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Vicious circle paradoxes are types of paradoxes that arise from a certain kind of self-reference. Here are some examples of these:

Many sets are not members of themselves. For example, the set of all elephants is a set, not an elephant. The set of all odd numbers is not an odd number. In contrast, it is possible to conceive of a set that is a member of itself; for example, the set of all sets is a set, and so would be a member of itself. Now, consider the set of all sets that are not members of themselves. Is this set a member of itself? If it is, then it isn't. If it isn't, then it is.
In a town, there is a single barber shop. The (male) barber claims that he shaves all men in the town who don't shave themselves, and only those men. Now, does the barber shave himself? If he does, then by his rule he doesn't, and if he doesn't, then he does. While not strictly a logical paradox (it's not logically impossible for the barber to ignore his own rule), it can help to make Russell's paradox easier to understand.