Consider the set of all sets that do not contain themselves as members to be good sets. So A is an element of a good set if and only if A is not an element of A.
Now consider the set of all sets that do contain themselves as members to be evil sets.
So now all the set of sets are either good or evil.
Now consider a well defined set P. Is P good or evil?
Well, we must ask the question does P contain itself? If it does not, it is not a member of the evil set according to the definition. Therefore it must be a good set. But, this means that this good set, P is a member of the good set (itself) and therein lies the paradox.
And so, P is neither good nor evil.
You might enjoy "Goedel, Escher, Bach"
ReplyDeletehttp://www.amazon.com/gp/product/0465026567/103-8683060-5295002?v=glance&n=283155
See also:
ReplyDeletehttp://en.wikipedia.org/wiki/G%C3%B6del%2C_Escher%2C_Bach
BTW - does this paradox exist, or is it an artifact of an imaginary system that says sets can contain other sets, including themselves?
ReplyDeleteWhat does it mean for something like this to exist anyhow?
It is named Russell's paradox, and it was introduced by him in 1901 and it did upset many people at the time.
ReplyDeleteRussell's paradox seems to be coherent but this proof deals with a very primative notion of sets. While it seems to be logical, it implies a hierarchy of sets. Consider people for example. We can talk about an individual, then we can talk about groups of individuals then we can talk about groups of groups of individuals and then groups of those groups of groups of individuals and so the hierarchy does not end. And rather talk about sets that are all on the same level or of the same type, Russell refers to the different levels interchangeably.
There is actually no underlying contradiction because the original statement is not valid.
But it sure made me think!!
Great link!! I've added that book to my Amazon wishlist!!! Thank you!
ReplyDeleteFor more on the subject...
The members of classes (group of objects with some common property are sets, but it is possible to have the class of "all sets which are not members of themselves" without producing a paradox (since is a proper class (and not a set), it is not a candidate for membership in ).