We will define a paradox as a statement that contradicts itself, or a statement that can be proved to be both true and false at the same time Let A be the set {1, 2, 3}. Clearly, A ∉ A. Now think of the set of all such sets that are not elements of themselves and call that set S. That is, S = {X : X is a set and X ∉ X}. Now we ask the question, “Is S ∉ S? ” First assume that S ∉ S. What must you conclude? Now assume that S ∉ S. What must you conclude
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