You don't need any sort of "repetition" of Dedekind cuts to construct the real numbers, it's actually fairly straightforward. The set of all Dedekind cuts is a

You don't need any sort of "repetition" of Dedekind cuts to construct the real numbers, it's actually fairly straightforward. The set of all Dedekind cuts is a subset of the power set of rational numbers (i.e. each Dedekind cut is a set of rational numbers) satisfying the following properties: for each Dedekind cut A 1. A is not the empty set 2. A is not the set of all rational numbers 3. A is closed downwards, meaning if x is in A and y < x, then y is in A 4. A has no greatest element, meaning for all x in A, there is a y in A such that y > x. And each Dedekind cut is in one-to-one correspondence with a real number. You can define the usual arithmetic operations on them, show that every rational number has a corresponding Dedekind cut (for any rational q, we have {x in Q | y < q } as the corresponding cut). I haven't seen a proof of the uncountability of Dedekind cuts using the diagonalization argument, but you can prove there is a one-to-one correspondence between Dedekind cuts and Cauchy sequences of rational numbers, which is another construction of the reals. And there is a fairly straightforward proof that those are uncountable using diagonalization. But as you can see, you don't need any sort of repetition to do the construction, it's just a set of sets of rational numbers satisfying a few simple properties. For more information, check out https://en.wikipedia.org/wiki/Construction_of_the_real_numbers https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...