> I am also convinced that there cannot be more distinct Dedekind cuts > than distinct rational numbers. Just drawing a sketch of some Dedekind > cuts convinces me. The Dedekind cuts are 1) nonempty and 2) totally > ordered by the relation "is a proper subset of". For finite sets it is > easy to see, and prove by induction, that for such a collection of > sets there are no more sets than elements. But I do not know how to > make this a transfinite induction.