This is the paradox that Russell shocked the Vienna Circle with, and so far as I know it has not been cracked before now, but I've discovered a crucial fault in it:-

**The set of all sets that don't contain themselves.**

The set of mice does not contain itself. Why? Because the set of mice is not a mouse. Many other sets do not belong in themselves for the same kind of reason, so we can make a set of them. Once we have collected them all into this new set, we need to look at this new set and see if it contains itself. Does it qualify? It doesn't contain itself yet, so yes. We put it in itself and now it no longer qualifies, so we have to take it back out, but then it qualifies again and we have to put it back in, etc. - we have a paradox.

However, there are problems with sets and the rules are actually being broken. Let me show you how.

Let's start with the set of all things, and in it we'll place the set of all animals, and inside that we'll put the set of all rodents, and inside that we'll put the set of all mice. Clearly mice belong in the set of animals, and they are indeed inside it, but the set of mice is not an animal, so what's it doing in the set of animals?

Now let's look at the set of all sets. Inside it we have all possible sets, including the set of mice. The set of mice is indeed a set, so it belongs in the set of all sets, but a mouse is not a set, and we've just put it inside the set of sets. Now, we could deal with this by declaring that everything in a set is itself a set, so that problem can be resolved. But there is an ugly infinite recursion involved when the set of all sets has to be put into itself. A proper analysis of sets actually reveals that no set is allowed to contain itself, as I will now prove.

We need to stop and think about what a set actually is. How can we resolve the problem of the set of mice not being an animal but being contained by the set of animals? Well, the set of mice = the set of all mice = all mice. All mice are animals. Some animals are mice. All mice belong in the set of animals, and there is no separate thing called a set sitting in the set of animals that isn't itself an animal or collection of animals. That is how that problem is resolved.

Now that we have this proper understanding of what a set actually is, we can apply it elsewhere. When we put the set of mice into the set of all sets, we are actually just putting all mice into the set of all sets, and the set we're putting them into is really just the set of all things (all things being sets). So, we have put a collection of all mice into a collection of all things. That works fine. Once all things (whether real or imaginary) are in this collection, all the many sets they belong to are automatically in this set. Now we ask the crunch question, should we put the set of all sets into itself? What is the set of all sets? The set of all sets = all things. Should we put all things into our collection of all things? They're already in the collection, so there is no need: it turns out that the question is a nonsense. You cannot put a set into itself because you're working at a level of abstraction that hides the reality of what you're actually trying to do. All things are already in the collection of all things.

"The set of all sets that don't contain themselves" should be translated into its alternative form: the collection of all things that don't contain themselves. Is the collection of all things something that doesn't contain itself? Yes. It is itself. It includes itself and nothing but itself, but it cannot contain itself without part of itself (a container) doing the containing and not being contained.

Also, "does the set of mice contain itself?" should be translated into its alternative form: does the collection of all mice contain the collection of all mice? No, but it is the same collection, and it includes itself.

So, no set can contain itself, but all sets must include themselves. The set of sets that don't include themselves is necessarily an empty set (and an empty set is not a real set, but a hypothetical set). The set of sets that don't contain themselves is the set of all sets, whether or not they exist (non-existent ones will automatically speak of their non-existence by failing to appear in the actual collection), and the set of sets that don't contain themselves is already included in itself without any paradox being generated.