How many points are inside the hole revealed by the Dedekind cut?
It is illustrated by the figure that the set of the rational numbers is not a continuum, there are holes in it, one question is how many points are inside the hole? From a geometrical perspective, I think treating real numbers as shorthands of Dedekind cuts of the set of rational numbers cannot convincingly answer the question - how many points are inside the hole revealed by Dedekind cut not produced by a rational number like \(\begin{matrix} L & \ = \{ x \mid x \in \mathbb{Q},x \leq 0\} \cup \left\{ x \mid x \in \mathbb{Q},x > 0,x^{2} < 2 \right\} \\ U & \ = \mathbb{Q} - L = \left\{ x \mid x \in \mathbb{Q},x > 0,x^{2} > 2 \right\} \\ \end{matrix}\) . Every such Dedekind cut can be conceivably illustrated on the number axis, without a sound proof, to say such a cut corresponds with one and only one point on the number axis is far from trust , however, which is what exactly required by Cantor-Dedekind axiom 1 , 2 . Unless a convincing proof is given, otherwi...