Natural Numbers as the Least Inductive Subset of Real Numbers

Before stating the principle of mathematical induction formally, it helps to ask a more basic question: what actually characterises the set of natural numbers $N = \{1, 2, 3, \dots\}$ inside the much larger set of real numbers $\mathbb{R}$?

A subset that starts at $1$ and keeps stepping forward. Call a subset $S \subseteq \mathbb{R}$ inductive if it satisfies two closure conditions: (i) $1 \in S$, and (ii) whenever a real number $k$ belongs to $S$, the next number $k+1$ also belongs to $S$. Many subsets of $\mathbb{R}$ are inductive in this sense -- for instance $\mathbb{R}$ itself is inductive, and so is the interval $[1, \infty)$. But among all such inductive subsets of $\mathbb{R}$, the set of natural numbers $N$ is the smallest one: $N$ is precisely the intersection of every inductive subset of $\mathbb{R}$, or equivalently, $N$ contains $1$, contains $k+1$ whenever it contains $k$, and contains nothing else.

Why this matters for proofs. This least-inductive-subset description of $N$ is exactly what makes a two-step argument enough to establish a statement $P(n)$ for every natural number $n$. If a set $S = \{ n \in N : P(n) \text{ is true} \}$ can be shown to satisfy the two inductive closure conditions -- $1 \in S$ (i.e. $P(1)$ holds), and $k \in S \Rightarrow (k+1) \in S$ (i.e. $P(k)$ holds forces $P(k+1)$ to hold) -- then $S$ is itself an inductive subset of $\mathbb{R}$ contained in $N$. But $N$ is the smallest inductive subset, so $S$ cannot be a proper subset of $N$; it must equal all of $N$. In other words, $P(n)$ is true for every $n \in N$.

A step-ladder picture. It is often useful to picture the natural numbers as rungs of an infinitely tall ladder. If you know you can stand on rung $1$, and you know that whenever you are standing on any rung $k$ you can always climb to rung $k+1$, then you can conclude you can reach every rung, however high. This intuition -- a starting foothold, plus a reliable step that always carries you one rung further -- is precisely the content of the principle stated formally in the next section.