This exercise presentsRussell’s paradox. Let S be the set that contains a set x if the set x does not belong to itself, so that S = {x | x ∉ .x}.
a) Show the assumption that S is a member of S leads to a contradiction.
b) Show the assumption that S is not a member of S leads to a contradiction.
By parts (a) and (b) it follows that the set S cannot be defined as it was. This paradox can be avoided by restricting the types of elements that sets can have.
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