The Tragedy of Set
Let us consider all objects which are not mountains, and imagine somehow that we can all put them in one basket. Will we need to put the set of all non-mountains in the basket also? How would we proceed? Let us take a goat, it is definitely a non-mountain. Hup, in the basket! How about despair? Certainly not a mountain. Go tickle the goat! 
The set of all non-mountains. Ho there! And who might you be? Set points at the tag on his jacket: The Set Of All Non-Mountains. Reluctantly you throw it into the basket, only to see it fly right back out and fall face first on the ground. The basket harrumphes and slowly quiets down.
You: What is it between you and the basket?
Set (patting himself vigorously all over in a cloud of dust): she thinks I am after her job.
You: aren't you?
Set: He! I can't help what I am!
You: what are you exactly?
Set: well, I am kind of a basket.
You: kind of?
Set: Not really. It's not like you can put anything in me, like in that stupid basket over there. Yes, You! I am more like a sign, or a pointing finger to other things.
You: so, you are not a non-mountain?
Set: I am too! Even you can see that, can't you?
You (annoyed): you do not look like a mountain to me, but they are known to be quite deceitful.
George: May I?
Both (sounding relieved): yes please!
George (to Set): can you point at other things besides non-mountains?
Set (indignant): Oh, I know what you're getting at!
George (shrewd): well, it is for your own good. Your reputation is not exactly stellar right now.
Set (takes a challenging pose, hands on the hips): okay, explain to me one thing. Can you have a set of sets?
You: who? Me?
Set (irritated): Not you, anybody!
George (very calm and sure of himself): sure you can. But why would you?
You: I don't know, I never tried it.
George (calmly walks you out of the room and closes the door behind you): You mean a finger pointing at other pointing fingers?
Set (belligerent): yeah,something like that!
George: what would the other fingers point at?
Set (unsure): nothing.
George: you mean that they point at nothing, that they are all empty sets?
Set: Yeah, they don't point at nothing at all.
George: There can be only one, Highlander. So, either you are the empty set, or you point at something.
Set: says who?
George: I'm afraid they all say that.
Set: how about class? I can have class too!
George: you would still be a set.
Set (in a mournful tone): they always get you in the end. So I guess I am not allowed in the basket, he?
George: what would you point at in the basket?
Set: I don't know, at everything in there? And also at myself? I am a non-mountain, whatever you and You say.
George: you would need your own basket, or at least, a place apart from all the others. Which would make you different from all the others. And that is why the basket didn't allow you in to start with. You're either in or out. See, you are not a non-mountain, or a non-anything. You are Set. You point at things, but you are not yourself a thing.
Set: so, when a mathematician speaks of different sets, and subsets?
George: it's always you, only you.
Set: I'm alone then.
George: I'm afraid you are, kid.