假定有理數已經清楚了，才這麼做的。
你的“空隙”概念是誤導性的。
use a rational set to define the concept of cut.
Use cuts(class of cuts) define a new "number" system, called real number system, in which a rational number is now a cut, and the sub class of all rational cuts is asomorphic to the rational system.In other words, the rational system is embedded into the new system.In new system, non-rational cuts are irrationals.

Hope that this can help you to change your view of the concept and construction of real number system.

Whenver you gets through it, the gate to the modern math is open to you.