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

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!

• Because Dedekind cuts are based in set theory, that is the formal core to all the modern mathematics. Indeed Dedeking cuts are simpler and more beautiful than the other ways to construct real numbers. – Masacroso Sep 27 '16 at 18:42
• It is also possible to define the reals via Cauchy sequences (as equivalence classes of such actually, not drectly as their limits). And we also do this. It is mainly a matter of taste. – Tobias Kildetoft Sep 27 '16 at 18:42
• @John11 Depends on how one defines "shortest". – Tobias Kildetoft Sep 27 '16 at 18:50
• You can say that the Dedekind cuts are just a slight generalization of the Cauchy sequences construction of $\mathbb{R}$. – reuns Sep 27 '16 at 18:50
• To the above comments: all rigorous constructions are based on set theory. Many mathematicians do not see Dedekind's cuts as elegant. The claim that it is the shortest path to defining the reals is extremely shaky. And in which sense is that a generalization? – Ittay Weiss Sep 27 '16 at 19:00