Before I get on with my question, I'll have to make it clear that the only set theory I've encountered can be found in the first few pages of a high-school or college-level book. Now, let me come to the question. Looking for a mathematical explanation of the source of Russell's paradox, I went to this page: http://planetmath.org/encyclopedia/RussellsParadox.html I came across this -> [tex]S = \left\{x:x\notin x \right\}[/tex]. If I read the notation correctly, it says, let S be a set of all x, such that x does not belong to/is not a member of x. What exactly does that mean? How can an element of a set belong or, in this case, not belong to itself?
Yes, it is a strange thing to get your head round to begin with: how can a set contain, or not contain, itself as a an element. It is easier to write out a 'set' that *does* contain itself as an element. Let S be the 'set' S={The set of mathematical objects that can be described in fewer than 100 words} S is an element of itself since I described it in fewer than 100 words. I suppose the point is that if we allow ourselves to be careless when defining the rules for an element to belong to a set, we start to create things that really don't behave like a set should. Thus we have to disallow such things from being called sets. It doesn't mean they are any less existant than other collections of objects, just that we can't call them sets because they don't behave like sets ought to.
Matt, Before I actually delve into the depths of the paradox , I wanted to know about the notation [tex]x\notin x[/tex]. Shouldn't the letter on right be the symbol for a set rather than an element of a set?
x is a varaible denoting a set. In fact, in axiomatic set theory, we often have that everything is a set.
It is perfectly possible for something to be a set and an element of some set. In the example I gave it is 'true' that [itex] S \in S[/itex]. There really isn't much depth to the paradox at all. If one says that 'a set is a collection of objects satsfying some rule' always defines a set then you can create things that ought to be sets, but cannot be sets. The depth is in the measures one takes to avoid the paradox.
Okay, I think I'm getting a hold of this, though, very slowly. In the case of the Barber Paradox what do S and x denote?
There is no S nor an x on that page. The Barber's paradox is a similar paradox, but I would not say they are the same. They are similar because one is led to conclude that both some statement P, and its negation, must be true. In the latter it is that the barber shaves himself, and that he does not shave himself.
Actually, I only included the link for someone following this thread who may not be familiar with the paradox. The S and x can be found in the first link. :) So if they are not similar, could you please give me an example for [tex]S = \left\{x:x\notin x \right\}[/tex]? I think it'll be easier [for me] to understand with an example.
Eh? The S of yuor post S is the set of all sets that do not contain themselves. I can't do more than give an example than tell you what it is, can I? I gave an example of a set (S in my first post, which is not the same as your S) that does contain itself, and it is trivial to give sets that do not contain themselves. The set {x : x not in x} is the collection of the latter type of 'sets'.
Yes, of course. Thanks for the help, matt. This discussion only makes me want to learn more. Are there books (print/online) that you would recommend on the basics of set theory that go beyond the intuitive intros found in most books?