In article <email@example.com>, firstname.lastname@example.org wrote:
> Set theorists claim that all rational numbers can be indexed by all natural > numbers.
Mathematicians, a group to which WM notoriously does not belong, have proved that the set of all positive rational numbers, Q+, and the set of all rational numbers, Q, can be well-ordered so as to have a first member and for each member a unique successor member, just like the set of all natural numbers is naturally ordered.
Since these well-orderings clearly establish bijections between N and eithre Q+ or Q, WM;s continues claims that no such bijections can exist in his worthless world of WMytheology reveal that world's anti-mathematical attitude.
Whether actual or potential, any well-ordered set with a unique non-successor and no fixed last member bijects with the well-ordered set of naturals with each position in the well-ordering determining its corresponding natural.
And both Q+ and Q are thusly ordered by:
Write each positive rational as p/q where naturals p and q have no common factor (other than 1). Order them by ascending value of (p+q), then within each set of p+q values, order by ascending p.
So you get: 1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1..., ad infintum, which is a well-ordering of Q+
Then well-order Q by putting zero at the start and interleaving each negative after its corresponding positive.
> In § 533 I have shown not only that every natural number n fails but > even that with increasing n the number of unit intervals of rationals without > any rational indexed by a natural less than n increases without bound, i.e., > infinitely.
What WM has shown is that infinitely many finite initial sets of naturals each fail to do what one infinite comlete set of naturals easily accomplishes (as the above proof demostrates N can do).
> Since nothing but finite natural numbers are available for > indexing, and provably all fail,
The issue is whether any SET of natural numbers can index the SET of ratinal numbers, so any argument that does not treat properties of the entire set of naturals is irrelevant.
The essential properties of the ordered set of all naturals, N, are (1) that there is first member, (2) that every member has a unique successor member, (3) that any set containing that first member and the successor member of each of its members contains N as a subset, and (4) any set with the above three properties bijects with the set of all naturals. > > I don't know what goes on in the heads of matheologians.
The mathematics that WM derides as matheology is far more clear to everyone other than WM than whatever passes for thought in the head of WM. -- Virgil "Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)