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.

History

Exactly when Russell discovered the paradox is not clear. It seems to have been May or June 1901, probably as a result of his work on Cantor's theorem that the number of entities in a certain domain is smaller than the number of subclasses of those entities. In Russell's Principles of Mathematics (not to be confused with the later Principia Mathematica) Chapter X, section 100, where he calls it "The Contradiction", he says that he was led to it by analyzing Cantor's proof that there can be no greatest cardinal. He also mentions it in a 1901 paper in the International Monthly, entitled "Recent work in the philosophy of mathematics" Russell mentioned Cantor's proof that there is no largest cardinal and stated that "the master" had been guilty of a subtle fallacy that he would discuss later.

Related Topics:
1901 - Cantor's theorem - Principles of Mathematics - Principia Mathematica - International Monthly

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Famously, Russell wrote to Frege about the paradox in June 1902, just as Frege was preparing the second volume of his Grundgesetze. Frege was forced to prepare an appendix in response to the paradox, but this later proved unsatisfactory. It is commonly supposed that this led Frege to completely abandon his work on the logic of classes.

Related Topics:
1902 - Grundgesetze

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While Zermelo was working on his version of set theory, he also noticed the paradox, but thought it too obvious and never published anything about it. Zermelo's system avoids the difficulty through the famous Axiom of separation (Aussonderung).

Related Topics:
Zermelo - Axiom of separation

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Russell, with Alfred North Whitehead, undertook to accomplish Frege's task, this time using a more restricted version of set theory that, they thought, would not admit Russell's Paradox, but would still produce arithmetic. Kurt Gödel later showed that, even if it was consistent, it did not succeed in reducing all mathematics to logic. Indeed this could not be done: arithmetic is "incomplete."

Related Topics:
Alfred North Whitehead - Kurt Gödel

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