> > Although this reply contains much interesting info, it doesn't answer my main concern. [Totally my fault for not expressing it directly enough, in the first place.] My point was meant to be that definitions of groups, rings, fields, topological spaces etc. generally begin with something like "Let X be a set.." > Why? Why not say "Let X be a class..." It would be more general.
Often one studies all groups, or all groups of a certain kind. Are those collections classes?
> Perhaps it might lead to Russell-style paradoxes. But, if that is the risk, why is it ok for surreal numbers to form a proper class which isn't a set?
Because there is just one structure of surreal numbers, people do not speak of the class (or whatever) of all such structures or all such structures of a certain kind.
Confession: my claim is a guess.
I suspect that when an algebra text says "Let X be a set...", the author is not thinking in terms of a formal theory of sets or of sets and classes.
Also, I have seen "Fields" with a capital "F" for (e.g.) the field of surreal numbers.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting