Dedekind Cuts and Rationals I have a question regarding how you might construct the reals from the rationals by taking Dedekind cuts. 
My basic understanding is that a Dedekind cut is a bipartition of the rationals such that the two partitions $X, Y$ satisfy certain properties: 
$\forall x \in X,\; \exists y \in X \text{ such that } x < y$
$\forall x \in Y, \; \exists y \in Y \text{ such that } y < x$
$\forall x,y, \; x<y \text{ and } y \in X \Rightarrow x \in X$
$\forall x,y, \; x<y \text{ and } x \in Y \Rightarrow y \in Y$
$\forall x,y, x<y \Rightarrow \text{ either } x \in X \text{ or } y \in Y$
$X \text{ and } Y$ are disjoint
$X \text{ and } Y$ are both non-empty.
A Dedekind cut is then used to represent a real number $z$ that can be see as the "mid-point" of this bipartition of the rationals in the sense that $\forall x \in X, \forall y \in Y, \; x < z < y$
In most descriptions of defining the reals as the set of all Dedekind cuts of the rationals, these inequalities are strict. However, doesn't that mean that the rationals cannot be defined in this way? Am I meant to take this to mean that the irrationals are defined using these Dedekind cuts and that the reals are to be treated as the union of this set of irrationals and the set of rationals?
If that is the case, doesn't that mean the irrationals and rationals in this construction of the reals are different kinds of objects in that they would have different ranks? I would imagine that is something inconvenient and would ideally want to be avoided. 

The problem is that these inequalites are not necessarily strict. For instance, it is stated here that the second set (your $Y$) may have a smallest (rational) number $q$, in which case the Dedekind cut $(X,Y)$ correspondes to the rational number $q$.
