Historical Developments in Set Theory

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HistoricalDevelopments in Set Theory

Themost famous of all set theoretical paradoxes is the Russell’sParadoxes, discovered by Bertrand Russell in 1901. It was discoveredwithin the naïve set theory by making the consideration that not allsets are sets of themselves. The naïve theory of sets defines a setas any well defined collection of items/ objects, otherwise referredto as members or elements of the set. An item/ object can assume anyform ranging from numbers or people. For example, the number 6 is anelement in the set of all even integers. A paradox, on the otherhand, refutes the notion that one can freely collect items thatsatisfy a particular property into a set without restriction. Anexample of this is assuming that all mathematical objects with aparticular characteristic can be put together as a set is false.(Stanford.edu).

Russell’sparadox is one of the contradictions to the set theory. Thefoundation of Russell’s paradox is based on an illustrationinvolving a group of barbers. The common factor amongst the barbersis that they only shave the men that do not shave themselves.Supposing that in this group of barbers, there is one who does notshave himself, this fact is in contraction with the commoncharacteristic of the group because in order to be part of thecollection that one must be in a position to shave himself. However,none of the barbers in the collection can shave themselves, thereforecoming to the conclusion that he is a man who shaves men who shavethemselves (Iep.utm.edu).

AfterRussell discovered the paradox, it became immediately clear to Fregethat it had a destructive outcome on his system. According to Frege’sphilosophy, a class was the extension of a concept, with conceptshaving the closest correlation to properties in his metaphysics. Aconcept is believed to exist for every condition that can bespecified. Therefore “thereis a concept of beinga class that does not fall under its defining concept”(Woods, 132).According to Russell’s paradox, the Frege’s aspect of conceptswas made questionable.


Woods,J. Paradox and Para consistency: Conflict Resolution in the AbstractSciences. Cambridge University Press. 2003. Print

StanfordEncyclopedia of Philosophy. Russell’sParadox.Jun26, 201. web.http://plato.stanford.edu/entries/russell-paradox/. Accessed16 August 2016

Klement,Kevin.Internet Encyclopedia of Philosophy: Russell’s Paradox. Web.http://www.iep.utm.edu/par-russ/.Accessed16 August 2016

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