On Tuesday, 25 June 2013 17:07:36 UTC+2, dull...@sprynet.com wrote: >> There are not uncountably many reals, neither in the natural order nor in the well-order.
> That's very funny.
That's easily provable:
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.