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.
Best Answer
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:
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:
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:
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.