Let’s discuss about Russell’s Paradox:
The paradox arises within naive set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.
(
http://plato.stanford.edu/entries/russell-paradox ).
The accepted solution for most mathematicians is ZF(C), which avoids the paradox by disallow sets as their own members.
In ZFC, given a set A, it is possible to define a set B that consists of exactly the sets in A that are not members of themselves.
(
http://en.wikipedia.org/wiki/Russell's_paradox )
ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, and is therefore not a ZFC set. In some extensions of ZFC, objects like R are called proper classes.
(
http://en.wikipedia.org/wiki/Russell's_paradox )
Proper classes are: "objects having members but that cannot be members." (
http://en.wikipedia.org/wiki/Von_Neumann-Bernays-Godel_set_theory )
ZF(C) framework arbitrarily avoids the paradox by disallow sets as their own members.
This paradox is possible because naive set theory does not distinguish between given set X, and set X as a member of set X.
By understanding the difference between given set X, and set X as a member of set X, both Russell’s Paradox and proper classes, are avoided.
For example, the empty set is defined by emptiness but it is not identical to emptiness, otherwise we have emptiness (nothing) to deal with. Since there are no many cases of emptiness, then the empty set is an existing and unique mathematical element.
Given any non-empty set, it is defined by its members but it is not identical to them because its existence is stronger than the members’ existence exactly as the empty set extends is stronger than the existence of emptiness.
By understanding the consistent extension of a given set w.r.t its members (whether it is empty, or not) we are able to distinguish between set’s definition by its members and set’s identity w.r.t its members.
Two sets that have the same members are identical, but the difference between a set and its members holds , whether the set is empty or not.
Some analogy:
Two identical trees are actually the same tree, but it does not mean that the absence or the existence of the branches of that tree is identical to the trunk of that tree.
By that analogy, the trunk’s existence as a living organism is stronger than the existence of the branches, such that cutting the branches does not kill the tree, but cutting the trunk kills the tree.
Exactly as (x>0)/0 or (x>0)/1 is an existing closed 1D path (whether there is or there is no element along it) a set exists independently of its members, and therefore it is defined by them (exactly as a trunk is defined by its branches or their absence) but it is not identical to them (no branch or its absence is identical to the trunk).
EDIT:
Here is an informal presentation taken from (
http://en.wikipedia.org/wiki/Russell's_paradox )
Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is "normal". On the other hand, if we take the complementary set that contains all non-squares, that set is itself not a square and so should be one of its own members. It is "abnormal".
Now we consider the set of all normal sets, R. Attempting to determine whether R is normal or abnormal is impossible: If R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if it were abnormal, it would not be contained in the set of normal sets (itself), and therefore be normal. This leads to the conclusion that R is both normal and abnormal: Russell's paradox.
By understanding the difference between given set X, and set X as a member of set X, the difference between “abnormal” and “normal sets is insignificant.
