#### What is Russell’s Paradox of set theory? Description and information on Russell’s Paradox.

**Russell’s Paradox Of Set Theory**

Among the paradoxes in set theory discovered about 1900, Bertrand Russell’s is the simplest to describe. In fact, several questions of the parlor-game type are based on it. In an axiomatic development of set theory, such paradoxes are avoided by somehow restricting the size that a set can attain.

Since the elements of a set can be objects of any kind whatever, these elements may themselves be sets. For example, the elements of the set = {X | X ⊂ S} are sets. Thus the possibility arises that a set A might be an element of itself, that is, A ∈ A. As an example, the set consisting of all those objects describable in exactly 12 words would be a member of itself. Sets coming readily to mind do not have this property. For example, the set N of all positive whole numbers is not, itself, a whole number, so N ∉ N. A set S might be called an ordinary set if S t S, and an extraordinary set if S ∈ S.

Russell’s paradox is concerned with the set R of all ordinary sets; that is, R={S|S ∉ S}. The question arises: Is R ordinary or extraordinary? These two possibilities may be considered in turn. First, suppose R is an ordinary set, that is, R ∈ R. But this is exactly the condition required for membership in the set R={S|S ∉ S}; that is R e R. Thus, this possibility has led to a contradiction, and it must be rejected. Second, suppose R is an extraordinary set; that is, R t R. But then R must satisfy the condition required of all objects that are elements of R; that is, R ∉ R. This possibility has also led to a contradiction. But there are no other possibilities, so either the first or second possibility (and not both) must describe the actual situation. The resolution of this difficulty lies in asking: Is R a set? In an axiomatic development, a set is required to satisfy certain conditions, or to be the result of applying certain prescribed processes. The object R considered above would fail to satisfy these requirements. It is not a set, so there is no reason to expect it to behave like a set. In particular, the relation e of set membership has no meaning in connection with R.

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