There is a lot of confusion about what counts as a paradox. For some, a paradox is just some any conclusion that seems absurd and that has an argument to support it. Others talk of a person being in a paradox or facing a paradox when from assumptions that they believe, by methods of reasoning they find acceptable, they establish conclusions that they find unacceptable, perhaps because the conclusions are absurd. From this second view, if you don't accept the assumptions, or the methods of reasoning or the absurdity of the conclusion, then there is no paradox for you to face. Paradox, on this view, is in the eye of the beholder. There are those that distinguish between a paradox and an antinomy. For these folks an antinomy is a paradox that is absurd because the conclusion is a self-contradiction.
Before I go further, a little history is in order. What has been called Russell's paradox/antinomy was discovered by Russell in 1901 and communicated by letter to Frege in 1902. There is a discussion of this discovery in Russell's Principles of Mathematics published in 1903. In 1908, Ernst Zermalo claimed to have discovered Russell's paradox/antinomy independently of Russell and to have written to Hillbert (among others) prior to 1903. Zermalo uses the term 'antinomy' rather than 'paradox'. Interestingly, no where in the 1902 letter nor in the 1903 book does Russell use the terms 'paradox' or 'antinomy' in discussing his discovery. In fact the heading in the section of Principles of Mathematics in which the discovery is discussed is 'The Contradiction'.
In Grundgeesetze der Arithmetik, Frege presented the principle that, in modern terminology, is given by: every condition determines a set. Russell reasoned that this principle was false. His reasoning amounts to the following; consider the condition "is a set and is not a member of itself "; by Frege's principle, there is a set whose members are those sets that satisfy this condition(i.e. are not members of themselves); either this set belongs to itself or it does not; in either case there is a contradiction. Hence, there is no such set and Frege's principle is false. Russell is very clear in the letter and in the 1903 book that this reasoning was intended to establish that this principle articulated by Frege was false.
Russell's paradox/antinomy has come to be associated not with Russell's reasoning in the 1902 letter or the 1903 book, but with the claim that there is a set whose members are exactly those sets that are not members of themselves. This practice ignores the evidence from the relevant documents. Russell's paradox/antinomy was neither to Russell.
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