real number system 5 (Dedekind cut)

Given a real number line $(-\infty, \infty)$, for example a coordinate axis, we are able to cut the line in two at an arbitrary point. The point will correspond one to one with a real number. Let the left side of the point on the line be $S$, and the right side be $T$. $S$ and $T$ are nonempty sets of real numbers such that
\[ S\cup T=\mathbb{R}, \quad  S\cap T=\phi .  \]
That is, all real numbers are mutually exclusive and collectively exhaustive in $S$ or $T$. Furthermore, if $s\in S$ and $t\in T$, then $s<t$. It is said 'Dedekind cut ' expressed by $(S,T)$.

In general, a cut of one ordered set into two sets $S,T$ logically has one of following opportunities.

(1) There exist both $\max S$ and $\min T$.
(2) There exists $\max S$, and does not $\min T$.
(3) There exists $\min T$, and does not $\max S$.
(4) There do not exist both $\max S$ and $\min T$.

You should note that we have to use $\max, \min$ in substitution for $\sup, \inf$. The reason is because conditions $\max S\in S$ and $\min T\in T$ should be met.

Any Dedekind cut of the real number line can not bring the result (1) in the dense field. Because, if we put $a=\max S$, and $b=\min T$,  then $a<b$. However, as real numbers are dense, there is a real number $x$ such that $a<x<b$. it is the contrary of $S\cup T=\mathbb{R} $. Hence, the result (1) is cleared out.

Suppose that $c$ is a point of 'Dedekind cut '. If $c$ is in $S$, the result (2) occurs, or if $c$ is in $T$, (3) does. As $c$ could be any real number, an irrational number $c=\sqrt{2}$  is available.

In the past, R.Dedekind explained real numbers were a continuum, since (2) or (3) was true by assuming that irrational numbers would exist in addition to rational numbers on real number
system.Then, although we intuitively feel it true, as it is impossible to reject the result (4) from all Dedekind cut in a dense ordered field, especially real numbers, we have to decide not to accept (4).

[41''] Dedekind cut at every point of a real number line brings just only the result (2) or (3).

It's assertion is equivalent to the axiom for completeness of real numbers [41] and  means the continuity of real numbers.

How do you feel about all of this?