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This could be a big post, so I'll try to summarize my thoughts and divide them into several questions.

When working in category theory, I used to choose the following definition. A category $C$ is the data of a set $Ob(C)$ and for each $X,Y \in Ob(C)$ a set $Hom(X,Y)$, such that etc ... This definition quickly leads to introduce the concept of Grothendieck universe. Because the "collection of all sets" is not a set, one can't define the category $Set$ in this context. Given a fixed universe $U$ it is however possible to define the category $U-Set$ whose objects are sets belonging to $U$. Similarly we work with $U-Gr$ for groups whose underlying set belongs to $U$. Nevertheless, one can be unsatisfied with this formalism because it is a bit "frustating" to not work with (for example) all groups. One way to avoid Grothendieck universes is to switch to Kelley-Morse set theory. In this context we allow $Ob(C)$ and $Mor(C)$ to be proper classes. It is now perfectly allowed to deal with the category of all sets or all groups but one should be constantly careful because intuitive theorems about sets are false for proper classes.

Actually, neither of these two theories seems to be perfect and they both focus on the size of the objects. Informally Kelley-Morse theory makes a distinction between small collections and big collections, and Grothendieck universes allow to stratify sets on several level, also depending on their size. This leads me to think about the victory of "limitation by size" in our current methods to do mathematics. When I write "victory", I just want to say that most mathematicians work with ZFC wich is the most famous example of "limitation by size"-theory. After doing some research (included MO related questions) I haven't been able to highlight the fundamental reason of this victory. I know there were other attempts to deal with the fundational crisis of mathematics such as NF of Quine or positivism and, if the failure of NF seems pretty clear to me, I can't say the same about positivism wich is philosophically very rich. Actually I've found plenty philosophical books about positivism but not really a pure mathematical book. The only think I could perceive is that this theory seems to be "too strong" for sets theorists.

I realize that this subject could be a philosophical question but I think it could be interesting to understand why size is such "an obsession" for mathematicians. Maybe this is related to physics where size is of course a natural way to control and study objects but one can legally think that abstract mathematics should be over these considerations. Now I'll try to formulate several questions, please feel free to answer at only one of them.

For category theorists

Question 1 : Why do you choose to work with Grothendieck universes or Kelley-Morse proper classes ? Is there any successfull third point of view ?

For sets theorists

Question 2 : Is there any good mathematical reference about positivism ? And more generally about theories wich failed against the "limitation by size" concept. Any mathematical history document explaining the deep reasons of this victory would be much appreciated.

Question 3 (maybe the most important for me) Are there current research works in set theory about a new theory mainly based on an other concept that limitation by size ?

I hope this text will be readable for you (I'm not native english speaker) and I thank you for any explanation or reference.

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    $\begingroup$ You seem to think that you have to go to Kelley-Morse set theory to refer to the class of all groups, but of course one can refer to that class even in ZFC, since it is after all a definable class. In particular, one can handle it satisfactorily even in ZFC, or in Goedel-Bernays set theory, which handles proper classes as objects, without going all the way to Kelley-Morse set theory. $\endgroup$
    – Joel David Hamkins
    Feb 9, 2016 at 1:30
  • $\begingroup$ One can avoid material set theory and use a foundation like in Bénabou's article Fibered Categories and the Foundations of Naive Category Theory (J. Symbolic Logic Volume 50, Issue 1 (1985), 10-37; artscidirectory.case.edu/wp-content/uploads/2013/07/…), in which one chooses a base topos E, then works with that as the 'category of sets', and large categories are categories over E. You can ignore, and I recommend it, the appendix (section 12). You can let the base topos satisfy one of a number of conditions analogous to Replacement, eg autology, if that bothers you $\endgroup$
    – David Roberts ♦
    Feb 9, 2016 at 3:52
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    $\begingroup$ @DavidRoberts but there is still a size distinction: the small categories are the ones internal to E and the large ones are the fibrations over E. $\endgroup$
    – Mike Shulman
    Feb 9, 2016 at 5:55
  • $\begingroup$ @MikeShulman Sure, there's always going to be a distinction in size, even if one is assuming a hierarchy of universes. I was just mentioning that one doesn't have to assume a universe or use material class-set theory to get one's hands on large/small categories. $\endgroup$
    – David Roberts ♦
    Feb 9, 2016 at 7:51
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    $\begingroup$ @C.Dubussy no, not really, but then most category theorists (in my experience) don't think too hard about foundational size issues, since they can be solved in any number of ways, and all of them are pretty much equivalent. There are of course very interesting questions one can ask, and people do study them, but day-to-day it's not an issue. $\endgroup$
    – David Roberts ♦
    Feb 9, 2016 at 22:44

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