In article <email@example.com>, firstname.lastname@example.org 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)