Russell's paradox explanation

Question

The Russell's paradox deals with the set of all sets that do not contain themselves. So I want a example of a set that do not contain themselves. I got a examples of set of turtles.It will contain turtles, but I want to understand how should I understand the set of all turtles is not member of the set.

Answer 1

Think of a set as a box. But it's a special kind of box, because it's defined entirely by what's inside it. I can put a label on the box to tell me what it is, but if I have two boxes that have exactly the same things inside them then they're actually two copies of the same box.

So you show me your box, and it's labelled "All Turtles". I look inside it, and indeed everything in the box is a turtle, and in fact every possible turtle is inside the box. And I can ask you - does the box contain itself? Well, the only thing in the box is turtles, so if the box isn't a turtle then it can't contain itself.

Then I give you another box, and it's labelled "All Boxes". Does the box contain itself? It should - every box that can possibly exist is inside this box, which includes itself. So I can open this box, find a box labelled "All Boxes" in it, and it would be indistinguishable from the original one. There would also be a box in there labelled "All Turtles", and it would contain all the turtles in it. But we would not say that the "All Boxes" box contained turtles, because we never go more than one step down to determine what's inside a box.

Russell's paradox comes about when we use more complicated labels for the boxes. Maybe I have a box labelled "All Boxes that don't contain turtles". And another one labelled "All Boxes that contain exactly one box", and one that's "All Boxes that don't contain any boxes with turtles in them". And then finally "All Boxes that don't contain themselves", and it all breaks down and we have to set rules about what kinds of contents boxes are actually allowed to have.