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Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A.
In Cantor's system, M is a well-defined set. Does M contain itself? If it does, it is not a member of M according to the definition. On the other hand, if we assume that M does not contain itself, then it has to be a member of M, again according to the very definition of M. Therefore, the statements "M is a member of M" and "M is not a member of M" both lead to contradictions.
In Frege's system, M corresponds to the concept does not fall under its defining concept. Frege's system also leads to a contradiction: that there is a class defined by this concept, which falls under its defining concept just in case it does not.
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 earlier 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.
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 completely to abandon his work on the logic of classes.
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).
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."
There are some versions of this paradox which are closer to real-life situations and may be easier to understand for non-logicians: for example, the Barber paradoxThe Barber paradox is a paradox that relates to mathematical logic and set theory. The paradox considers a town with a male barber who shaves daily every man who does not shave himself, and no one else. Such a town cannot exist: If the barber does not sha supposes a barber who shaves everyone who does not shave himself, and no one else. When you start to think about whether he should shave himself or not, the paradox becomes obvious.
Similarly, Russell's paradox proves that an encyclopedia entry titled "List of all lists that do not contain themselves" must be either incomplete (if it does not list itself) or incorrect (if it does).
As illustrated below, consider five lists of encyclopediaEncyclopaedia Britannica An encyclopedia (alternatively encyclopaedia is a written compendium of human knowledge. The term comes from the Greek words , enkyklios paideia ("in a circle of instruction"). From , circuit shaped from circuit and , meaning inst entries within that same encyclopedia:
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List of articles about people:
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List of articles about computer science: |
List of articles about places: |
List of articles about Japan: |
List of all lists that do not contain themselves:
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If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself.
While appealing, these "layman's" versions of the paradox share a drawback: an easy refutation of, for example, Barber's paradox seems to be: "Such a barber does not exist". The whole point of Russell's paradox is that the answer "such a set does not exist" means that the definition of "set" within a given theory is unsatisfactory. Notice the subtle difference between the statements: "such a set does not exist" and "such a set is empty".
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