In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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.
But every real number between 0 and 1 defines an infinite 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. 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.
> > 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.
If it were of any importance at all, many others would have found it and made use of it long before WM came up with it.
But it in no way counters Cantor. At least not outside Wolkenmuekenheim. > > Both theorems contradict each other
Not outside of Wolkenmuekenheim! And if they do inside Wolkenmuekenheim, they are far from the only contradictions that flourish there. --