Taking a proper class as a model for Set Theory When I am reading through higher Set Theory books I am frequently met with statements such as '$V$ is a model of ZFC' or '$L$ is a model of ZFC' where $V$ is the Von Neumann Universe, and $L$ the Constructible Universe. For instance, in Jech's 'Set Theory' pg 176, in order to prove the consistency of the Axiom of Choice with ZF, he constructs $L$ and shows that it models the ZF axioms plus AC. 
However isn't this strictly inaccurate as $V$ and $L$ are proper classes? For instance, by this very method we might as well take it as a $Theorem$ in ZFC that ZFC is consistent since $V$ models ZFC. However this is obviously impossible as ZFC cannot prove its own consistency. I highly doubt that Jech would make a mistake in such classic textbook, so I must be missing something.
How could we, for instance, show Con(ZF) $\implies$ Con(ZF + AC) without invoking the use of proper classes? I imagine, for instance, that we would start with some (set sized) model $M$ of ZFC and apply some sort of 'constructible universe' construction to $M$. 
 A: What is shown in the cases you mention is not that the model is a model of ZFC, made as a single statement, but rather the scheme of statements that the model satisfies every individual axiom of ZFC, as a separate statement for each axiom. 
The difference is between asserting "$L$ is a model of ZFC" and the scheme of statements "$L$ satisfies $\phi$" for every axiom $\phi$ of ZFC. 
This difference means that from the scheme, you cannot deduce Con(ZFC).
For the proof that Con(ZF) implies Con(ZFC), one assumes Con(ZF), and so there is a set model $M$ of ZF. The $L$ of this model, which is a class in $M$ but a set for us in the meta-theory, is a model of ZFC, since it satisfies every individual axiom of ZFC. So we've got a model of ZFC, and thus Con(ZFC).
A: Yes, that is true. But note that in its nature statements like $\operatorname{Con}(T)$ are meta-theoretic statements. So when we say that $V$ is a model of $\sf ZF$, we mean that in the meta-theory it is a model of $\sf ZF$.
This is often something which is not stressed enough in introductions to $V$ and relative consistency results: when we prove that $L$ is a model of $\sf ZFC$, we do not "just prove a meta-theoretic result", we in fact prove a stronger statement:

There is a formula $L$ in the language of set theory which defines a class that is provably transitive and contains all the ordinals, and for every axiom $\varphi$ of $\sf ZFC$, $\sf ZF\vdash\varphi^\it L$.

So not only you have this model, but in fact $\sf ZF$ itself prove that each axiom of $\sf ZFC$ holds in $L$.

Let me also share, in my first course on axiomatic set theory, which was given by the late Mati Rubin, we had proved that $\sf ZF-Reg$ and $\sf ZF$ are equiconsistent by practically proving that $\sf PRA$ proves that if there is a contradiction in $\sf ZF$, then there is one in $\sf ZF-Reg$.
Of course, the same can be done with $\sf ZF$ and $\sf ZFC$. And it is much more annoying than using the model theoretic approach. Sometimes with impunity when it comes to class models.
A: The class $V$ of all sets is not a model of $ZFC$, because it is a proper class, not a set.
A model of $ZFC$ is a set (or small class) $U\in V$ which satisfies all the axioms of $ZFC$ when these axioms are restricted or relativized to $U$, even though $U$ does not include all the sets of the $ZFC$ universe.
If a model $U$ of $ZFC$ exists within the universe $V$ of $ZFC$, then its cardinality is "inaccessible" with respect to the universe $V$. Conversely, if an inaccessible cardinal exists in the universe $V$ of $ZFC$, then a small class, or set $U\in V$ exists, which is a model of $ZFC$.
The existence of a model $U$ within the universe $V$ of $ZFC$ implies that $ZFC$ is consistent. $ZFC$ would be inconsistent if its axioms could prove the existence of a model of itself within itself.
I fear that the meta-theorems, meta-theory, and other meta-language are unnecesssary and ill-defined, unless in the universe $V$ of $ZFC$ we are speaking of the properties of a possible model $U, U\in V, U\subsetneqq V$ where the original axioms of $ZFC$ have been restricted to a $U$.
