Dedekind Cuts & the Real Line: A Countable Set?

[cragar]

If every Dedekind cut is at a rational it seems that these cuts would only produce a countable set and would not produce the whole real line. So how should I think about it.

 

[mathman]

If every Dedekind cut is at a rational it seems that these cuts would only produce a countable set and would not produce the whole real line. So how should I think about it.

The point is that there are Dedekind cuts that are NOT at rationals.
For example, take any real number A and consider its decimal expansion. A sequence of rational numbers A_n can be defined by taking n terms of the expansion. Let the cut be defined by all rationals less than any term in the sequence. This cut gives the real number A.

 

[bobby2k]

Even though all Dedekind cuts consist of only rational numbers, all are not rational cuts.
T = { x \in Q: x^2 < 2 or x < 0 } is a dedekind cut, you can check that all the properties hold, but it can not be a rational cut.