In article <firstname.lastname@example.org>, email@example.com wrote:
> On Thursday, 7 August 2014 21:31:09 UTC+2, Zeit Geist wrote: > > > > > If infinity can be finished, then N is ""all natural numbers. I have > > > assumed that and have shown that then rationals remain uncounted.
In any proper set theory, N would be the SET of all natural numbers, not merely '"" all natal numbers'. > > > Only for All Finite Sets do Any Rationals Remain Uncounted. > > What else but finite numbers can index rationals?
The Infinite Sets of finite natural numbers can can index the infinite set of rational number when that infinite set of rational numbrs has been properly well-ordered as follows:
Once again, since WM is having so much trouble understanding it:
Each member of Q has UNIQUE representation as m/n, with m being an integer, n being a positive integer, and with m and n having no common factor greater than 1. Order them by increasing values of abs(m)+n, and within equal values of abs(m)+n by increasing values of m, if any. Note that for positive m, m/1 has successor -(m+1)/1. For any other form, m/n will have successor of form (m + k)/(n - k) for some natural k with 0 < k < n. This is a well-ordering of Q with a first rational, 0/1, and for each rational a uniquely defined successor rational, and with no rationals left out. Thus each rational is now enumerated by the natural number marking its position in the above well-ordering, at least everywhere outside of WM's worthless world of WMytheology. > > > > > > > > > But since it is impossible to identify remaining rationals
In the above well-ordering, there aren't any "remaining" un-well-ordered
> In every case naturals remain.
Name one! 1 is used in the well-ordering of Q, and for every n used, n+1 is also used, so which naturals have not been used? -- Virgil "Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)