TITLE: Why is something not a field if it's a proper class?
QUESTION [11 upvotes]: Why is it a convention to say that, for example, the nimbers and surreal numbers aren't fields because they aren't sets? Is it just pedantry?

REPLY [4 votes]: Yes, it's pedantry, but of the sort that's necessary for avoiding wrong conclusions.
If you call the surreal numbers a field -- let's call them $\mathbf S$ -- then you would expect to be able to do the things with them that you can do with fields in general. And sometimes those theorems are proved using the assumption that the field is indeed a set, and they don't necessarily hold for the "large" fields you propose.
For example you would expect to be able to form the vector space $\mathbf S^\omega$ of finitely supported sequences of surreal numbers -- so far so good; we can certainly speak about functions with domain $\omega$ whose values are surreal numbers and finitely many nonzero; each of theses is even a set and $\mathbf S^\omega$ becomes a proper class itself.
Now, how about the dual space of $\mathbf S^\omega$? Things begin to creak here, because officially that's supposed to be the collection of all linear functionals on $\mathbf S^\omega$, and since $\mathbf S^\omega$ is proper-class big, each linear functional is a proper class, and then $(\mathbf S^\omega)^*$ is not even a proper class.
Fortunately, the the usual argument will go through that alinear functional of $\mathbf S^{\omega}$ always corresponds to an element of $\mathbf S^{\mathbb N}$, and those things are sets, so $\mathbf S^{\mathbb N}$ is a proper class.
However, now things break down completely. $\mathbf S^\omega$ is certainly infinite-dimensional, and for an infinite-dimensional vector space that is a set, it is easy to prove that $V$ is not isomorphic to $V^*$. But this proof proceeds by counting the elements of the two spaces ...
Edit: Bzzt. No it doesn't. Counting dimensions is enough. Example dies.

Thus we need to keep careful track of which properties of our structures hold in general, and which ones are only for "small" structures whose collections of elements are sets. The "large" structures will generally satisfy fewer properties than the "small" ones will.
In some cases we do indeed do so while developing the theory -- most significantly in category theory and when speaking about models of set theory. In these particular cases, the benefit of being able to speak about "large" structures justify developing a theory of them.
However, for most kinds of structure -- such as fields or semirings -- the need to speak about large instances has not been so great that people usually bother about keeping score about what can be said about them.