What sets does $\mathbb{N}$ include?

Question

My text states that the set $\{1, 2, 3...\}$, and the set $\{101, 102, 103, 104...\}$ are elements of $\mathbb{N}$.

Doesn't this imply that $\mathbb{N}=\{1, 2, 3... 101, 102, 103, 104...\{1, 2, 3 \}, \{101, 102, 103, 104\}...\}$ ?

I had presumed that only numbers comprised the set of natural numbers (i.e. the set of natural numbers did not include other sets). Evidently, that's false.

What does $\mathbb{N}$ include, and why must it include what it does?

---- Edited--- I suspect I've misinterpreted the text, so I've inset a clip of it below Thank you,

-Hal

Answer 1

Copying from my comment, the line

then $S \in \mathbb{N}$.

should read

then $S = \mathbb{N}$.

It's clear that this is a typo because the following paragraph states that the only inductive set containing 1 is $\mathbb{N}$, which reiterates that $S = \mathbb{N}$. Unfortunately, this typo is pretty bad because it invalidates the statement of the principle of mathematical induction.