According to Georg Cantor, a set was a “well-defined collection of objects”. This definition is a little bit “loose”, since it uses intuitive concepts such as “collection” and “well-defined” and thus involves the risk of running into logically paradox situations.
Imagine, Cantor’s definition describes the mathematical object “set” sufficiently, or, as mathematicians say, makes it “well-defined”. Then – again using this definition – we can construct a “set of all sets which do not contain themselves.” Now, does this set contain itself? If so, then it contradicts its defining property. Therefore, it has must not contain itself. But if it does not contain itself, then it has to be its own element, by definition. This leads again to a contradiction.
Bertrand Russell stumbled upon this paradox in 1901 when he examined one of Cantor’s proofs. However, he failed to realize its importance for a year, until he sent a letter in which he mentioned it to Gottlob Frege, a logician who intended to derive arithmetics only by the application of the laws of logic. After 10 days, Russell received an answer from Frege after 10 days, who was devasted and who realized that the paradox caused his lifetime achievement to falter. Russell had a long correspondence with Frege and hoped he could resolve the paradox in time. But he failed to do so. This encouraged him to collaborate with Alfred Whitehead and to create a very complex, three-volume work with a logical system correcting this paradox, The Principia Mathematica.
| | | | created: 2018-12-09 22:38:41 | modified: 2020-08-06 10:11:58 | by: bookofproofs | references: , 
 Hoffmann, Dirk W.: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011
 Kohar, Richard: “Basic Discrete Mathematics, Logic, Set Theory & Probability”, World Scientific, 2016