[zeroca]

Originally Posted by subversion

Zeroca, I completely agree that you can not take the set of all sets which do not contain themselves. That is because it is the empty set no? It cannot be the full set because the full set contains itself. Therefore I think that the set of all sets which contain themselves is the complete set and the set of all sets which do not contain themselves is the empty set. I think this is a very useful way to define it. Don't you?

I don’t think so: we can put it this way: if you can’t collect all elements of any set (i.e. if you can’t collect the whole quantity of them, but only part of them), it surely mustn’t be an empty set. You can call empty set only a set without any single element.
I’ll try to be more precise with my explanations:
If you collect green things into the bag, so accordingly you have a set of green things, then regarding to the matter of “self-containness” you have two possible alternatives: either this bag enters among the elements or not:
If the bag is green so you have self-contained set, if the bag isn’t green, the set is self-not-contained. I’d like to emphasize that you aren’t obliged to analyze all qualities of bag: is it big or small; is it red, black, or yellow; is it new or old. It’s enough for you to check if it is green, if not – then all other infinite qualities of bags enter within quality “not-green”.
Let’s take refrigerator, which’s turned on. It contains cool things, but it itself isn’t necessary cool, especially if refrigerator is somewhere in Africa, i.e. if we collect the set of some cool things, this refrigerator doesn’t enter in requirement, so the set refrigerator in this particular case isn’t example of self-contained sets, but if we place it on the pole, it at once takes up position within self-contained sets, as it is also cool, together with its content.
Let’s take all refrigerators in Africa. Their number is definite, as there’s not infinite number of refrigerators in Africa, so the set of all turned on refrigerators in Africa is the example of set, which is all self-not-contained sets in Africa in regard to refrigerators (i.e. within some definite region with some definite qualities of set).
The trick of Russel’s paradox is to thrust infinity within consideration. I.e. we must collect not definite number of self-not-contained sets, but all of them! I.e. Russell complicated the task for us, but I don’t think so, as it’s enough to seek one example of self-not-contained set, which for some reason (for any reason) doesn’t enter these requirements, that the task is almost solved, so cited by me above example of set “Mama” in previous post is good example of it:
The set “Mama” is the part of “home”, which is self-contained, but the same “Mama” ” is the part of “Unity”, which is self-not-contained.
And it doesn’t’ matter how you call any set. With infinity you are free to choose any variant, because infinity can contain all credible ones: You can call unity as unito, as unitu and despite their being the same set with different names, anyway they are considered as different sets.
I showed above (in previous post), that
Any set (be it self-contained, or self-not-contained) is represented in two different forms of sets within universal set: within some sets- as part of self-contained, within another sets- as part of self-not contained.
That’s enough to assert that we can not collect separately all self-not-contained sets into one single set, as the same, infinite quantity of the same sets at once remain as elements of single "container", defined as set of all self-contained sets and this quantity of sets is the same as that of universal set, which as you see, is of binary nature, I.e. it contains each consisting set in two different forms: as self-contained and as self-not-contained ones.