I don't like the way it's written, but what they are talking about is completeness in the sense of "Dedekind completeness"; i.e., that given any two sets A and B with everyone in A below everyone in B, there is some number which is simultaneously an upper bound for A and a lower bound for B.
Note that this fails for the rationals: e.g., if we let A be the rationals below sqrt(2) and B be the rationals above sqrt(2).
In school, we talked about “Dedekind cuts” but we never formalized the definition. Kind of disappointed now because your explanation is very simple and elegant.
Note that this fails for the rationals: e.g., if we let A be the rationals below sqrt(2) and B be the rationals above sqrt(2).