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Topic: § 534 Finis
Replies: 61   Last Post: Aug 13, 2014 3:18 PM

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 Virgil Posts: 10,821 Registered: 6/8/11
Re: � 534 Finis
Posted: Aug 6, 2014 11:44 AM

mueckenh@rz.fh-augsburg.de 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)