Information On The Russell’s Paradox Of Set Theory

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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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