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| CANTOR IS WRONG! |
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| in a sci.math posting, david petry wrote: "i believe that the attack on cantor's theory will not end until someone has found a way to drive a stake right through the heart of the theory." * permit me to take a cranky stab at it... two finite numerals represent different numbers if they differ in at least one digit, as 101 is different from 111 (in base 2)...this principle is not valid with infinite numerals, since 0.111... is not different from 1.000... (again base 2)...thus, cantor's claim that the diagonal real is different from every listed real fails...yes, it differs in one digit from every listed real, but no, this does not mean it is different from every listed real... again, allowing infinite numerals precludes the possiblilty of comparing digits, and after all, cantor's proof is strictly speaking about numerals, not numbers... the patch to this defect is to "normalize" the decimal expansions of listed reals...for example, wilfrid hodges, in his interesting paper "an editor recalls some hopeless papers", puts it like this, using for his list base 10: "where it applies, we choose the expansion which is eventually 0, not that which is eventually 9." ** unfortunately, this is closing the barn door after the horse has left...cantor's diagonal real may be missing from the list simply because it is not in normalized form...and if it were, it very well might be on the list...from his construction of the diagonal real, there is no way to tell... in summary: (1) if the listed reals are un-normalized (i.e. can end in "111..."), cantor's diagonal real can differ from every listed real, yet still be equal to one of them...on the other hand (2) if the listed reals are normalized, a real missing from the list could be so simply because it is un-normalized, and in its normalized form may indeed be listed...either way, cantor has not contradicted the assumption that all reals were listed... ---cobb stregagobo 1/6/02 * http://mam2000.mathforum.org/epigone/sci.math/queldspangchal/7b34gn$soc$1@remarQ.com ** http://www.math.ucla.edu/~asl/bsl/0401/0401-001.ps & a special thanx for the link at http://www.crank.net/maths.html ...check zem out.... |
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