It is often said that unsuperstable theories do not admit a classification in the sense of Shelah. Why exactly is this so? And also what exactly does in the sense of Shelah mean? It is hand waved in a lot of places and it is hinted that the dividing line is having few models. 


However the only definition that I know of that works (See here for a definition that fails [Number of non-isomorphic models](https://mathoverflow.net/questions/210926/number-of-non-isomorphic-models)); having the maximal number of models would not exclude the possibility of a classification (see for example the definition in Hart, Hurshovski, Laskowski; <http://arxiv.org/pdf/math/0007199.pdf>
of classifiable, i.e. superstable, has prime models over pairs, and does not have the dimensional order property). 


So to re-iterate, what is "admits a classification in the sense of Shelah"? and why does it fail for unsuperstable theories? Is there no hope for recovering any sort of classification in this case? And is there any place where this is discussed in any depth?



Classification Theory
The idea behind "admits a classification in the sense of Shelah" is that the class of models of $T$ in a given cardinal $\aleph\_\alpha$ can be described by bounded many numercial invariance (generalizing the situation that a vector space is determined by its dimension). In the case of a countable first-order theory $T$, if $T$ admits classification then the number of isomorphism types of models of card $\aleph\_\alpha$ is bounded by $\beth\_{\omega\_1}(|\alpha|)$. When $\alpha$ is such that $\aleph\_\alpha$ is much bigger than $\alpha$ then $\beth\_{\omega\_1}(|\alpha|)<< 2^{\aleph\_\alpha}$.


Thus $I(\lambda,T)=2^\lambda$ for all uncountable lambda implies that $T$ is not classifiable.


