Timothy Little a écrit : > Lee Rudolph wrote: > >>Is it possible to say anything interesting about "how different" the >>usual ordering of the reals is from a well-ordering (where "a >>well-ordering" might be *any*, or *every*, according to taste)? > > > One way to investigate this question would be to consider how large an > ordinal one can embed in the usual ordering on the reals. Fairly > clearly, we can only embed a countable ordinal in this way. We can > associate with each rational the least element of the embedding > greater than it, hence defining a surjection.
And the reciprocal is well-known ; any countable ordinal can be embedded in the rationals (actually, any countable total order can be embedded in the rationals)
> > Consequently for any well-ordering of the reals we can only extract a > countable subset that agrees with the usual ordering. > > As to the number of pairs of reals (r1,r2) in the well-ordering (R,W) > for which r1 < r2 and r1 W r2, that has cardinality equal to that of > the reals. > > > - Tim