A Problem with Real Numbers as Cuts

The Diagonal Argument

Cantor's diagonal argument is a famous piece of mathematics that is typically used to show how between the numbers 0 and 1, there are an uncountably infinite number of real numbers; that is, you could never assign each of these real numbers a natural number in a list, you would always have more reals than naturals. This argument can of course apply to any two rational numbers, 0.9 and 1, for example. Or 0.99 and 1, or 0.999 and 1, or 0.999... followed by another billion nines, and 1; you can always just shift the argument over by a number of decimal places.

Dedekind Cuts

Dedekind cuts are supposedly a way of constructing the real numbers that involves defining a real number as the set of all rational numbers that fulfill some condition. For example, defining 2 as the set of all rational numbers a such that a2 < 2 (and a < 0 since this is quadratic and you want all negatives to be in the set, even the ones that are > 2 when squared).

If you define a rational q via Dedekind cuts, you define it as the set of all rationals less than q. Initially this may look acceptable, ignoring that roots are a conveniently easy and uncomplicated example, but consider how a regular old rational number is defined through cuts: a rational q is defined as the set of all rationals a < q.

But between any arbitrarily close rationals p and q, there are infinitely many reals (via the diagonal arugment). So if you define any rational number with these cuts, there must be infinitely many reals that share this definition! Technically this set/cut also must have no greatest element, so you may think that this means there can be no reals between the set and the rational as we can always get closer to q, but remember the set is comprised in its totality of rationals (you can't construct real numbers with sets of real numbers, obviously), and it doesn't matter how close two rationals are, there are always uncountably infinitely many reals between them.

Thus using the system of Dedekind cuts, any definition of a rational (at least) is shared by an uncountably infinite number of other, somehow different reals that can be seen as occupying the space on the number line around the rational.

Norman Wildberger has made excellent video lectures on the difficulty of constructing the real numbers in general and if this topic interested you I would recommend you start here.