Russell’s Paradox – A wake up call for set theory

Before the philosopher Bertrand Russell began really thinking about the nature of sets, sets were defined informally. Set theory before various axiomatic definitions is now referred to as naïve set theory. Russell’s paradox, discovered by Bertrand Russell in 1901 was also discovered a year earlier by Ernst Zermelo but never published. This paradox shows that ideas in naïve set theory led to contradictions, to avoid these set axioms were put in place. These include axioms such as the Zermelo- Franekel axioms (ZF).

Russell considered the set which contains all sets which do not have themselves as an element.

(1)    \begin{equation*} R=\begin{Bmatrix}x \left | x \ \text{is a set,} \ x \notin x\end{Bmatrix} \end{equation*}

An interesting question to ask is, is R contained within itself?

Let the answer be yes, then R \in R. This implies R \notin R by the way in which R was defined. Suppose the answer is no, R \notin R, then the definition of R implies R \in R. Either answer leads to a contradiction!

This contradiction can be avoided if sets such as the one above are axiomatically defined to not be sets. Similarly the set of all sets and other such “weird sets” are not allowed in axiomatic set theory. Some examples of the Zermelo- Franekel axioms which don’t allow sets such as R include the axioms of extensionality, regularity, pairing, union and a few more. These will be discussed further in other articles and real analysis notes.

This article was inspired by 1^{st} year mathematical analysis lectures from Professor Toby Wiseman at Imperial College London. 


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