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Russell’s Paradox Jami Durkee Valerie Toothman Jason Prindell

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What is it? Russell's paradox (also known as Russell's antinomy) was discovered by Bertrand Russel in 1901. It showed that the naïve set theory created by Georg Cantor (which states any definable collection is a set) leads to a contradiction.

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Russell discovered the paradox in May or June 1901. By his own admission in his 1919 Introduction to Mathematical Philosophy, he "attempted to discover some flaw in Cantor's proof that there is no greatest cardinal". He did on the other hand discover that the axioms Frege was using to formalize his logic were inconsistent. Russell questioned his theory to the point where Frege eventually felt forced to abandon many of his views about logic and mathematics. There have been many attempts in the last century to avoid it.

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The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea. By 1908, Zermelo created the Zermelo set theory that avoids the paradox.

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Bertrand Arthur William Russell May 18, 1872-Feb. 2, 1970 He was a British philosopher, logician, essayist and social critic best known for his work in mathematical logic and analytic philosophy. He was noted for his spirited anti-war and anti-nuclear protests. He remained a prominent public figure until his death

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His family His paternal grandfather, John Russell, 1 st Earl Russell, was the third son of John Russell, 6thDuke of Bedford, and had twice been asked by Queen Victoria to form a government, serving her as Prime Minister in the 1840s and 1860s. Russell's mother, Katharine Louisa was the daughter of Edward Stanley, 2 nd Baron Stanley of Alderley, and the sister of Rosalind Howard, Countess of Carlisle. Kate and Rosalind's mother was one of the founders of Girton College, Cambridge.

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Deaths In 1874 his mother died of diphtheria and sister died 1876 his father died of bronchitis after a long period of depression His grandparents then got custody of him and his brother 1878 his grandfather died By order of his grandmother, instead of being sent to school he was taught by governesses and tutors, and thus acquired a perfect knowledge of French and German.

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Wives 1894 marries Alys Pearsall Smith 1921 divorce from Alys and marries Dora Black In 1927 they started a school for young children, which they carried on until 1932. 1935 divorces Dora 1936 marries Patricia (Peter) Helen Spence 1952 divorce from Patricia (Peter) and marries Edith Finch

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School and Honors He entered Trinity College in 1890 He was fined 110 pounds and dismissed from Trinity College as a professor as a result of anti- war protests in 1916. He was elected Fellow of the Royal Society in 1908 In 1931 he became the third Earl Russell upon the death of his brother. He was awarded the Order of Merit in 1949 and the Nobel Prize for Literature in 1950

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Politics 1907, 1922, 1923 runs for parliament and is defeated 1918 he was imprisoned for five months as a result of anti-war protests. 1961 imprisoned for one week in connection with anti-nuclear protests.

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Ernst Friedrich Ferdinand Zermelo July 27, 1871-May 21 1953 He was a German mathematician whose work had major implications for the foundations of mathematics and philosophy. He is known for his role in developing Zermelo- Fraenkel axiomatic set theory and his proof of the well-ordering theorem.

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What is Russell’s Paradox? Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox.

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Definitions Set: Georg Cantor said “a set is a gathering together into a whole of definite, distinct objects of our perception and of our thought – which are called elements of the set.” – Example of a set is C = {4, 2, 1, 3} Contradiction: a statement or proposition that contradicts or denies another or itself and is logically incongruous. biconditional statement: A biconditional statement is defined to be true whenever both parts have the same truth value. – The biconditional operator is denoted by a double-headed arrow

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Barber Example Russell's paradox is based on examples like this: Consider a group of barbers who shave only those men who do not shave themselves. Suppose there is a barber in this collection who does not shave himself; then by the definition of the collection, he must shave himself. But no barber in the collection can shave himself. (If so, he would be a man who does shave men who shave themselves.)

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Has it been solved? Creating a solution to Russell's paradox does not involve 'fixing' the paradox. The paradox will always lead to a contradiction. In order to 'solve' the paradox, set theory must deal with the issue by voiding the premise of the paradox.

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Dealing with the paradox Several methods have been used to try to deal with the paradox, two of which are: – Russell's type theory – Zermelo's set theory

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Russell’s Type Theory attempts to tackle the paradox by creating 'types‘ The lowest tier (Tier 1) of types would be primitive objects, including numbers, shapes, ducks, etc. The next tier would be sets of Tier 1 objects (e.g.- sets of numbers, sets of ducks) The third tier would be sets of sets of objects (sets of sets of ducks). Defining types this way removes the possibility of self-referencing sets R, where R = { x : x is not in x }.

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Zermelo-Fraenkel set theory (ZFC) The accepted set theory modified version of Zermelo set theory to deal with Russell’s Paradox created from several axioms The axiom that deals with Russell's paradox is the axiom of separation

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