TITLE: Example of set of cardinality $\aleph_2$
QUESTION [5 upvotes]: I am looking for an example of a set of cardinality $\aleph_2$, such as the continuum is an example for cardinality $\aleph_1$.

REPLY [13 votes]: If you think of "the continuum as an example for cardinality $\aleph_1$" then I'm guessing that you (or your professor) are (at least implicitly) assuming the so-called Generalized Continuum Hypothesis. Nothing wrong with that; the GCH is a perfectly cromulent foundation for mathematics, even if the set-theory experts around here are more interested in various exotic alternatives.
In that case, the set of all functions $f:\mathbb R\to\mathbb R$ is a good example for cardinality $\aleph_2$. However, it is important to realize that we are considering arbitrary functions. If you only count continuous functions, there are only $\aleph_1$ of those. So most of those $\aleph_2$ functions are very ragged, and their graphs do not look like anything you would call a "curve".
You can also get $\aleph_2$ as the number of functions $f:\mathbb R\to\{0,1\}$ which is the same as the number of subsets of $\mathbb R$, since a subset $X$ can be coded by a function $f$ such that $f(x)=1$ if $x\in X$ and $f(x)=0$ if $x\notin X$.