We would have studied "A Set is a collection of well defined objects". This is one of the primitive definitions in high school mathematics. Thus, we can speak of "set of cats", "set of books" etc. Once we define a set we can notice that, a set of objects is not that object anymore. For instance, a set of cats is no more a cat, a set of books is no more a book. Thus, a set of things is no more that particular thing.
Is the set of all sets, a set ? Why ? Why Not ?
10 comments:
Universal set :)
it should be a set, since all objects in a set needs to be unique and non-repetitive.. so in that sense, all that qualified to be a set would be unique..
;-).. i am sure you would be driving a philosophical point home in one of the comments to this post..
kasthuri,
your post is decptively simple
hmm either you are considering the mathematical concept of a universal set or you are trying to
think of the philosophy behind mathem,atics, I wouldn't be surprised if your next post was on linking this concept with that of God. all the best
hey regarding the idea that vedanta can be presented in the language of modern science and formal logic I think it should be given a genuine and full fledged try!
I would love to be part of such an effort.
can we cosider this at greater lenghth?
looking forward to your reply on this one!
Dude,
Your factory is working at amazing speed - i can't catch up with your pace of production.
I am still thinking about your previous posts...
:-(
It is not for us to think. Why would anyone other than the famous mathematician Kurt Godel be interested in 'set of all sets'.
Anyway, I would feel uncomfortable if set of all sets exists; because this set must be in itself.
I feel that this 'set of all' is just a confusion created by mankind: 'set of all ...' makes sense if you substitute any plural noun; but what about 'cat of all cats', 'book of all books'? Any reasonable man will tell that these statements are absurd.
@ioiio :
Actually the definition of a set preceeds the concept of a universal set. Universal set depends upon the context we are talking about.
@tt_giant : Good attempt. This was a great paradox that once baffeled almost all mathematicians.
@anand :
Yes, this question has deep philosophical implications. I've pointed out one in my latest post.
As far as having a rigourous framework for Vedanta - that is my passion in life and I am working on it. Yes, I'll be too happy to discuss things in length. Godel's theorem may prove very benificial here. I have a post on this
http://kasthurisrinivasan.blogspot.com/2005/07/godel-vedantin.html
@arvind : Hope the factory doesn't go on strike or crazy in this context.
@lavanya : Yes...what you say is true. See my latest post.
Kasthuri - By your definition of a Set, the set of "all" sets is a set...
Is there a problem with that?
@Swami : If we say set of all sets is a set, then we mean a set is well defined. Is the notion of a set well-defined, that is my question ? We'll never know...Or in otherwords, my question is whether is it proper to speak of "set of all sets", what does that mean ?
Hmm... I dont know what you are getting to..
But if you ask whether it is proper to speak of a "set of all sets", then I would say it is as proper as speaking of a "set of sets".
Which inturn is as proper as speaking of a "set".
I know the notion of "set of sets" leads to paradoxes... but i feel the problem lies in the definition (if any) of a "set" rather than "set of all sets".
In other words, the notion "set of all sets" is as clear and as vague as a "set".
That's exactly right Swami, the notion of a set is vague as well as clear. I mean what do we mean by "well-defined" when we define a set. If we assume the notion of a set is clear then set of all sets makes sense, otherwise set of all sets doesn't make sense. Thus the ambiguity lies in the definition of a set. But we take it for "granted" in math that a set is well-defined. That's why I raised the question about set of all sets.
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