Linked Questions

91 votes
10 answers
14k views

Reflection principle vs universes

In category-theoretic discussions, there is often the temptation to look at the category of all abelian groups, or of all categories, etc., which quickly leads to the usual set-theoretic problems. ...
Peter Scholze's user avatar
32 votes
4 answers
6k views

Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in \...
David Roberts's user avatar
  • 33.8k
38 votes
2 answers
3k views

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 ...
user avatar
14 votes
1 answer
1k views

"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 ...
Gro-Tsen's user avatar
  • 29.9k
11 votes
1 answer
4k views

What is the use of Grothendieck universes in category theory?

First of all, I have to mention that I'm truly sorry if this question would seem inappropriate for this site for some people. Still, I think it is better to ask here rather on math.stackexchange. I ...
Jxt921's user avatar
  • 1,085
8 votes
1 answer
601 views

Abandoned LCAs on Cantor's Attic : Grand Reflection cardinals, universe cardinals, weak universe cardinals

Cantor's Attic is a really great website for the various descriptions of large finite numbers, large countable ordinals, and large cardinal axioms. However, after looking through the archives of the ...
Zetapology's user avatar
28 votes
0 answers
2k views

Is Feferman's unlimited category theory dead?

In 2013 Solomon Feferman in Foundations of unlimited category theory: what remains to be done (The Review of Symbolic Logic, 6 (2013) pp 6-15, link) laid out three desirable axioms for "...
ziggurism's user avatar
  • 1,436
3 votes
1 answer
578 views

Very weak notions of Universes in ZFC

Let us work in ZFC set theory. 1: We name "Very very weak universe (VVWU)" a set u such that if a and b are two member sets of u, then every function between a and b is also a member set of u; The ...
Gérard Lang's user avatar
  • 2,617
7 votes
1 answer
428 views

Can we have more malleable proper classes without sacrificing conservativity?

NBG is a conservative extension of ZFC that includes a concept of "proper class." Now I like the conservativity, since it means anytime I want to prove something in ZFC, I am free to work in NBG. ...
goblin GONE's user avatar
  • 3,693
6 votes
2 answers
364 views

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 ...
user267839's user avatar
  • 5,948
2 votes
1 answer
1k views

On the foundations for large categories

There are some basic discussions on the motivations of large categories and small categories: On the large cardinals foundations of categories, Large cardinal axioms and Grothendieck universes, Small ...
Tom's user avatar
  • 179