Is aleph-$0$ a natural number? Would I be right in saying that $\aleph_0 \in \mathbb N$? 

Or would it be a wrong thing to do?
$\aleph_0$ is not a natural number. It is the cardinality of the set of natural numbers - each individual natural number is finite, but the set of all natural numbers is infinite.



Based on the comments below, let me share an anecdote: I was once part of a medical survey on Things Neurological. One of the tasks they had us do was come up with short definitions of common words, on the spot. Beforehand, I (talk about tempting fate) joked about how easy this would be. The very first word they gave me: "Number." For the life of me, I couldn't come up with anything. The doctors gave me very weird looks.  Finally I sputtered something like "An element of a formal system," which didn't help with the weird looks but did let us move on to other, simpler words. 

The Moral: Man, number is weird.
No.


1 is a natural number.
A number made by taking a natural number and adding 1 to it is a natural number.
There are no other natural numbers than the ones whose existence is implied by 1 and 2.


$\aleph_0$ is not equal to 1. Therefore it can only be a natural number if there is some natural number $x$ such that $x+1 = \aleph_0$. If you think it is a natural number, you have to be able to say what $x$ is.

In fact, there is no such $x$.