I have been trying to understand some elementary set theory recently, and am trying to understand how the real number line can be defined using the set of rational numbers. In particular, I am trying to understand Dedekind cuts. I understand the definition of a Dedekind cut, and I have no trouble identifying if a cut is Dedekind or not. But I have troubles understanding how this definition is useful.
In a video I saw by Dr. Peyam, he claimed that $\displaystyle \sqrt[3]{2}$ can be defined the following way:
$\displaystyle \sqrt[3]{2}=\{ r\in\mathbb{Q}:r^{3}<2 \}$
I understand that this set is in fact a Dedekind cut because it has (i) no maximum, (ii) contains all rationals less than $\sqrt[3]{2}$ and (iii) is a real, nonempty subset of $\mathbb{Q}$. But I do not understand how we can define a singular number as a set containg infinitely many numbers.
Sometimes the best way to understand a new definition is by understanding its equivalence with an old definition that you are already familiar with. I will assume that you are familiar with the definition of a real number (say, between 0 and 1) as an unending decimal $a=0.a_1a_2a_3\ldots$ (a technical issue here is that one must also identify 1 with $0.999\ldots$ and all similar cases). Here is an observation that may help: the number $a$ splits (or "cuts") $\mathbb Q$ into two subsets: $A_L$ consisting of all the rationals less than $a$, and $A_R$ consisting of all the rationals greater than $a$ (I will ignore for the moment the issue with $a$ itself in the case when it happens to be rational). Now we have $\mathbb Q$ split into two subsets such that if $x\in A_L$ and $y\in A_R$, then always $x<y$.
Conversely, if one has a splitting of $\mathbb Q$ into a "left" set and a "right" set having the property above, then the splitting uniquely determines a real number.
This is all that is meant when one says that a real number "is" a cut on the rationals. You can actually think of it as the pair $(A_L, A_R)$, but this is merely an issue of set-theoretic formalisation that does not affect the way you actually use the real numbers.