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So the definition of a Dedekind cut is: a = sup{r $\in$ $\mathbb{Q}$: r < a} for each a $\in$ $\mathbb{R}$. Subsets a $\in$ $\mathbb{Q}$ having the form {r $\in$ $\mathbb{Q}$ : r < a} satisfy three properties:

(i) a $\neq$ Q and a is nonempty.

(ii) if r $\in$ a, s $\in$ $\mathbb{Q}$, and s < r, then s $\in$ a.

(iii) a contains no largest rational.

What I'm not understanding is 4 things:

1) Why can't a contain a largest rational number? I mean, on one hand, if it does, then the set is bounded? But the set is bounded.

2) Is a a set or is a some number in the real numbers line? But if it's a real number what I'm confused is; it's a set. It's a set that satisfies a property?

3) So a $\in$ $\mathbb{R}$, but is a $\in$ $\mathbb{Q}$?

4) How is $\mathbb{Q}$ a subset of $\mathbb{R}$ just from the definition of a Dedekind cut?

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    $\begingroup$ Dedekind cut has not been defined properly. A Dedekind cut is a certain kind of set of rationals. $\endgroup$ – André Nicolas Aug 13 '15 at 3:21
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That is not the definition of a Dedekind cut. Rather, it's a proper subset $A\subset \mathbb{Q}$ that's downward closed (that is, if $x\leq y$ with $y\in A$, then $x\in A$) such that $A$ contains no greatest element. (There are a couple of equivalent definitions that are also often used; see, for example, Wikipedia for some alternatives.) It looks like you're confusing the set $A$ with the real number it represents. Identifying $A$ with a real number is the point of this construction, but it's not immediate; we're trying to construct $\mathbb{R}$ from just $\mathbb{Q}$, and $A$ is a subset of $\mathbb{Q}$.

To answer your specific questions:

1) It's not a crucial part of the construction, but it does help with the uniqueness. Ultimately, we want an embedding $\mathbb{Q} \to \mathbb{R}$ such that a Dedekind cut $A$ is just $A = \{q\in \mathbb{Q}:\, q < A\}$. Of course, that expression is currently meaningless; I haven't put an order on $\mathbb{R}$ or described that embedding. But intuitively, the Dedekind cut corresponding a number $x$ on the real number line is the set of rationals $q < x$. Making that intuition rigorous is the point of this construction. Remember, we don't even have the real number line yet; we just have $\mathbb{Q}$.

2) It's a set of rationals. Forget anything about the real number line; the point of Dedekind cuts is to rigorously construct it from $\mathbb{Q}$.

3) A Dedekind cut $A$ is a subset of $\mathbb{Q}$. In this construction, $\mathbb{R}$ is a set of subsets of $\mathbb{Q}$. (For right now, forget the usual embedding of $\mathbb{Q}$ as a set of points on $\mathbb{R}$. It takes a little bit of work to produce that from this particular construction of $\mathbb{R}$.)

4) It's not quite immediate from the definition, but the embedding maps $q\in \mathbb{Q}$ to the Dedekind cut $A_q = \{q'\in \mathbb{Q}:\, q' < q\}$.

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the point is to create a collection of objects that we can map the rationals into injectively.

A cut is supposed to model a real number $x.$ The cut is $\{ q < x \}.$ (Of course, we can't actually use $x$ since that's what we are trying to define.)

If $x$ is rational then we could end up with two sets representing $x$: $\{ q < x \}$ and $\{ q \leq x \}$ by banning sets with a maximum we outlaw the second case.

We define $\mathbb{R}$ to be set of such cuts. We can map $\mathbb{Q}$ into $\mathbb{R}$ by the map $$ a \mapsto \{ a < q\}. $$ It is then typically identified its image under this map.

( I have a lot of discussion of the motivation for this definition in my book "Proof patterns").

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