I want an intuition of how this set is constructed more than a formal proof. At first I thought that the set was simply defined axiomatically but further reading showed me that there had been attempts to construct the set explicitly like the construction from Cauchy sequences. So what are the real numbers ? An axiomatically defined set? the completion of the rational numbers? Both? Something else?.
Thanks a lot in advance!
[Math] How are real numbers constructed
real numbersreal-analysis
Best Answer
In mathematics, we often don't really care what something "is" in some fundamental sense, but what its properties are. In this way, we may view the real numbers as any complete, ordered field $\mathbb{R}$ which contains the rational numbers as an ordered subfield.
At this point, you may have two questions: does any set of real numbers $\mathbb{R}$ exist and, if so, are there more than one which are "different" from each others. As you alluded to, there exist several constructions, most famously in terms of Dedekind cuts and Cauchy sequences of rational numbers, which show the existence of a set of real numbers. It is also true that the real numbers are unique. That is, if I have two complete, ordered fields $\mathbb{R}, \mathbb{F}$ containing the rationals as an ordered subfield, there there exists an order-preserving isomorphism between them. In other words, $\mathbb{R}$ and $\mathbb{F}$ are the "same".
Note that the Cauchy sequence construction shows that the real numbers are the metric completion of $\mathbb{Q}$ under the metric $d(x,y) = |x-y|$, so we may additionally view $\mathbb{R}$ as the metric completion of $\mathbb{Q}$.
What is worth noting is that, for any practical matters, it doesn't matter what model of the real numbers you use. Whether real numbers are equivalence classes of Cauchy sequences, Dedekind cuts, or something else doesn't effect the actual properties of the real numbers that you care about for analysis.