Well ordering principle for rationals Why can positive rationals be not well ordered?  If we define the relation to be greater than(>), then every subset will have a least element.  Or why are positive or even integers not well ordered?  By the same logic we can always find a least element in any subset.  I know I am wrong at some very fundamental point, but please explain it to me.

Positive integers are well ordered but positive rationals are not because for well ordered, every non empty subset must have least element( least element must belong to subset and there is difference between least element and greatest lower bound). There are many subsets which have no least point in positive rationals like the subset ${1, 1/2, 1/4, 1/8, 1/16, ...}$ has no least element or the set of all positive rationals greater than any irrational number.
