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.

Why is the word associative used to represent the concept of the associative property?

What is the difference between a supremum and maximum; and also between the infimum and minimum?

Log of a negative number

How do I find the diameters of the circles in this geometry puzzle?

What is Validity and Satisfiability in a propositional statement?

Given a data set, how do you do a sinusoidal regression on paper? What are the equations, algorithms?

Proof of L'Hospitals Rule

Integral $\int_0^\infty \log(1-e^{-a x})\cos (bx)\, dx=\frac{a}{2b^2}-\frac{\pi}{2b}\coth \frac{\pi b}{a}$

What does this (double absolute value like) notation mean?

Evaluating the improper integral $\int_0^\infty \frac{x\cos x-\sin x}{x^3} \cos(\frac{x}{2}) \mathrm dx $

Find $ \lim\limits_{{n \to \infty}} \frac1{2^n} \sum\limits_{k=1}^n \frac1{\sqrt{k}} \binom nk$