Russell's paradox

Russell's paradox (also known as Russell's antinomy) is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory.

Set-theoretic responses

After this paradox was described, set theory had to be reformulated axiomatically as axiomatic set theory in a way that avoided this and other related problems. Russell himself, together with Alfred North Whitehead, developed a comprehensive system of types in his work Principia Mathematica. This system does indeed avoid the known paradoxes and allows for the formulation of all of mathematics, but it has not been widely accepted. The most common version of axiomatic set theory in use today is Zermelo-Fraenkel set theory, which avoids the notion of types and restricts the universe of sets to those which can be constructed from given sets using certain axioms. The object M discussed above cannot be constructed like that and is therefore not a set in this theory; it is a proper class.

Related Topics:
Axiomatically - Axiomatic set theory - Alfred North Whitehead - Types - Principia Mathematica - Universe - Proper class

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Other approaches have been proposed, such as New Foundations.

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Responses illustrated

Some of the various set-theoretic approaches to address and circumvent Russell's paradox can be illustrated in the context of Wikipedia, respecting the requirement that the content of each entry must be correct according to its entry name, and allowing the possibility of its entire contents to be correctly linked in turn:

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• either by of entry content to the same entry being discouraged; together with noting that the entity through which all Wikipedia entries are necessarily linked, namely Wikipedia as a whole, is itself not just an entry, but an entire web site. Accordingly, no entry would contain and link to itself; and the entity containing all entries (which don't link to themselves) is identified as the whole Wikipedia;
• or instead by requiring that the name of any entry which concerns listing, inclusion or linking must be explicit and decisive about the inclusion of the entry itself. Names such as "list of all lists which do not contain themselves, but including this one itself", and "list of all lists which do not contain themselves, except this one" could be correctly and legitimately articulated as Wikipedia entries (although doing so is nevertheless not advisable, if their contents may be obtained in Wikipedia more efficiently otherwise).

: In context of the Barber example, the latter requirement would ensure the consideration instead, for instance, of a barber who shaves everyone who does not shave himself, as well as the barber himself; perhaps along with a town sheriff who may arrest all those who cannot arrest themselves, with exception of the sheriff.

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