Why do we use Dedekind cuts to define the real numbers?

Question

I have recently embarked on reading some more advanced set theory, and in particular an article on alternative set theories which opens with a description of how one can obtain the reals via Dedikind cuts: link here.

I understand this description, but I wonder why we use Dedikind cuts, as it seems quite difficult to explain, and it's not immediately obvious that the set of cuts is the set of real numbers. Is it not possible to define the set of reals as the completion of the rationals, i.e. the set of limits of rational Cauchy sequences?

It is easy to think of sequences that tend towards each irrational number, for example $\pi$ is the limit of $a=\langle 3,3.1,3.14,3.141,\ldots\rangle$ and $\sqrt 2$ is the limit of $b=\langle 1,1.4,1.41,1.414,\ldots\rangle$. Then arithmetic is easy to define, because of the basic properties of limits, e.g. $\pi\cdot\sqrt2=\lim_{i\to\infty}c_i$ where $c_i=a_i\cdot b_i$ for all $i$.

Thanks!

Answer 1

Firstly, note that the completion if a metric space cannot be defined as the set of limits of Cauchy sequences in that space before you actually constructed all of those limits. So, it's not so easy to give a short and sweet construction of the reals. In fact, historically, defining the reals rigorously posed a huge challenge even to such great minds as Weierstrass, illustrating how non-trivial that matter is.

Nowadays, the two mot commonly met constructions of the reals are Dedekind's construction by cuts and Cantor's construction by Cauchy sequences. There are many criticisms of these constructions, primarily pedagogical ones. It is also possible avoid any construction at all and simply list the axioms (which are categorical, if second order). Another option is to view the reals as a completion of the rationals as a uniform space and apply the general machinery of topology. This is Bourbaki's approach, which is extremely elegant but from the perspective of this question one must realize this just shifts one's attention away from a construction which still must be given - e.g., in terms of minimal Cauchy filters.

In short, it is unavoidable that any construction of the reals will be technically involved, simply because the reals are so incredibly more compilated that the ratioanls. A survey of more than a dozen of constructions of the reals can be found in here and a detailed construction of the reals by means of filters, together with a historicl account of the reals including detailed criticims of Dedekind's construction and of Cantor's construction, are given here.

And so, to answer why do we define the reals as Dedekind cuts I'd say it is one of several alternatives, by fear not the optimal one, and historically the first rigorous one to be published (Cantor's and Bachmann's constructions were given very close in time, and all three constructions should be considered to share the same 'birthdate').