Cardinality natural numbers Is the cardinality of the natural numbers a natural number?
$|\mathbb{N}| \in \mathbb{N}\text{ or } |\mathbb{N}| \notin \mathbb{N}$, that is the question.
 A: $|\mathbb{N}| \notin \mathbb{N}$
$\mathbb{N}$ doesn't contain any kind of $\infty$.
Yes, $\mathbb{N}$ contains huge numbers, but not infinity.
A: Problem is that what "$\mathbb{N}$" and the "cardinality" is.
If you look at them with their definitions, the answer will be very easy.
In the set theory, natural numbers are defined like this:
$0 := \emptyset$
$1 := {0 \cup \left\{ 0 \right\}}$,
$2 := {1 \cup \left\{ 1 \right\}}$
$\dots$
and finally, the whole set of natural numbers $\mathbb{N}$ is the smallest "set" $X$ with the property "$\emptyset\in X$ and if $x\in X$ then $x\cup\left\{x\right\}\in X$" (we are assuming the existence of such sets in $\mathbf{ZFC}$). Notice that there is no largest element of $\mathbb{N}$ since every member of $\mathbb{N}$ has a successor in $\mathbb{N}$.
More generally, a set $\alpha$ is an ordinal number if $\alpha$ is strictly-well-ordered by the binary relation "$\in$". Ordinal Number (Wikipedia page) can read for more detailed information, especially for ordinal arithmetic.
For example, $\mathbb{N}+1:=\mathbb{N}\cup\{\mathbb{N}\}$ is also an ordinal.
Now, cardinality of a set $X$ is the smallest ordinal bijective to $X$, and an ordinal is also a cardinal if its cardinality is equal to itself. For example $\mathbb{N}$ is also a cardinal, and it is the smallest infinite cardinal. $\mathbb{N} + 1$ is not a cardinal. Cardinal Number (wikipedia page) can read for more detailed information.
If we look natural numbers as a set, we write $\mathbb{N}$. If we look them as an ordinal, we write $\omega$, and for cardinal, we write $\aleph_0$.
Then the answer is very easy, since no ordinal is an element of itself. 
