What is a natural number?
We can define the real numbers axiomatically to be the unique (up to isomorphism) complete ordered field. In this setting, $(\mathbb{R}, +, \cdot, \le)$ must satisfy the properties
$(\mathbb{R},+,\cdot)$ is a field (i.e. both of the operations are commutative and associative, multiplication distributes over addition, every element has an additive inverse, and every nonzero element has a multiplicative inverse).
$(\mathbb{R},\le)$ is a totally ordered set (i.e. the relation $\le$ is reflexive, transitive, and antisymmetric, and for any $a,b\in\mathbb{R}$ with $a\ne b$, then exactly one of $a\le b$ or $b\le a$ holds).
The order is compatible with the field structure in the sense that $a \le b$ implies that $a+c \le b+c$ for all $c$, and if $0 \le a,b$, then $0 \le a\cdot b$. (We could actually deduce the properties of an order from these properties plus the comparability axiom above, but I personally find it easier to attack it this way.)
$(\mathbb{R},+,\cdot,\le)$ is complete, in the sense that if a nonempty subset of $\mathbb{R}$ has an upper bound, then it has a least upper bound.
In this context, we probably have to take the existence of the real numbers as an axiom (i.e. the existence of at least one totally ordered field), as well (thank you for pointing that out, DRF). Alternatively, see the addendum.
It is a bit of work to show that if $(X,+',\cdot',\le')$ satisfies these axioms, then $X$ is isomorphic to $\mathbb{R}$, but it can be done. In any case, we can (and often do) define the real numbers axiomatically in this manner.
Note that there is nothing in the above about natural numbers, rational numbers, Dedekind cuts, or Cauchy completeness. The real numbers are given to you on a silver platter. From this, it is then possible to define the naturals in (more or less) the manner described in the question.
Addendum: Alternatively, instead of declaring the existence of the reals by fiat (item 5, above), we could build them up from scratch via the usual process (i.e. build the integers as equivalence classes of naturals, then the rationals as equivalence classes of integers, then the reals as Dedekind cuts). This construction of the reals will satisfy the above axioms. Then, you can build the real natural numbers via the above definition. If you don't like having two different sets of objects called "natural numbers" running around, you can go ahead and show that the two definitions are isomorphic (i.e. indistinguishable in every way that matters).
That being said, this comes from a book on analysis, not axiomatic set theory. If you had to start from $\mathsf{ZFC}$, you could never get to the actual results of analysis in a reasonable page count. You have to build a foundation somewhere.
Suppose we did start with some notion of "natural number" which we used to construct a model of the real numbers.
Then even in this setting, the quoted definition is still not circular, because it's defining a new notion of "natural number" that will henceforth be used instead of the previous notion of "natural number".
We could give the new notion a different name, but there isn't really any point; the new version of "natural numbers" has an obvious isomorphism with the old version so it's not really any different from the old one in any essential way.
There are a number of reasons why an exposition of real analysis might construct the natural numbers from the real numbers; the two most prominent are:
- It is technically convenient to have the natural numbers be a subset of the real numbers
- It makes the exposition somewhat more agnostic about foundations; it simply needs the real numbers as a starting point
You have to start somewhere.
And in mathematics, you have to start from some axioms.
If you choose to start from the axioms for the real numbers, then using them you define the natural numbers exactly as you quoted, and using that that definition you can prove that the Peano axioms are true.
If instead you choose to start from the Peano axioms for the natural numbers, then using them you can define the real numbers, and using that definition you can prove that the axioms for the real numbers are true.
Either one of those provides an answer to your question. You may not like that answer, but that's how things go. What you cannot do is to avoid choosing your axioms.
Nonetheless, perhaps there is still a way to break this vicious loop?
Well, actually, there is, by making a different choice of axioms. You can, instead, choose the basic axioms for set theory, also known as the Zermelo-Frankel axioms or ZF for short.
If you choose to start from the ZF axioms, then using them you can define the natural numbers using the Von Neumann definition. Using that definition, you can then prove that the Peano axioms are true. And then, as already said, you can use the Peano axioms (which in this scheme are now theorems instead of axioms) to define the real numbers and use that definition to prove that the axioms for the real numbers are true.