Virgil
Posts:
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Registered:
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Re: Mueckenheim's Theorema Egregium
Posted:
Aug 19, 2014 12:25 PM


In article <fccf852ae8ed49599716f02909d5123e@googlegroups.com>, mueckenh@rz.fhaugsburg.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 wellordered by:
Here is a straightforward way to construct a wellordering 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)

