I have not dealt professionally with set theory, so excuse me if my way of formulating this question does not completely follow standard terminology. Actually, my question is about whether or not the following idea can be formalised:
The question is whether it makes sense to talk about a model for ZFC where the number of real numbers is maximal. I know that Paul Cohen has proved that there exist models for ZFC where $\Bbb R$ has an arbitrarily large cardinality. However, as Löwenheim–Skolem shows us, cardinality within a model is not the same as the cardinality of the number of elements in a model. So would it be possible to construct a model $M$ for ZFC in which, for any model $M'$ and a real number $r'$ in $M'$, there exists a real number $r$ in $M$ which has the same numerical value, i.e. the same digits?
The reason I would think this could actually be the case is this: We choose to construct the real numbers $\Bbb R$ as the set of maps $r\colon\Bbb Z\to\{0,1,2,\ldots, 9\}$ with $r(n) = 0$ for all sufficiently large $n$. Intuitively, $r_n:=r(n)$ is the $n$th digit of $r$, i.e. $r = r_N\ldots r_0.r_{-1} r_{-2}\ldots$ for some $N\ge 0$. Of course we shall have to apply an equivalence relation because, for instance, $0.999999\ldots$ must be equal to $1$. Now, this is where it goes vague, but what we do is to include any such sequence of numbers in $\{0,1,2,\ldots,9\}$. I mean literally that any such combination must correpsond to a well-defined map $\Bbb Z\to \{0,1,2,\ldots, 9\}$.
But is this idea of "any combination" possible to formalise in model theory? In this case, is my maximal model for $\Bbb R$ possible to construct formally?