Let me state initially that I have no formal advanced qualification in mathematics. My education is high-school/college level, very much an average student. My interest in mathematics and the like is very much at the lay-person's level of knowledge and enthusiasm. I do not see how the resolution presented below fails, but understanding that realistically this is due limitations of my understanding, an explanation of why it fails will be welcomed and appreciated.
So with that said...
The supposed paradox arises from the unrestricted comprehension principle of Set Theory, which essentially states that the members of a particular set are those objects that satisfy the criteria of the set. As in, the members of the set of European capital cities are the capital cities of Europe; the members of the set of all the names for shades of the colour red are all the names for shades of the colour red.
A Set contains members X where X meets criteria Y.
The paradox occurs when you consider the set of sets that do not contain themselves and ask 'Is this set a member of itself?' - if it is a member of itself, then it contains itself and should not be a member of itself - if it is not a member of itself, then it does not contain itself and should be a member of itself, and so on. The paradox being that the set is simultaneously a member of itself and not a member of itself.
I disagree. I suggest that there is no paradox.
The set of 'all sets that do not contain themselves' is not actually a set. The 'set' has no criteria that can be met or not met to determine if an object should be a member of this set. Sets that are not members of themselves are not simply not members of themselves. They are not members of themselves because the set concept/structure does not meet the specific criteria of that set.
The set of all breeds of dog does not contain itself because the set *is not a breed of dog*. The set of all diamonds and all clowns does not contain itself because the set *is not a diamond or a clown*.
The full name of 'the set of all sets that do not contain themselves' is '*the set of sets that do not contain themselves because the set itself does not meet the criteria of that set'*. When you consider this set itself and try to determine if it should be a member of itself, it is impossible. There is no criteria that the set, the container, can meet or not meet to make that determination. It is, indeed, impossible to construct such a set as you would need to assign two distinct forms of criteria to a single set. That is not a single formula of criteria that refers to more than one quality (objects that are X and/or Y) but two distinct forms of criteria that apply at the same time (objects that are X - objects that are Y).
For example, to construct the set (A) in a way that meets the requirements of being a set, the criteria of A would have to be the set of all English language adverbs (the set would not be a member of itself) and *simultaneously* the set of sets that are not members of themselves.
Without set A having a criteria that determines what is and is not a member of set A, it is impossible to determine if set A should or should not be a member of itself.
The set of sets that are not members of themselves is not a set, it is a *list*, an informal collection, of those sets that are not members of themselves. It is not a set because it has no criteria and, not being a set, no paradox arises.