An algebraic introduction to mathematical logic page 46 has the following:
the proof continues on, but it seems to me we can stop here. Every consistent theory has a model, and we've just proven $T'$ consistent, so $T'$ has a model; since $| P(V_0, R) |=\aleph$ the model is of Cardinal $\aleph$. qed.
What am I missing?
here is the rest of the proof:
I managed to take a look at the book and if I understand correctly, $P(V_0,\mathscr R)$ is the set of formulae with variables in $V_0$ and relation symbols in $\mathscr R$. From the point of view of the theory $\mathscr T'$, it is the set of sentences (i.e. formulae without free variables) expressible in the language of $\mathscr T'$.
So your claim is that a model of a theory must be of the cardinality of the sentences expressible in the underlying language. This is false! Otherwise, every group would be countable for example...
The only thing you can say is that a model of $\mathscr T'$ has cardinality greater than $\aleph$. Indeed, the theory $\mathscr T'$ stipulates that the interpretations of the elements of $V_0 - C$ are pairwise distincts, and there is $\aleph$ such elements. But the model can be significantly larger!
What the rest of the proof do is actually constructing a model that is not signifcantly larger. Morally, the problem is that a model can offer numerous witnesses for a theorem of the form $\exists x q(x)$. The construction of the proof inductively add just one witness $t_q^{(n)}$ for each such formula and the axiom that goes with it $q(t_q^{(n)})$ (saying that it is indead a witness for $q$). But then, by enlarging the language and the theory, we might have created new existential theorems which now await for a witness, so we do it again, and again, and again... until we did it $\aleph_0$ number of times.
I must agree with the comment of Alex Kruckman: this does not seem to be a good book to learn first-order logic. It certainly is interesting to read afterwards to get a glance at a purely algebraic treatment of the syntax (free-ich of the usual inductive definitions), but it is probably a bad idea to approach logic for the first time with this point of view.
And if you want to learn some model theory, just toss the book away: it doesn't even introduce elementary embedding...
I would recommend Model theory: an introduction by David Marker to begin with. And when you are more confortable with the subject, you can pass to the marvelous book of Bruno Poizat A course in model theory (if you read french, the original version is a lot of fun).