Talk:Grothendieck universe

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It's said here that every Grothendieck universe has the cardinality of a strongly inaccessible cardinal, and that, therefore, the existence of Grothendieck universes can't be proved within ZF (if ZF is consistent). However, both the empty set and the set of all hereditarily finite sets would be examples of Grothendieck universes, according to the formulation of this article, and, of course, can both be proven to exist in ZF. Indeed, the article seems to acknowledge at one point the possibility that a universe might be empty. Is this business about Grothendieck universes having strongly inaccessible cardinalities made under some additional assumption (such as, for example, that the universe contains an infinite set)? -Chinju 19:44, 18 April 2006 (UTC)

The definition explicitly says that U is a non-empty set. So you are wrong about the empty set satisfying the definition. As the definition stands, you are correct that V_ω (the set of hereditarily finite sets) would satisfy the definition. I do not have access to the original definition of a Grothendieck universe, so I do not know whether that is a mistake (leaving out the axiom of infinity) or whether the error is in the statement about strong inaccessibles. But there are no other problems asside from V_ω. Any U satisfying this definition is either V_ω or V_κ for a strong inaccessible κ; and any such V_κ is a Grothendieck universe. JRSpriggs 07:07, 19 April 2006 (UTC)

Ah, whoops, I just looked at the four numbered properties and missed the opening bit about U necessarily being non-empty. (I was also kinda misled by the sentence starting "In particular, it follows from the last axiom that if U is non-empty..."). Thanks for clearing things up about V_ω. -Chinju 18:53, 19 April 2006 (UTC)

My guess is that V_ω is excluded by hand, in the same way that $\aleph _{0}$ is excluded from being an inaccessible cardinal (it satisfies the two defining features). If anyone has access to the proper literature and can confirm this nuance of the definition, please update the article thus. --expensivehat 22:47, 26 May 2006 (UTC)

See Bourbaki's article. Firstly, "L'ensemble vide est un univers noté U₀". Secondly, he defines a strongly inaccessible cardinal as a regular strong limit cardinal, with no uncountability assumption. The article has been made consistent with modern usage. 141.211.62.20 20:36, 31 May 2006 (UTC)

Omissions

1. It is not clear from the article why the cardinality of a Grothendieck universe must be strongly inaccessible. Perhaps it is obvious but there is no explicit statement about that.
2. You have to prove (U) entails (C) for any cardinal, taking a strongly inaccessible cardinal is not enough.

Yecril 11:06, 30 March 2007 (UTC)

The article was unclear to the point of verging on error. Strong inaccessibility is true, but not obvious. (U) => (C), fortunately, is not too hard. You should try reading Bourbaki's article. It's really not that bad. 141.211.120.175 21:42, 6 June 2007 (UTC)

The other way round

The text says:

Since the existence of strongly inaccessible cardinals cannot be proved from the axioms of Zermelo-Fraenkel set theory, the existence of universes other than the empty set and Vω cannot be proved from Zermelo-Fraenkel set theory either.

It is exactly the other way round. By Gödel's 2nd incompleteness theorem, the consistence of Zermelo-Fraenkel set theory (and therefore the existence of a model) cannot be proved from Zermelo-Fraenkel set theory. Now, every strongly inaccessible cardinal would give rise to such a model. Therefore the existence of a strongly inaccessible cardinal cannot be proved.--91.23.239.166 (talk) 11:38, 22 December 2007 (UTC)

Explanation for "expert-subject" template: Replacement axiom

Doesn't the definition of Grothendieck universe include some formulation of the Axiom schema of replacement too? Functor salad (talk) 12:47, 13 April 2008 (UTC)

Isn't that precisely what property (4) is? Benja (talk) 20:06, 5 May 2008 (UTC)

Oops, I see how it isn't quite; I missed the fact that the union of the $x_{\alpha }$ is taken. But I still believe replacement is implied: Let I be an element of U, and let F be a function from I to U; then consider the family $\{\{F(\alpha )\}\}_{\alpha \in I}$, i.e., the family that assigns to each $\alpha \in I$ the singleton $\{F(\alpha )\}$. Then the union of these singletons is the image of F under I, right? Benja (talk) 20:16, 5 May 2008 (UTC)

