My professor defined Dedekind cuts in the following way:
"Given two nonempty sets E,F $\subset \mathbb{R}$, we say that the pair (E,F) is a Dedekind cut of $\mathbb{R}$ if
- $E \cap F = \emptyset $
- $E \cup F = \mathbb{R}$
- $x < y$ for all $x \in E$ and for all $y \in F$"
However, I was under the impression that Dedekind cuts could only be taken of the rational numbers, in order to construct the real numbers. Can you take a Dedekind cut of the Real line?
Edit: I should mention that he defined a cut this way in order for him to state the following Dedekind Axiom:
"For every Dedekind cut (E,F) of $\mathbb{R}$ there exists a unique $L \in \mathbb{R}$ such that $x \leq L \leq y$ for all $x \in E$ and for all $y \in F$"
Best Answer
You can take Dedekind cuts of any linearly ordered set and use them to form the order completion of that set. If you start with a complete linear order, however, like $\langle \Bbb R,\le\rangle$, you get nothing new: the Dedekind completion is order-isomorphic to old one. (By the way, it’s usual to add one more condition, either that $E$ has no maximum element or that $F$ has no minimum element.)