Linked Questions
16 questions linked to/from Reflection principle vs universes
65
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When size matters in category theory for the working mathematician
I think a related question might be this (Set-Theoretic Issues/Categories).
There are many ways in which you can avoid set theoretical paradoxes in dealing with category theory (see for instance ...
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52
votes
2
answers
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What interesting/nontrivial results in Algebraic geometry require the existence of universes?
Brian Conrad indicated a while ago that many of the results proven in AG using universes can be proven without them by being very careful (link). I'm wondering if there are any results in AG that ...
- 20k
35
votes
6
answers
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Is there a metamathematical $V$?
As with many of you, I've been following Peter Scholze's recent question about universes with great interest. In ring theory, we don't often have to deal with proper classes, but they occasionally ...
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45
votes
3
answers
4k
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Do Grothendieck universes matter for an algebraic geometer?
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 ...
user138661
25
votes
2
answers
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Foundations and contradictions of Scholze's work: the category of presentable infinity categories contains itself
Preface: I am not an expert in the work of Scholze, or anything for that matter.
Question
Has Scholze stated what axioms he is using to develop his theory of motives and analytic geometry. In the ...
- 485
17
votes
2
answers
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Why do we care about small sets?
I have been a user of category theory for a long time. I recently started studying a rigorous treatment of categories within ZFC+U. Then I become suspecting the effect of the smallness of sets.
We ...
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12
votes
4
answers
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On the large cardinals foundations of categories
(This question was posted on math.SE over two weeks ago, but received no answer. I am therefore posting it here as well.)
It is well-known that there are difficulties in developing basic category ...
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22
votes
1
answer
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What is known about the relationship between Fermat's last theorem and Peano Arithmetic?
As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem?
In particular, what is known about the arithmetic systems $PA + \...
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13
votes
3
answers
1k
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Elementary theory of the category of groupoids?
One axiomatisation of set theory, the Elementary Theory of the Category of Sets, or ETCS for short, comes from category theory and states that sets and functions form a locally cartesian-closed, ...
user173426
2
votes
2
answers
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Downgrading from ZFC with universes to ZFC
Is the following correct?
If something is provable for small sets in ZFC with Axiom of Universes ("For any set x, there exists a universe U such that x∈U.") then it is provable for any sets in ZFC (...
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14
votes
1
answer
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"Largish" cardinals
In what follows, $\mathsf{ZCKP}$ refers to the subset of $\mathsf{ZFC}$ consisting of the axioms of Zermelo set theory with choice and foundation ($\mathsf{ZC}$) plus those of Kripke-Platek set theory ...
- 41k
10
votes
1
answer
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Practical Benefits of HTT/univalent foundations for assisted proofs
I'm trying to understand what the claimed practical benefits of HTT/univalent foundations are for doing computer assisted proofs and while I've seen a lot of claims of benefits they all seem to be ...
- 4,079
5
votes
2
answers
486
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Size issues in localization $\mathcal{C}[\mathcal{W}^{-1}]$ category
When one starts with a locally small category $\mathcal{C}$ and wants to localize it at an appropriate choosen collection of morphisms $\mathcal{W}$, then in general one faces some size issues in the ...
- 4,246
11
votes
1
answer
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Case study: what does it take to formulate and prove Quillen's small object argument in ZFC?
I'm getting a bit lost over at Peter Scholze's interesting question about removing the dependence on universes from theorems in category theory. In particular, I'm being forced to admit that I don't ...
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11
votes
0
answers
480
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Feferman's universes for proof assistants?
This question was prompted by a discussion from another MO question about the consistency of ZFC. There are some mathematicians who are comfortable with ZFC but uneasy with large cardinals. For them,...
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