I'm trying to understand this definition. Actually I have seen a lot of questions that describes this topic from the set theoretical point of view, but I would to ask it again from the point of view of humble real analysis.
The stuff is taken from Stromberg on p. 14.
Notice that $\mathbb{R}$ is an inductive set and so is $\lbrace t\in \mathbb{R}: t\geq1 \rbrace$ (this is clear). Let $\Theta$ denote the family of all inductive sets of $\mathbb{R}$ and let $\mathbb{N} = \bigcap \Theta$.
I think the questions is, what is the proof that $\mathbb{N} = \bigcap \Theta$? If it's too involved, is there an intuitive argument why this is so?
I suppose that you agree if $M\in\Theta$ (that is, if $M$ is inductive), then $\mathbb N\subset M$. Therefore, $\mathbb N\subset\bigcap_{M\in\Theta}M$.
On the other hand, since $\mathbb N\in\Theta$, $\mathbb N\supset\bigcap_{M\in\Theta}M$. But the only subset of $\mathbb N$ which is inductive is $\mathbb N$ itself (this follows from the induction principle). And it should be clear that $\bigcap_{M\in\Theta}M$ itself is inductive. So, $\bigcap_{M\in\Theta}M=\mathbb N$