Why do Dedekind cuts use sets?

Question

As far as I understand Dedekind cuts, they are really about ranges, not sets. In other words, they are about all of the values greater than or less than some number. But by using sets, I'm led to thinking about odd sets of numbers like "all the numbers with a 3 as the third decimal digit", or "all of the numbers that are the cosine of an angle that is a power of two". Is this generality necessary in defining Dedekind cuts? And if so, why?

Answer 1

"Ranges" are just sets; the "range" of all numbers less than $1$ is (in $\Bbb R$) the set $(-\infty,1)$. On these grounds, I fail to see the issue; I somewhat agree that Dedekind cuts are really about ranges, but there is no issue in encoding those ''ranges" as sets (what else would you do?).

On a different note, you remark that "[Dedekind cuts] are about all of the values greater than or less than some number". This is subtle; what is a "number"? If you already have real numbers at your service, then the technical definition of a Dedekind cut might feel a bit silly, since any cut $\mathfrak{p}$ "is" the interval $(-\infty,x)\cap\Bbb Q$ for some real $x$. But the whole point is that we don't yet have real numbers at our service. The set $\{q\in\Bbb Q:q<\sqrt{2}\}$ is a true Dedekind cut but it makes absolutely no sense unless you already have a real number field $\Bbb R$ that contains $\Bbb Q$ as a subset... which defeats the point of trying to construct the real numbers (it is horribly circular). In this construction, $\sqrt{2}$ itself is a Dedekind cut, e.g. the cut $\{q\in\Bbb Q:q^2<2\vee q<0\}$; we can't describe all Dedekind cuts as intervals using just the rational numbers. The technical definition is designed to capture the idea of "interval" without specifying any explicit upper bounds. The irrational numbers are then the 'strange' cuts which can't be written as rational intervals. A good thing is that we don't need to explicitly define all of our Dedekind cuts; we just define what a cut is and then the basic set-theoretic axioms kick in and give us a set of all cuts, which we call $\Bbb R$. We can then discover that there must exist irrational cuts.

I guess that does mean "this generality [is] necessary"; using only the concept of an interval, you would not be able to define $\Bbb R$ in this way since you would not be able to write down an interval (in the theory of $\Bbb Q$) that describes $\sqrt{2}$, whereas using the technical definition of a cut we can sit back and relax; let our universe provide all the extra, weird, irrational cuts for us and give us the real numbers.