I'm currently going through the introductory chapters of Munkres' Topology, and in chapter 1 section 4 of the second edition, Munkres attempts to briefly establish some Mathematical foundations for the study of Topology. In particular, he assumes a set of axioms for $\mathbb{R}$ (i.e. the so-called Field axioms), and from them he is able to obtain the set $\mathbb{Z}$ of integers. He does so as follows:
A subset $A$ of real numbers is inductive if it contains the number $1$ (whose existence is given by one of the field axioms) and if for every $x \in A$, $x+1$ is also in $A$.
Then, letting $\mathcal{A}$ be the collection of all inductive subsets of $\mathbb{R}$, he defines $\mathbb{Z}_+$ as:
$$ \mathbb{Z}_+ = \bigcap_{A \in \mathcal{A}} A$$
Intuitively this makes sense since if $A$ is inductive, $1$ is necessarily in $A$ (by definition) and so is $1+1 = 2$ and $2+1 = 3$ and so on.
However, for a novice such as myself, this begs the question, why didn't we just define $\mathbb{Z}_+$ as $\{1, 2, 3, \ldots \}$ in the first place (indeed, many texts introducing elementary set theory do just this). What am I missing here?
Best Answer
In Volume 2 of Ewald's "From Kant to Hilbert" you can find a paper on formalism written by Hilbert. He is very clear that constructive accounts of mathematics are deprecated in formalism. Things like Dedekind cuts and Cauchy sequences are representable in formalist real numbers. They just do not have any significance in the sense of logical or constructive priority.
If you read Munkres with this in mind, there is no "cheating" and it is perfectly understandable. He introduces the real numbers as those objects being discussed. He associates the membership relation with extensions obtained by comprehension of properties. And, he applies this account of how he is using logic to the domain of objects under discussion -- the reals as given under his fomalist axiomatization.
To obtain the positive integers, he defines a class of classes -- the class of classes satisfying his account of an inductive class. He then quantifies over this class in order to take the intersection yielding the positive integers.
So, this is like "third-order" operation, or something like that.
The hint that makes this somewhat apparent is in the exercises. In one exercise he points out that it is not possible to generate Russell's paradox with the approach he has taken.
My copy of Munkres is pretty old (1975?). But I would bet that particular exercise can be found in later editions.