Singularity Notes

One of the most celebrated paradoxes in set theory is the Russel's paradox. The story behind the paradox and subsequent developments is rather interesting. Consider a set of the kind S = {a, b, S}. It seems somewhat unusual because S is a set in which S itself is a member. If we expand it, we get S = {a, b, {a, b, {a, b ....}}} leading to an infinite membership chain. Suppose we want to express the class of sets that don't have this "foundationless" property. Let us call this set as the set of all "proper" sets, that is, sets that don't contain themselves. We can express this set of sets as: X = {x | x is not a member of x} Now this begs the question whether X is a member of itself. If X is a member of itself, then by the definition of X (set of all sets that don't contain themselves), X should not be a member of itself. If X is not a member of itself, then by the definition of X (set of all sets that don't contain themselves), X should be a memb