Number of non-isomorphic models Let $C$ be the class of cardinals. Define by recursion $C_0 = C$, $C_\alpha = C_\beta\cup P(C_\beta)$ if $\alpha=\beta+1$ and $C_\alpha = \bigcup_{\beta<\alpha}{C_\beta}$ for limit $\alpha$ (Here $P(C_\beta)$ is the class of subsets of $C_\beta$).
We say that a complete theory $T$ has an invariant system of rank $\alpha$ iff there is some (class) function $f$ associating to each model of $T$ an element of $C_\alpha$ such that given models $A, B$ of $T$; $f(A)=f(B)$ iff A is isomorphic to B. And a theory $T$ is classifiable iff it has a invariant system of some rank $\alpha$.
I recently read that having $2^\lambda$ models for each uncountable cardinal $\lambda$ meant that you could not have an invariant system of the above style. I would like to know the reason why. 
I know that classification theory is very complicated and technical, I would be quite happy with an intuitive answer that doesn't have all the details for this question.
 A: The idea here is that we want to classify models of $T$ up to isomorphism by some "bounded amount of dimension information". 
The simplest classification scheme is when the models of $T$ are determined by a single cardinal dimension. This is the case for the theory of $K$-vector spaces, for example, or more generally for any uncountably categorical theory. Every $K$-vector space has a dimension, which is some cardinal number, and two $K$-vector spaces have the same dimension if and only if they're isomorphic.
Most theories don't admit such a simple classification, but sometimes we can still classify their models, not just a single cardinal, but by a set of cardinals, or maybe a set of sets of cardinals, or a set of sets of sets of cardinals, etc. But as long as you can bound the complexity of the information you need (for example, no matter how large the size of the models we're considering, we can still classify each of them by a set of sets of cardinals - an invariant system of rank $2$) then you have some understanding of what all the possible models look like.
However, I have to admit that I don't entirely understand the framework that Marcja and Toffalori are using here. I particular, I don't understand their assertion that "some cardinal and ordinal computations exclude that such a $T$ [with $2^\lambda$ models of size $\lambda$ for all uncountable $\lambda$] has an invariant system of any possible rank".
There are a lot of cardinals, and it doesn't seem obviously inconsistent to me that there could be a theory $T$ with the maximal number of models ($2^\lambda$) in all uncountable $\lambda$ with an invariant system of rank $0$. 
In practice, though, classification schemes like this usually come with a bound on the cardinals involved, and bounds like this will easily imply that the theory doesn't have the maximal number of models in all uncountable $\lambda$. For example, in the case of uncountably categorical theories, the dimension assigned to a model of size $\lambda$ must be at most $\lambda$. Maybe Marcja and Toffalori meant to include some bounding information like this in the definition.
