Question

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Can somebody make a simple yet thorough explanation of Russell’s Paradox. It should contains the definition, explaining it, and as if you are teaching someone who has never heard of Russells paradox and include an example problem of it. Thanks.

Answer 1

Russell's paradox is one of the most famous set theoretical paradox. To understand this paradox let us understand the following first. "A book contains pages, but it is not a page by itself". "A library consists of books, but it is not a book".

Russell's paradox: Let be the set of all sets which do not contain themselves that is .

In other words, is the set of set which are not the members of themselves.

For example, Set of all integers is not itself an integer that is , and hence the set of integers lies in this set.

Now it is impossible to define whether or .

If , then the defining property of implies that .This contradicts our assumption that .

If , then then the defining property of implies that .This again contradicts our assumption.

So, the only possible conclusion is that the property cannot define a set. This contradiction is the main essence of Russell's paradox .

In order to avoid this paradox, we may always have to ensure that any set that we talk about, is not a member of itself. It is convenient to choose a 'Largest' set in any given context, called the UNVERSAL SET.