of the set of rational number
s into two pieces, in such a way as to uniquely define a real number
Intuitively, the number line is divided into disjoint left and right parts. The point at which the two parts meet is a real number. For any given real number, the segments to the left and right of it are unique, so it is well-defined.
If the point at which the cut is made is in either the left or the right half, it is rational. If it is in neither, it is irrational. Remember that the cut is a partition of just the rational numbers. If it is exactly at some rational point, that point belongs to either the left or the right, but not both. But if the cut is at an irrational point, neither subset of the rationals contains it.
More formally, let QL and QR be nonempty subsets of the rationals Q such that
- QL ∪ QR = Q
- if a ∈ QL and b < a then b ∈ QL
- if c ∈ QR and d > c then d ∈ QR
- if x ∈ QL and y ∈ QR then x < y
Now QL is bounded above, because QR is nonempty, so has some element y, and for every x in QL, x < y. In general, however, it does not have a least upper bound, unless the cut happens to be at a rational point (a member of either QL or QR: in either case that point is the l.u.b.).
Any such pair (QL, QR) defines a real number. The real number is the l.u.b. of QL and also the g.l.b. of QR, whether or not it belongs to either of them. Equivalently, any QL defines it, because obviously QL defines QR = Q - QL.
But this doesn't say anything sensible (anything constructive) about the contents of QL, if all we've got to build from is rational numbers. It says QL is some set of rationals but doesn't tell you any of them. So let a Dedekind cut be made at some rational point q1. This constructively defines some QL1. Then take q2, q3,... to create QL2, QL3,... with the proviso that there exists some rational y that is a common upper bound, i.e. q1 <= y and q2 <= y and q3 <= y and...
It need not be the least upper bound: if all we have is rationals there might be no such least number. Informally, the qn approach a limit, either y or some "number" smaller than y. Formally, let QL be the union of all the sets QL1 ∪ QL2 ∪ QL3 ∪... . Then the same boundedness can be shown to hold for QL, which is now a constructed set.
The real number can be defined to be this set. If from some point n onward, Qn = Q(n+1) = Q(n+2) =... then it corresponds to the rational number that is the endpoint of these; otherwise it has no rational counterpart.
All this is is a formalization of the fact that every real number is a limit of rational expansions, either 2.5, 2.50, 2.500,... or 3.1, 3.14, 3.141, 3.1415...