# Interactive Real Analysis - part of MathCS.org

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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
• A = { {1}, {1,3}, A }
• B = { {1}, {1,3} }
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:
• Is S an element of itself or not ?
But this question will give a contradiction, because:
• If S is an element of itself, then - since by definition S contains those sets only that are not part of itself - S is not an element of itself. That's not possible.
• If S is not an element of itself, then - since S does contain those sets that are not part of itself - S is a member of S. That's not possible either.
Hence, we have arrived at a logical impossibility, and the set S does indeed not exist.
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