Understanding of real numbers based on Dedekind cut?

Question

I can not understand the construction of real numbers by Dedekind cuts. Can somebody help me with understanding? The problem which I have is that the cardinality of rationals is $ \aleph_0 $. Base on that, my assumption is that I can cut this only in $ \aleph_0 $ places in such a way that $ \forall a \in A, b \in B : a \lt b $ and $ A \cup B = \mathbb{Q} $. If it is true then it could find only $ \aleph_0 $ real numbers from $ \mathbb{R} $. Please point me where is a logic mistake.

Answer 1

The mistake lies in your assertion that you can cut only in $\aleph_0$ points. Why? The set $\Bbb R$ is uncountable and, for each $x\in\Bbb R$, if you consider the cut$$\Bbb R=\bigl((-\infty,x)\cap\Bbb Q\bigr)\cup\bigl([x,\infty)\cap\Bbb Q\bigr),$$you get different cuts for distinct values of $x$.

Answer 2

The mistake may be that you are trying to extrapolate from simpler linearly ordered sets. If, instead of $\Bbb{Q}$ you had a finite linearly ordered set in which every element has an immediate successor (and all but the first element is a successor of a single element) then the number of cutting points would, indeed, be more or less equal to the number of elements ($\pm1$ depending on whether you include the ends). Every possible cut occurs between an element $x$ and its successor $s(x)$. The same holds for infinite sets that have a similar successor function. For example, you can cut $\Bbb{N}$ at only countably infinitely many points – the two "halves" being $[0,n]$ and $[n+1,\infty)$ for all $n$.

However, the ordering of $\Bbb{Q}$ is more complicated. For one, it is dense. Between any two rational numbers there are infinitely many others. This means that there cannot be a successor function. Implying that you can keep on adding more cuts between any two earlier cutting points. This is in sharp contrast to cutting $\Bbb{N}$. It is easy to see that this leads to uncountably many cuts, but the details of that argument may be better explained by other means (such as Cantor's diagonal argument).