What does it mean for a number to be in a set? Frustratingly my book gives me several examples of a number in a set but offers no explanation at all.
Anyways what is going on here? According to the book $2$ is not an element of these sets:
$$\{\{2\},\{\{2\}\}\}$$
$$\{\{2\},\{2,\{2\}\}\}$$
$$\{\{\{2\}\}\}$$
What is going on? Clearly $2$ is in all of those sets. Or are they saying that $2$ isn't in any of these sets but a set is in all these sets and in that set is $2$? Which really seems like a logical fallacy because $2$ is in those sets contained in a set means the set has $2$ even if it is behind a layer of sets.
For example you wouldn't say that there are no cars in a neighborhood if all the cars in in a garage, so why does math take this approach?

You can think of 'is an element of' as stripping off a single layer of set braces.
$2$ is not an element of $\{\{2\}\}$, because removing one layer of braces, you get $\{2\}$, the set containing $2$, which is different from $2$.
Also, if this wasn't the case, then there would be no way to distinguish between, for example,
$$\{\{2\}, \{2,3\}\} \text{ and } \{\{2,3\}\},$$
which are clearly different sets, even though each of their elements only contain $2$s and $3$s.
