We use Zermelo-Fraenkel-Choice (ZFC) axioms to define natural numbers, then integers and then rational numbers. Then, we define real numbers as limit points of rational Cauchy sequences. I have two questions about it:
Thank you.
On
We don't want to define real numbers as Cauchy sequences, but only use them to construct a "continuum" that satisfy our intuition about continuity (from measuring space and time). Cauchy sequence is not the only way to define real numbers, we can also use e.g. Dedekind cut.
We want to make a system of numbers that fulfills the whole geometric "line" without holes. The Greeks thought $\mathbb Q$ is enough, then they discovered $\sqrt{2}$, etc. Dedekind made the observation to cut a continuum into two pieces, you must hit a number/point on it.
To quote Dedekind himself,
"If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions." ...every one will at once grant the truth of this statement; the majority of my readers will be very much disappointed in learning that by this commonplace remark the secret of continuity is to be revealed.
It "happens" that the different constructions or different types of completeness axioms (Cauchy completeness, minimal upper bound exists, etc.) all lead to the same objects. The idea of Cauchy sequences can used in other cases so it became more popular and standard in textbooks today, while Dedekind cut seems to be an isolated example with no further development, but TBH, Dedekind cut is way more satisfying for revealing the nature of continuity.
On
The "completeness axiom" is an axiom, when you begin your study of real analysis at the starting point of the reals. The "completeness axiom" is a definition under ZFC (or a theorem derived from an equivalent definition).
The reals can be defined as "new numbers" that satisfy the least upper bound property of subsets of rationals (essentially, Dedekind cuts). Or the reals can be defined as "new numbers" that give limit points to all rational cauchy sequences. In either starting point, one statement is taken as a definition, and the other is a theorem derived from that definition.
I am sure there are other equivalent ways to define the reals from the rationals, but the notion is the same. All definitions are equivalent and derivable from each other as theorems. In real analysis we don't bother building ZFC from scratch and simply take the reals themselves as an axiom (i.e., there exists a set called R, with the least upper bound property).
It seems you are confusing two separate ways to define the real numbers.
First, there's the axiomatic approach, where we define a set of axioms (of the real numbers) that the real numbers should satisfy, and work with the object defined by these axioms.
Second, there's the constructive approach, where we work in a stronger theory, like ZFC, to construct an object that we then declare to be the real numbers (i.e., we construct natural numbers, then rationals, and define the reals as (equivalence classes of) limits of Cauchy sequences).
However, it is easy to get confused here, since we talk about the axioms of ZFC in the constructive approach as well as about the axioms of the real numbers in the axiomatic approach: the term "axiom" is used in a different way between these to: in the constructive approach the axioms tell us what construction steps are allowed, while in the axiomatic appreach the axioms tell us what properties the reals should satisfy. To avoid confusion, I will use "axiom" to refer to the axioms of ZFC, and "property" to refer to the axioms of the real numbers.
It is a bit of a perspective issue, see also my answer on this question.
We don't use the property of completeness (of the real numbers) in the construction of the real numbers; it is a property that the reals should satisfy, but not a tool that helps us build them. Instead, we construct the real numbers with Cauchy sequences, and then we prove using the axioms of ZFC that these constructed real numbers satisfy the property of completeness (and all other properties of the real numbers).
What this shows, is that the constructed real numbers form a model of the theory that is given by the properties of the real numbers. There exist several models for the real numbers; instead of using Cauchy sequences, we could define the reals using Dedekind cuts, or as a subset of the Surreal numbers. However, the nice thing about the properties of the real numbers, is that we can prove (in ZFC) that any two sets that satisfy the properties of the real numbers are isomorphic to each other. Therefore, up to isomorphism, we will always construct the same object.
This is different from, for example, the properties of a field, since we can construct several fields that are not isomorphic to each other (the rationals are not isomorphic to the reals, for example). The proper term for this is to say that the properties of the real numbers are categorical. We can therefore talk about THE real numbers, instead of SOME (set of) real numbers, since the method of construction does not really matter (up to isomorphism).