At the beginning of the twentieth century, Russell and Whitehead decided to strengthen the foundations of mathematics with their monumental treatise Principia Mathematica, and this work is based on set theory.
Woolly thinking about sets has been responsible for some famous mistakes and misunderstandings in mathematics, perhaps the best example being Russell's identification of a fundamental logical flaw in the treatise which represented the culmination of Frege's life's work (Grundgesetze der Arithmetik) in June 1902 while the book was at the printer. As Frege said (in German) in a doleful addendum, "Hardly anything more unwelcome can befall a scientific writer than that one of the foundations of his edifice be shaken after the work is finished. I have been placed in this position by a letter of Mr Bertrand Russell just as the printing of this volume was nearing completion."
When I was a child I liked to collect things. Still do. Do you children have any collections?
Charlotte: I've got a collection of soft toys, and a collection of books about animals.
Owen: I've got a collection of bongs.
Yes. Well, philosophers through the ages have spent a lot of time thinking about collections. How to count the things in them, and so forth.
Do you think you could have a collection of collections?
Owen: Of course. You could have a collection of apples and a collection of cars and a collection of dolls. Then you could have a collection of the three collections.
Very good. Do you think a collection could contain itself?
Charlotte: You could have a collection of books, and one of the books could be a catalogue which listed all the others.
Owen: That's not the same thing. The catalogue doesn't actually contain the collection of books. It just describes the collection.
That's right. Even if the catalogue listed itself, it would not be true to say it actually contained the collection of books.
Owen: What about a collection of just one thing? A collection containing one apple and nothing else is just the same as the apple. In other words, the collection contains itself.
Actually, there has been a lot of discussion on this precise point. But if you think about it, I hope you will be able to convince yourselves that a collection of one apple is not the same as the apple. Suppose we have the following collections: (a) a collection of two oranges and an apple; (b) a collection of one orange and one apple; (c) a collection of an apple and a banana. Consider the collection of these three collections. Let's call it the big collection. What does it contain?
Charlotte: Three oranges, three apples and a banana.
Not exactly. Consider the collection of one orange and two apples. Does the big collection contain this collection?
Charlotte: Yes. They're all in it.
No. You're confusing the collections with the individual pieces of fruit which they contain. We didn't put a collection of one orange and two apples in the big collection. So it isn't there. This shows how careful you must be in thinking about collections.
If we accept that a set can contain itself, we can find ourselves in real trouble. For if a set can contain itself, then we may envisage two other sets: A the set of all sets which contain themselves, and B the set of all sets which do not contain themselves. Does B contain itself? The point is sometimes made in the form of the "barber" paradox: Sam the barber shaves all the men in town except the ones who shave themselves. Does Sam shave himself?
For further information on this subject, see Potter's Sets. An Introduction.