> >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. 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.