Is aleph-$0$ a natural number?
$\aleph_0$ 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.