still won't have anything useful. Marshall
From: John Jones on 27 Dec 2009 00:38
zuhair wrote: > On Dec 25, 7:40 pm, John Jones <jonescard...(a)btinternet.com> wrote: >> zuhair wrote: >>> Hi all, >>> One of the facts of set theory is that the primitive relation >>> "e" is totally vague relation, there is no way of understanding this >>> relation at all, even at informal level! Which is actually a great >>> drawback, since in set theory we are building hierarchy of objects >>> using this relation, a relation that nobody understand what it stands >>> for? >>> Here I will present a simple trial to understand this relation "e" >>> and to give it some informal intuitive background. >>> First I shall speak about the concept of set as used in ordinary >>> language. >>> When we say we have a set of objects x,y,z , then we usually mean that >>> there is an entity that represent the *Whole* of >>> x,y,z, so although x,y,z are distinct objects, but yet >>> the *whole* that is constituted from these objects is considered the >>> set of these objects, or the *collection* of these objects. >>> However this *Whole* of x,y,z is in a sense these objects themselves. >> So far so good, but that last statement is incorrect. There's no >> relationship between parts and their whole. > > Well the wrong thing is that I used the word "whole", if you examine > carefully what I wrote, you will see that I am not advocating > Mereology here. So what I am speaking is not about "whole" and "part". > > Zuhair Then you must have meant "totality" or summation. |