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used the barber paradox to explain a more serious paradox called Russell’s paradox . This is more abstract than the other self-referential paradoxes we saw and is worth pondering. Consider different sets or collections of objects. Some sets just contain elements and some sets contain other sets. For example, one can look at a school as a set containing different grades, where each grade is the set of students in the grade. Some sets even contain copies of themselves. The set of all sets described in this book contains itself. The set of all sets with more than five elements contains itself. There are, of course, many sets that do not contain themselves. For instance, consider the set of all red apples. This does not contain itself since a red apple is not a set. Russell would like us to consider the set R of all sets that do not contain themselves. Now pose the following question:
    Does R contain itself?
    If R does contain itself, then, by definition of what belongs to R, it is not contained in R. If, on the other hand, R does not contain itself, then it satisfies the requirement of belonging to R and is contained in R. We have a contradiction. This can be visualized in figure 2.3 .
    Figure 2.3
    Which part contains R?
    This paradox is usually “solved” by positing that the collection R does not exist—that is, that the collection of all sets that do not contain themselves is not a legitimate set. And if you do deal with this illegitimate collection, you are going beyond the bounds of reason. Why should one not deal with this collection R? It has a perfectly good description of what its members are. It certainly looks like a legitimate collection. Nevertheless, we must restrict ourselves in order to steer clear of contradictions. The obvious (and seemingly reasonable) notion that for every clearly stated description there is a collection of those things that satisfy that description is no longer obvious (or reasonable). For the clearly stated description of “red things,” there is a nice collection of all red things. However, for the seemingly clear description of “all sets that do not contain themselves,” there is no collection with this property. We must adjust our conception of what is obvious. 5
    Russell’s paradox should be contrasted with the other paradoxes. There are simple solutions to the barber paradox and the reference-book paradox: those physical objects simply do not exist. And there is a simple solution to the heterological paradox: human language is full of contradictions and meaningless words. We are, however, up against a wall with Russell’s paradox. It is hard to say that the set R simply does not exist. Why not? It is a well-defined idea. A collection is not a physical object, nor is it a human-made object. It is simply an idea. And yet this seemingly innocuous idea takes us out of the bounds of reason.
    The liar paradox was summarized by one sentence:
    This sentence is false.
    It can also be summarized by the following description:
    The sentence that denies itself.
    Similarly, the other four self-referential paradoxes can be summarized by the following four descriptions:
    â€¢ “The villager who shaves everyone who does not shave themselves.”
    â€¢ “The word that describes all words that do not describe themselves.”
    â€¢ “The reference book that lists all books that do not list themselves.”
    â€¢ “The set that contains all sets that do not contain themselves.”
    As you can see, all these descriptions have the exact same structure (as do figures 2.1 through 2.3 ). Every time there is self-reference, there are possibilities for contradictions. Such contradictions will have to be avoided and will require a limitation. We explore such limitations throughout the book.
    Before moving on to the next section, there is an interesting result that demands further thought. One might think that every language paradox has some