In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 13 Apr., 01:23, Virgil <vir...@ligriv.com> wrote: > > > > 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. > > How many others have found that it is impossible to prove the parallel- > axiom in the 2000 years before Gauss?
It is still impossible without first assuming something equivalent to it. > > If d is nothing but its fis, then there is a contradiction. And if you > claim that d is more than its FISs, then there is a contradiction too.
But that alleged "contradiction" does not exist outside Wolkenmuekenheim, no no one but WM will ever be bothered by it. > > The first contradiction does not contradict Cantor's proof but what > has been interpreted into his result: > Cantor showed: for all k exists n : d_n =/= q_kn & n =< k > This has been interpreted as > For all k: d =/= q_k.
What Cantor showed was there is a d such that for all k : d_k =/= q_k_k, therefore d =/= q_k
Anything beyond that was not part of his proof. > > The second contradiction is this: > If d (that has aleph_0 digits with aleph_0 > n for all n) is not in > the sequence of all its FISs, then the union of all FISONs has not > aleph_0 elements.
Again WM's incapacity reveals itself!
The set of all FISONs is not the same thing as the sequence of all FISONs.
The sequence, being strictly increasing by inclusion, has a limit larger than any member, and that limit is the union of all FISONs, which cannot itself be a FISON --