Why does my topology textbook (Munkres) define positive integers as the intersection of all inductive subsets of the reals? This is how the topology textbook I'm reading (Munkres) defines integers:

A subset of the real numbers is "inductive" if it contains 1 and $1+x$ for all $x$ in the subset. The intersection of all inductive subsets of the reals is the set of positive integers.

Why take this route involving the intersection of so many sets? I could define the positive integers given reals as $1$ along with any sum of positive integers and get the same set much more easily.
 A: You appear to be asking why we can't simply define $\mathbb N$ as an inductive set, i.e. a subset of $\mathbb R$ satisfying the following two axioms:

*

*$1\in\mathbb N$

*$(\forall x)(x\in\mathbb N\to x+1\in\mathbb  N)$
The issue is that there are many sets which satisfy these two axioms. While the usual natural numbers are indeed an inductive set, so is:

*

*The set of positive real numbers.

*The set of positive real numbers unequal to $1/2$.

*The real numbers themselves!

Therefore, we need to add a third axiom to make our definition of $\mathbb N$ workable. One possibility, which is very similar in spirit to Munkres' definition, is the following:


*$\mathbb N$ is the "smallest" set satisfying (1) and (2), i.e. if $X\subseteq\mathbb R$ is inductive, then $\mathbb N\subseteq X$.

However, we might find it difficult to rigorously prove that there is a subset $\mathbb N$ of $\mathbb R$ which satisfies these three axioms. To get around this issue, we could instead use Munkres' definition of $\mathbb N$ as the intersection  of all inductive sets, before proving that $1\in \mathbb N$ and $(\forall x)(x\in\mathbb N\to x+1\in\mathbb  N)$. Then, it immediately follows from the definition of intersections that $\mathbb N$ satisfies (3).
This is in fact exercise 25 of chapter 2 of Spivak's Calculus (4th edition), page 34.
