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Example 2.2.6: Logical Impossibilities - The Set of all Sets

Let S be the set of all those sets which are not members of themselves. Then this set can not exist.
This definition seems to make sense, because a set could be an entity of its own, as well as an element of another set. For example, we could define two sets Then A is a set that is also a member of itself, whereas B is not a member of itself. Therefore, we could consider the set of all those sets that are not members of itself. Call this set S. The above set A would not be an element of S, whereas B is an element of S. While this, albeit strange, does seem to make sense, we might ask: But this question will give a contradiction, because: Hence, we have arrived at a logical impossibility, and the set S does indeed not exist.
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