TITLE: What precisely is lost when considering proper classes rather than sets?
QUESTION [11 upvotes]: Motivated by concerns over the foundational issues vis-a-vis category theory. What is the essential useful characteristic of sets that is lost when instead considering proper classes? 
Referring to Hungerford's Algebra, it would seem that the notions of functions and surjection/injection carry over more or less identically. So what precisely is lost when we consider, for example, categories that are not locally small?
I understand totally the necessity of defining proper classes, that is to defined sets that would otherwise be self-referential in terms of membership.

REPLY [11 votes]: If you know a little set theory, there is a certain mental image that you can use to understand the difference between sets and classes.  Suppose that $\kappa$ is an inaccessible cardinal. Then we can make a model of set theory, called $V_k$, which consists of all sets whose rank is strictly less than $\kappa$. The subsets of this model are precisely the sets whose rank is strictly less than $\kappa+1$. The subsets of rank exactly $\kappa$ are proper classes from the point of view of $V_\kappa$. One can show that $V_k$ is a model of ZFC and $V_{\kappa + 1}$ is a model of Morse-Kelley set theory.
Now, if we have a set $a$ in this model (so $a \in V_\kappa$), we can do a lot of things to $a$ while staying in $V_\kappa$. We can take the powerset of $a$, and the powerset of that, and so on. We can even iterate the powerset $\alpha$ times for any ordinal $\alpha \in \kappa$, and we will stay in $V_\kappa$. 
This is not the case if we take a "proper class" $b$ of rank $\kappa$ in our model. We cannot take even one powerset, or do any other operation that increases the rank. So for example $\{V_{\kappa}\}$ itself, which has rank $\kappa + 1$. The answer by dfeuer has more examples. 
Now, the mental picture to have is that the entire universe of set theory is exactly like this, but viewed "from the inside" instead of being viewed from the outside as above. The class of ordinals is analogous to a large inaccessible cardinal $\Omega$. The objects of rank less than $\Omega$ are sets. The objects of rank exactly $\Omega$ (that is, the objects whose elements are all of rank $<\Omega$) are the proper classes. With sets, we can do many things and still have sets; with proper classes, we cannot do anything that increases the rank, because from our "inside" perspective the ordinals end at $\Omega$ (although we can easily imagine the ordinal $\Omega + 1$ and many more "small" extensions of $\Omega$). 
So the arguments that will go through for classes are the ones that do not rely on objects of larger rank. On the other hand, there are results for sets that necessarily use objects of much larger rank; the most famous of these if probably the Borel determinacy theorem.