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Topic: § 534 Finis
Replies: 30   Last Post: Feb 22, 2015 8:14 AM

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Posts: 10,821
Registered: 6/8/11
Re: � 534 Finis
Posted: Aug 8, 2014 12:13 PM
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In article <>, 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?
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)

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