## 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} }

**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 ?

- 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.

**S**does indeed not exist.