Eilenberg and Mac Lane formally defined categories in their 1945 paper General Theory of Natural Equivalences. Their definition of a category starts as follows:
"A category {A,a} is an aggregate of abstract elements A (for example, groups), called the objects of the category and etc."
When they consider their first example of categories on P. 239, namely the category of all sets, they immediately remark: "This category obviously leads to paradoxes of set theory."
Obviously, all sets collected together don't form a set. However, their definition does not require the objects of a category to form a set. They use the term "aggregate" in their definition, perhaps to avoid this particular issue. That is, a set can be an aggregate, a class (in the sense of NBG) can also be an aggregate. Since the "aggregate" of all sets is a class, I do not see the "paradox" that they say will arise from considering the category of all sets. Unless by "aggregate" they really meant set. So my question is: what is the paradox that they were referring to?
Please note, my question is specific to Eilenberg and Mac Lane's comment about the category of all sets. Obviously, there are other paradoxes caused by their definition of categories, and allowing the notion of a class doesn't eliminate all set-theoretic paradoxes from category theory. But I am not asking about these topics.
On the other hand, as pointed out on page 245 of R. Kromer's book (Tool and Object): "Eilenberg and Mac Lane use the term 'set' in the combination the set of all objects of [a] category, on page 238 of their paper."
Is this an evidence that by "aggregate" they really meant "set"?
Best Answer
I interpret the question as not being about the problems of size in category theory in general and how to deal with them (which are fairly well-understood and the subject of other questions on this site), but about what Eilenberg and MacLane actually meant in their original paper. The phrasing of that particular footnote is sloppy, but I think section 6 of their paper ("Foundations") suggests that what they meant is that "this category would lead to paradoxes if we required the objects of a category to form a set rather than something like a proper class".
My guess is that they used the word "aggregate" in the definition in section 2 as a nod to the fact that to be formal, one may want to take these to be proper classes (or something related), but assumed that the average mathematician reading the paper would interpret "aggregate" as "set" at least until they got to section 6. So they added a footnote pointing out that they were aware of the issue, but deferred a fuller discussion of it (and an explanation of what "aggregate" can formally be defined to mean, or other ways one can deal with the problem while still interpreting "aggregate" as "set") to the later section. For instance, in section 6 they wrote "we have chosen to adopt the intuitive standpoint, leaving the reader free to insert whatever type of logical foundation (or absence thereof) he may prefer".