In an assignment question I am asked to show that the cardinality of the set of all functions from $\mathbb{N}$ to $\mathbb{N}$ is equal to $2^{\aleph_0}$. To proceed with my proof I am trying to understand what exactly $2^{\aleph_0}$ is.
In my professor's lecture notes, he defines $2^A$ to be the set of all functions from $A$ to $\{0, 1\}$. Based on this, I am thinking that $2^{\aleph_0}$ is simply the set of functions from $\aleph_0$ to $\{0, 1\}$. However, I am confused because I am not sure how to think of $\aleph_0$--is it a set, a number, or something else entirely?
From what I understand, $\aleph_0$ is defined to be equal to the cardinality of $\mathbb{N}$ so I would think that $\aleph_0$ is just a number, but this seems naive to me--if $\aleph_0$ were a number, how could I define a map from it?
After doing some research on what $2^{\aleph_0}$ is I learned the astonishing fact that $|\mathbb{R}|=2^{\aleph_0}$ which has left me even more confused: If $|\mathbb{R}|=2^{\aleph_0}$ then isn't a $2^{\aleph_0}$ just a number? But if $2^{\aleph_0}$ is a number, how can it be a set containing functions (as per my professor's definition of $2^A$ where $A$ is a set).
I suspect that the significance of $\aleph_0$ transcends the meaning of $2^A$ in the context of “regular” sets because it is very special in set theory, in which case, what is $\aleph_0$ and is it possible to define a map from it?
When we think about $\aleph_0$, we really think about a set which represents all sets of cardinality $\aleph_0$. The point behind cardinal arithmetic is that it is invariant under bijections, so if $|A|=|B|$ then $|2^A|=|2^B|$.
So when you think about $\aleph_0$ you really have to choose a countably infinite set, and use it as the representative for all countable sets when it comes to cardinal arithmetic. It's just that sometimes it's easier to use one set, and at other time it's easier to use another set.
The canonical choice for a representative in the case of $\aleph_0$ is of course the natural numbers $\Bbb N$. So when we say that $|\Bbb R|=2^{\aleph_0}$ we say that there is a bijection between $\Bbb R$ and $2^\Bbb N$. And after you prove this, whenever you talk about the cardinal $2^{\aleph_0}$, you can think about the set $\Bbb R$ or the set $2^\Bbb N$ or so on.
(And this principle can, and is, extended to all cardinals. Not just $\aleph_0$.)