I am trying to get a feel for what kinds of sets are added by forcing. I apologize in advance, this question might be hard for me to put precisely into words. Let's give a very simple example:
Let $\mathbb{P}$ be the set of all finite partial functions $p:\omega\to 2$, ordering by reverse inclusion. If $G$ is a generic for $\mathbb{P}$ over $V$, then $F:=\bigcup G$ is the characteristic function for a cofinal subset of $\omega$, or alternatively a total function with an unbounded number of $0$'s and $1$'s.
In particular this set $X$ was not in the ground model. Let's say the ground model is the universe I happen to live in. So $F$ is distinct from any $\omega$-length sequence of 0's and 1's I could come up with! This is where I'm fuzzy on things (for example "come up with" is vague and probably the critical issue), can anyone give me some intuition here to go on?
This question is pretty much the heart of forcing, but is also somewhat difficult to answer. I'll stick with examples. (If this is insufficient, leave comments and I'll attempt to improve.)
First of all, after adding $G$ (and thus $F$), since the extension also satisfies $\mathsf{ZF(C)}$, all sets definable from $F$ (and elements of the ground model $V$) are added. Trivial examples are $\{ F \}$, $\{ \{ F \} \}$, $\langle F , \{ F \} \rangle$, etc. The sets $F^{-1} [ \{ 0 \} ]$ and $F^{-1} [ \{ 1 \} ]$ are added. For any $g \in {^\omega}2 \cap V$ you are going to add $F + g$ (pointwise addition modulo $2$). The set of all such reals (functions) will also be added. (This last set is added because $V$ is a definable class in $V[G]$, so one can speak of $\{ F+g : g \in {^\omega}2 \cap V \}$ in the extension.)
Note that if any of these sets happened to belong to $V$, then one could use that set to show that $F$ is also an element of $V$. ($F$ is the unique element of $\{F\}$. $F$ is the characteristic function of $F^{-1} [ \{ 1 \} ]$. $F$ is $(F+g)-g$ for any $g \in {^\omega}2 \cap V$.) This shows that these sets are necessarily added.
For nontrivial examples of sets added when adding a Cohen real, things get a little more complicated, and requires a finer analysis of the forcing to suss out. I'll just give a couple of examples of types of sets added:
In
it is proven that that adding a Cohen real also adds a subspace of ${^{\omega_1}}2$ with a particular property (it is a strong L-space).
In
it is shown that adding a Cohen real adds a Souslin tree.
(Neither of the above objects can exist assuming $\mathsf{MA} + \neg \mathsf{CH}$, so when you add one Cohen real to a model of $\mathsf{MA} + \neg \mathsf{CH}$ you destroy Martin's Axiom.)