In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> Am Montag, 2. Dezember 2013 16:53:04 UTC+1 schrieb christian.bau: > > On Monday, December 2, 2013 1:36:40 PM UTC, WM wrote: > > > > > Am Montag, 2. Dezember 2013 09:09:39 UTC+1 schrieb Zeit Geist: > > > > > > The proof proceeds by Contradiction. We assume the Set of Real Numbers > > > > is Countable, and thus can be exhausted in a Sequence. > > > > > > > > > > And that is the point at which further reading is absolutely useless. > > > > > > > > Please explain your reasoning. I didn't find anything wrong with the proof, > > that puts you in aline with thousands of mathematicians.
And only in conflict with one anti-mathematician, WM! > > > and if I missed something, I'm quite sure it's fixable. > > That is not an argument.
It is as good an argument as WM has ever produced. > > The set of positive rational numbers that is less than the natural number n > and has not been enumerated by the first n natural numbers grows with n.
But its size does not, as the "number" of ratioals in any interval of positive lenght is the same as in any other such interval.
> It > is impossible eneumerate all rational numbers, i.e., to remove all rationals > from the state of being not enumerated to the state of being enumerated.
It may be in WM's wild weird world of WMytheology, but not elsewhere, since bijections between |N and |Q abound outside of WM's wild weird world of WMytheology. > > It has been neglected that beyond every n there are infinitely > many following, such that never all can have been used.
One can well-order the rationals as follows:
Each rational, n/d, is represented by the quotient of an integer numerator, n, and a natural number denominator, d, with no common integer divisors greater than 1 then define a new ordering on the rationals so that n1/d1 > n2/d2 if and only if either | n1 | + d1 < | n2 | + d2 or both | n1 | + d1 = | n2 | + d2 and n1 < n2.
Then the set of all rationals reordered as above is order-isomorphic to the naturally well-ordered set of naturals, producing a natural bijection between |Q and |N.
But WM is incapable of understanding anything so straightforward and simple as the above, WM has to make things so complicated that no one can sort them out before he feels comfortable with them.
I dare WM to try to find any flaw in the above well-ordering of the rationals so as to have only one non-successor element preceding all others. --