Let M be a countable transitive model for ZFC, **P** is a partial order in M. Notions like "partial orders" and "dense" are absolute. Consider the following set
$S$={$D\in M: D$ is dense in $P$} = {$D: D$ is dense in $P$}$^M$, the superscript notion denotes relativization. The remark is the set is usually not countable in $M$. (Note: since $M$ is countable, from $V$, the class of all sets, anything lies in $M$ should be countable with respect to $V$). I know from Skolem's Paradox, countability is not absolute. However, if considering the following function:
f: $S$ $\rightarrow$ $\omega$ and f is in $V$ and f is injective. Since $f\in P(S\times \omega)$, and $S, \omega \in M$, therefore, by the fact that $M$ is a transitive model of ZFC $f \in M$. The only conclusion I can draw at this point is when relativized to $M, f$ is not one-to-one. However, I feel confused and could not see how this is the case. I am asking for somewhat better example illustrating the non-absoluteness of notion of countability.

By the way, the question originates from Chapter VII Section 2 on Kunen's Set Theory (1980).