>On 12 Apr., 16:14, dullr...@sprynet.com wrote: > >> >Cantor's list contains real numbers r as binary or decimal fractions. >> >Real numbers, however, are /limits/ of binary or decimal fractions. >> >> No, it _is_ a list of limits of decimal fractions. > >There is no such list possible unless you give finite definitions. >It is impossible to define a number by writing an iinfinite sequence. >> > >> It passes belief that a person could actually >> think that such a well known proof could contain >> such a simple error, without that error being >> noticed by any of the thousands of mathematicians >> who've studied the subject for a century or so. > >This assumption may be the reason that Cantor's "proof" has been >believed over 100 years.
Curiously you deleted my statement of exactly what error of yours I was referring to. Why was that? Wait, I know. Because you have no intellectual integrity.
>Try to understand the following. (If you are >not a too slow thinker, this will happen before midnight.) > >Consider a Cantor-list that contains a complete sequence (q_k) of all >rational numbers q_k. The first n digits of the anti-diagonal d are >d_1, d_2, d_3, ..., d_n. It can be shown *for every n* that the >Cantor- >list beyond line n contains infinitely many rational numbers q_k that >have the same sequence of first n digits as the anti-diagonal d. > >Proof: There are infinitely many rationals q_k with this property. >All >are in the list by definition. At most n of them are in the first n >lines of the list. Infinitely many must exist in the remaining part >of >the list. So we have obtained: > >For all n exists k: d_1, d_2, d_3, ..., d_n = q_k1, q_k2, q_k3, ..., >q_kn. >This theorem it is not less important than Cantor's theorem: For all >k: d =/= q_k. > >Both theorems contradict each other with the result that finished >infinity as presumed for transfinite set theory is not a valid >mathematical notion. > >Regards, WM >