The anon above is right when he says, "See Bourbaki's article". The definition is the very first statement in the article. Click the light blue link marked "Univers" in the first reference and you find the exact and precise and complete and total definition of a Grothendieck universe. The article is correct. So is Benja when he says that replacement is implied. Ozob (talk) 21:14, 29 July 2008 (UTC)

@Benja: How do you know that $\{\{F(\alpha )\}\}_{\alpha \in I}$ is in U ? Short of having property 4 as it was before, or equivalently that U is closed under Axiom schema of replacement, Grothendieck universes would be nothing more than models of Zermelo set theory. V_ω·2 satisfies the 4 properties as they are now, but not the previous version of point 4: $\bigcup _{n\in \omega }$V_ω+n = V_ω·2. I suggest the last change on the fourth point be reverted. Bruno.barras (talk) 18:14, 26 January 2012 (UTC)

You're right. I've undone the anonymous change that made the article incorrect. Ozob (talk) 00:10, 27 January 2012 (UTC)

Fourth point changed

If $\{x_{\alpha }\}_{\alpha \in I}$ is a family of elements of U, and if I is an element of U, then the union $\bigcup _{\alpha \in I}x_{\alpha }$ is an element of U.

I changed the fourth point with an equivalent but more elegant formulation, in the context of set theory. — Preceding unsigned comment added by 213.156.35.244 (talk) 14:33, 20 December 2011 (UTC)

Context for the reader to clarify the significance of Inaccessibles

In the section on Inaccessibles the presentation jumps right into the formal proof that a G-universe is equivalent to the existence of an Inaccessible. The narrative leaps over but forgets to explicitly mention, for the benefit of the earnest but naive reader, this key fact:

You cannot prove that a G-universe exists within the framework of ZF. So in other words the key concept of G-universes is that they are a larger set-theoretic universe than ZF. This already has philosophical implications for the foundations of math; since it's a practical fact that most (much?) modern math is conducted in a framework larger than ZF. So the hope of the early 20-th century set theorists that ZF is the best and proper foundation for the rest of math, is already beginning to crumble.

All this is implicit in the article, but not made explicit.

I'm happy to leave it to others whether this point should be explicitly included in order to help anchor the reader in the larger context of why G-universes are important in the first place. And that G-universes extend conventional set theory in a way that leaves 20th century foundations behind.

71.198.226.61 (talk) 01:19, 2 January 2013 (UTC)

I think this is a very good point for the article to make. Why don't you add it? You seem to know set theory; you'd be a better person than I would to write about this distinction (I'm an algebraic geometer, here only because of SGA...). Ozob (talk) 03:05, 7 January 2013 (UTC)

What has the existence of inaccessibles to do with this “key fact”? To prove the unprovability of the existence of G-universes in ZFC you do not need inaccessible cardinals, you just need Gödel’s incompleteness theorems. Most books using G-universes do not mention inaccessible cardinals at all. I do not see any philosophical implications of the equivalence. --Chricho ∀ (talk) 23:00, 10 January 2013 (UTC)

Couldn't c(U) be zero?

The article says:

" $\mathbf {c} (U)=\sup _{x\in U}|x|$

...

Then for any universe U, c(U) is strongly inaccessible "

Shouldn't this read "c(U) is either zero, or strongly inaccessible?"

Bayle Shanks (talk) 19:58, 18 September 2013 (UTC)

Yes, it's a holdover from when the article assumed that zero was strongly inaccessible (per Grothendieck, but clashing with modern terminology). I've tried to fix it. Ozob (talk) 23:21, 18 September 2013 (UTC)

What does a "family of sets" means here: set or class?

Actually, it doesn't matter in this particular case, but it should be specified for clarity anyway. — Preceding unsigned comment added by Georgydunaev (talk • contribs) 10:00, 5 August 2020 (UTC)