What is a natural number? According to page 25 of the book A First Course in Real Analysis, an inductive set is a set of real numbers such that $0$ is in the set and for every real number $x$ in the set, $x + 1$ is also in the set and a natural number is a real number that every inductive set contains. The problem with that definition is that it is circular because the real numbers are constructed from the natural numbers.

Defining the natural numbers (to within isomorphism) as a distinguished subset of a complete ordered field (COF) seems mathematically slick, but is unsatisfactory in two ways. First, definition should go from the simple to the complex and from the elementary to the advanced—not the other way round. Second, in order for the definition of a COF, and hence of the distinguished subset, to make sense, it must be proved that (A) any two COFs are isomorphic and (B) a COF exists. Let us grant A (which is not exactly trivial) and look at B. The simplest way to establish B is to construct $\Bbb R$ from $\Bbb Q$ via Dedekind cuts or equivalence classes of Cauchy sequences in $\Bbb Q$. But then what is $\Bbb Q$? The simplest way to define it is in terms of equivalence classes of pairs of elements of $\Bbb Z$. And $\Bbb Z$? Why, it is just a set of equivalence classes of pairs of elements of $\Bbb N$. Thus we have come full circle.
Now, it is true that all this set-theoretic structure is unwanted baggage when our aim is to get on and do real analysis, in which we would like to have simply $\Bbb N\subset\Bbb Z\subset\Bbb Q\subset\Bbb R\subset\Bbb C$ (the last one for good measure). But this is not really a problem. Having established existence from the bottom up, we can identify an isomorphic copy of each more basic structure within each more encompassing one, and agree henceforth to use these copies (in $\Bbb C$, say) rather than the originals, for convenience.
