This is a question pertaining to the construction of a function on page $7$ of T. Chow's " A beginner's guide to forcing." http://timothychow.net/forcing.pdf
construct a function $F$ from the Cartesian product $\aleph^{M}_2\times \aleph_0$ into the set $2=\{0, 1\}$. [$\aleph^{M}_2$ plays the role of $\aleph_2$ in the model $M$.] We may interpret $F$ as a sequence of functions from $\aleph_0$ into $2$. Because $M$ is countable and transitive, so is $\aleph^{M}_2$; thus we can arrange for these functions to be pairwise distinct. Now if $F$ is already in $M$, then $M$ satisfies $\neg CH$! The reason is that functions from $\aleph_0$ into $2$ can be identified with subsets of $\aleph_0$ and $F$ shows us that the powerset of $\aleph_0$ in $M$ must be at least $\aleph_2$ in $M$.
How does this function $F$ work. I can see that a function from $\aleph_0$ into $2$ "emulates" (if that is the appropriate word) the powerset $P(\omega)$ or $2^{\aleph_0}$ (if this is correct). But how do things work with the Cartesian product $\aleph^{M}_2\times \aleph_0$, supposing $\aleph_0$ is an index. And (presumably when I understand that) how that results in the powerset of $\aleph_0$ in $M$ being at least $\aleph_2$ in $M$.
Thanks
EDIT Just saw this related question Forcing Question about a sequence of functions from $\aleph_{0}$ into $2 = \{0, 1\}$
A function $F:S\times \aleph_0\to\{0,1\}$ is essentially the same thing as a family of functions $F_s:\aleph_0\to\{0,1\}$, one for each element $s\in S$. Explicitly, for each $s\in S$, we define $F_s(n)=F(s,n)$.
So such a function $F$ gives us an element of $P(\omega)$ for each element of $S$. Assuming these elements are all different from each other, this proves that $P(\omega)$ has cardinality at least $|S|$. In particular, when $S$ is $\aleph_2$ (in our model), this means $2^{\aleph_0}\geq\aleph_2$ (in our model).