history-page44 - Along with other mathematicians of the...

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Along with other mathematicians of the late 1800s he was interested in the question of representing functions with trigonometric series of the form f(x) = a 0 + s n = 1 [a n cos nx + b n sin nx]. In particular, he was interested in the question: Of what sets S is it true that if f(x) = 0 on S , then 0 = a 0 = b 1 = a 1 = b 2 = a 2 = ··· ? He knew it was true of S = R and of S = R minus a Fnite number of points. He knew of an operation H that turned one set into another and that was such that if it was true of S , then it was true of H(S) . Applying this operation repeatedly gives more and more sets, which Cantor needed to index somehow. His indices ended up being the ordinal numbers, including the transFnite ones. Cantor defned a set to be inFnite if its elements could be put in a one-to-one correspondence with the elements of a proper subset of itself. Two sets had the same
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This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.

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