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 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)