In the original abstract set theory formulated by Georg Cantor (1845–1918), a set was defi...

In the original abstract set theory formulated by Georg Cantor (1845–1918), a set was defined as “any collection into a whole of definite and separate objects of our intuition or our thought.” Unfortunately, in 1901, this definition led Bertrand Russell (1872–1970) to the discovery of a contradiction — a result now known as Russell’s paradox—and this struck at the very heart of the theory of sets. (But since then several ways have been found to define the basic ideas of set theory so that this contradiction no longer comes about.)

Russell’s paradox arises when we concern ourselves with whether a set can be an element of itself. For example, the set of all positive integers is not a positive integer —or Z+ ∉ Z+. But the set of all abstractions is an abstraction.

Now in order to develop the paradox let S be the set of all sets A that are not members of themselves—that is, S = {A|A is a set  A ∉ A}.

a) Show that if S ∈ S, then S ∉ S.

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b) Show that if S ∉ S, then S ∈ S.

The results in parts (a)and (b) show us that we must avoid trying to define sets like S. To do so we must restrict the types of elements that can be members of a set. (More about this is mentioned in the Summary and Historical Review in Section 3.8.)