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I recently learned that some parts of SGA require axioms beyond ZFC. I am just a simple algebraic geometer so I am trying to understand how can this fact impact my life (you may have engaged in a similar game with the axiom of choice at some point of your mathematical career). I think this was only needed in the volume developing cohomology of topoi, correct me if I am wrong.

The question: is there some statement about schemes not involving the word "topoi" that you know how prove using the additional axioms, but do not know how to prove without them? I believe this question has nothing to do with the notion of completeness in mathematical logic (because I am not asking if something can be proved in principle, just if there is an obvious argument). If there is some logical argument showing that the answer is negative, I would like to learn about that too.

Bonus points if the proof of the statement in your answer relies on a Weil cohomology theory, rather than something random (I do not know if this is even possible). To be more precise, a Weil cohomology theory is only defined for smooth projective schemes over a field, for which all of this should probably be irrelevant, so I mean that the cohomology theory you use restricts to a Weil cohomology theory for smooth projective schemes over a field.

P.S. There was similar discussion in the context of Fermat's theorem, and if I understand correctly, the user BCnrd insists that universes are useless for etale cohomology without giving a reference. I believe there are few people on this planet who know etale cohomology better than BCnrd so that's some useful information. The question is not, however, limited to etale cohomology.

It's inherently difficult to give a negative answer to a question like this, but here's a technical fact that pushes in that direction:

Let ZFC$_n$ be the subtheory of ZFC gotten by restricting Separation and Replacement to $Sigma_n$ formulas. By the reflection principle,$^1$ for each $n$ the theory ZFC proves that there is an ordinal $alpha_n$ such that $V_alpha_nmodels$ ZFC$_n$. That is: $$mboxFor each $ninmathbbN$, ZFC proves Con(ZFC$_n$).$$

We can think of the $V_alpha_n$s as "approximate universes" which behave like universes for all "sufficiently simple" formulas, the point being that if you specify a complexity level ahead of time you can always assume you have an approximate universe appropriate to that complexity level.

Now the compactness theorem now naively suggests that - since we can only ever use finitely many sentences in a given proof - any argument with universes whatsoever can be replaced with one involving just approximate universes, and hence a proof in ZFC. This is of course false, but counterexamples have to be "global" as opposed to "local" - they need to at some point refer to the whole of the universe in question as a single completed object.

For exampe, the way ZFC + universes proves the consistency of ZFC is by showing that a universe $U$ is a model of ZFC. The statement "$Umodels$ ZFC" is expressed in the language of set theory by talking about Skolem functions over $U$ (or something morally equivalent), and this takes place in the context of the powerset of $U$. But this sort of thing isn't to my knowledge how universes are applied in algebraic geometry - they instead use a universe to argue that a "sufficiently closed" object exists in that universe, and this "local" argument is exactly the sort of thing that the reflection principle tells us can generally be reduced to ZFC alone.

Basically, a candidate example needs to not just take place inside a universe, but rather over a universe.

That said, there is an obvious place to look for such: arguments using two (or $n$) universes. The larger universe does see the smaller universe as a completed object, so the coarse heuristic above suggests that we can replace only the larger universe with an approximate universe - that is, that arguments which are quickly phrased in terms of two universes can be directly translated to arguments involving only one universe. Now we can't cheat anymore - we nee actual arguments about algebraic geometry. My understanding is that we're still in a situation where universes are an unnecessary convenience, but now I'm far outside of my area of competence. Still, the above should give an indication of why a real essential use of universes in a concrete result (which will certainly only involve reference to a small fragment of the cumulative hierarchy) would be very surprising.

$^1$OK fine, the reflection principle is usually phrased for finite subtheories of ZFC. But $(i)$ that's not really any different as far as the heuristic is concerned, just more annoying to work with; and $(ii)$ the stronger version of reflection I've stated is also true (the point being that for each $n$, the schemes of $Sigma_n$-Separation and -Replacement can be expressed in the language of set theory by a single sentence, which in turn can be proved from finitely many of the ZFC axioms which we can bash with the usual reflection hammer).

And on that note, it's worth pointing out two facts about reflection which help flesh out the picture:

• First, given that ZFC proves the compactness theorem, we seem to be in tension with Godel's incompleteness theorem. What saves us is that "$forall n$" and "ZFC proves" don't commute (unless ZFC is inconsistent of course): while ZFC does prove each specific instance of reflection, it can't prove the full version (unless, again, it's inconsistent).




• It's also worth noting that a similar result holds for (first-order) Peano arithmetic (as does the analogous version of the previous bulletpoint), although of course we need to talk about mere consistent Henkinized complete theories as opposed to canonical-ish models. As a cute consequence, Kripke used this fact to give a purely model-theoretic proof of Godel's incompleteness theorem (in the absence of reflection, his argument would require the soundness of PA, similarly to how Godel's original argument assumed $omega$-consistency rather than mere consistency).