It is well known that it is not possible to define the "set of all sets" in standard ZF set theory without contradiction, as it breaks the axiom of foundation. However, I find this to be surprisingly limiting in many contexts.
For example, when dealing with the study of groups, I often feel tempted to consider the set of all groups under some constraint, for example, to try to apply something like Zorn's Lemma to this set, before realizing that the set is too big to be a group, since most groups can act over essentially any set.
I find this urge to consider "the set of all sets" crops up often for me, and is very tempting.
I feel as if I do not hear much about other mathematicians running into these kinds of limitations. But at the same time, hear whispers of those who have transcended these limitations by the definition of "Universes" like that of the Grothendieck Universe.
There are many candidates for "universes", such as Gödel's constructible universe, Von Neumann Universe, classes and the class of all sets, the aforementioned Grothendieck Universe, type theoretical universes, or just a plain universe (just fixing a set $X$ and working within that set). And I imagine there are others that I don't know about. But I see very little discussion of these outside the context of either foundational mathematics or, in the special case of Grothendieck Universe, Algebraic Geometry (which I do not understand).
So when I am writing proofs and I get that insatiable itch to define "the set of all sets" and work inside that universe, or its union, I have no idea how to work with these universes or to choose the right one. I have no reasonable intuition about how "big" set theory needs to be for my current domain, but I get the feeling that it does have a "size" in many cases, not in the sense of cardinality of course, but in the more vague, harder to pin-down sense of "the number of allowable sets is much bigger than the number of sets I need to consider right now, and if we could remove the unnecessary sets, things could be easier"
In other words, even though I feel like the ability to say "let $A \in U$ be any set", or "apply Zorns lemma on $A \subset U$" where $U$ is a universe seems very useful. But I feel woefully unequipped to even consider such an idea, because when thought about too much it is clear that contradictions pop up. E.g. $\mathcal{P}(U)$ would break the idea of $U$ being the "universe", so we can't exactly "construct" $A$.
Tl;dr
The study of different kinds of "universes" seem to be
- Restricted to foundational mathematics
- Scattered across many different sub-fields of foundational mathematics
But I would like to learn more about how to work within them in practice.
Are there examples or references that go more into depth into one or all of these different kinds of universes, their limitations, and techniques for how to use them in other fields like geometry, algebra, topology, analysis, or number theory, or am I thinking about this all wrong?