Russell's Paradox
A Stepping Stone Towards Modern Set Theory
Mathematics is a science based on logic. The foundation of the subject lies in its many axioms, theorems and lemmas. Some of these theorems have a proof, and some are still waiting to be proved or disproved. But none of them is quite like Russell’s paradox, a statement that is neither true nor false, because it contradicts itself.
A Real-Life Analogy to Russell's Paradox
Imagine a library that has catalogs for each section. It has a catalog for the science section, one for the British literature section, one for American literature, one for history and so on. These catalogs are also books in their own right, so they may also be listed in catalogs. Now it also has a master catalog, which lists all books which do not list themselves.
Now the question is, does the master catalog list itself? If it does, then on the premise that it lists those books that do not list themselves, it doesn’t list itself. If it doesn’t list itself, then by the same logic, it does.
A related, and perhaps a simpler analogy is the town barber analogy. A small town has only one barber. He shaves only those people who do not shave themselves. Does the barber shave himself?
Bertrand Russell’s Formulation
In 1901, British logician and mathematician Bertrand Russell discovered a contradiction in set theory. He saw that if sets are allowed to be members of themselves, we arrive at a contradiction. A set being a member of itself is different from a set containing itself. Al sets contain themselves, for they are subsets of themselves. But if a set includes itself as an element, then it is a member of itself.
Any set may or may not be a member of itself. Constitute a large set that contains all sets which do not contain themselves. Does this large set contain itself? If it does, then by definition, it doesn’t. If it doesn’t, then it does, according to its own defining property.
Russell discusses this paradox at great lengths in his 1903 book, The Principles of Mathematics.
Zermelo Set Theory Avoids Russell's Paradox
In 1908, Ernst Zermelo formulated Zermelo Set Theory with seven axioms. His paper was published in Mathematische Annalen, a Mathematical journal published by Springer Media, in 1908. His axiom of separation disallows composing a set using arbitrary elements, like the set that Russell composed. The axiom says that if there is a proposition that is well defined for the members of a certain set M, then a subset M’ can be composed with all those elements of M for which the proposition holds true.
In Russell’s formulation, the property of a set based upon whether or not it contains itself is not well defined for all sets, and hence the collection of sets satisfying this property is not actually a set in accordance with this axiom.
Russell’s Paradox acted as a significant stepping stone towards the formulation of modern set theory. It has helped mathematicians formulate their definitions and axioms in better, more systematic ways to avoid contradictions.
Bertand Russell in 1907
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