It's still in reference to the OP so its not really a new question.

justchillin, I'm not sure what you are getting at with density. As far as the

being unique, the idea behind these cuts is to construct the real numbers partitions of the rational numbers. In other words for each real

the corresponding "cut" can be defined to be the rational numbers less than

. We can then define addition and multiplication and get something isomorphic to the real numbers as we know them. Then the uniqueness property now makes sense. We want each cut to uniquely represent each real number. I hope that clarifies things somewhat.