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Topic: Mueckenheim's Theorema Egregium
Replies: 5   Last Post: Aug 19, 2014 12:35 PM

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Virgil

Posts: 10,821
Registered: 6/8/11
Re: Mueckenheim's Theorema Egregium
Posted: Aug 19, 2014 12:25 PM
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In article <fccf852a-e8ed-4959-9716-f02909d5123e@googlegroups.com>,
mueckenh@rz.fh-augsburg.de wrote:

> But if actual infinity is assumed, then infinite sets can be exhausted. Then
> |N is exhausted before Q+.


Name one member of Q+ that does not get matched to a member of N when
Q+ is well-ordered by:

Here is a straightforward way to construct a well-ordering of all
positive rationals. Write each one 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...
A first member, and for each a unique successor, just like N.

You can then do the whole set of rationals by putting zero at the start
and interleaving each negative after its corresponding positive.
--
Virgil
"Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)



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