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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)? -
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182:(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
331:
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
949:
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
914:
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.
907:
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?)
167:
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
162:
Using
Internet Explorer 6.0 (like approximately 90% of web users), symbols such as &isin are not displayed correctly (I assume a membership symbol was intended). If there is a means by which this significant problem can be cured, readers of this (and similar) pages should be informed.
207:
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
903:
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:
932:
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...).
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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
256:
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.
908:
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.
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572:. 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
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275:". 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.
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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.
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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
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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.
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The article was unclear to the point of verging on error. Strong inaccessibility is true, but not obvious. (U) =: -->
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I changed the fourth point with an equivalent but more elegant formulation, in the context of set theory.
307:(C), fortunately, is not too hard. You should try reading Bourbaki's article. It's really not that bad.
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You have to prove (U) entails (C) for any cardinal, taking a strongly inaccessible cardinal is not enough.
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Actually, it doesn't matter in this particular case, but it should be specified for clarity anyway.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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satisfies the 4 properties as they are now, but not the previous version of point 4:
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The anon above is right when he says, "See
Bourbaki's article". The definition is
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You're right. I've undone the anonymous change that made the article incorrect.
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Doesn't the definition of Grothendieck universe include some formulation of the
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Oops, I see how it isn't quite; I missed the fact that the union of the
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Shouldn't this read "c(U) is either zero, or strongly inaccessible?"
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Context for the reader to clarify the significance of Inaccessibles
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existence of a strongly inaccessible cardinal cannot be proved.--
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not see any philosophical implications of the equivalence. --
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All this is implicit in the article, but not made explicit.
717:. I suggest the last change on the fourth point be reverted.
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Explanation for "expert-subject" template: Replacement axiom
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is taken. But I still believe replacement is implied: Let
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Then for any universe U, c(U) is strongly inaccessible "
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What does a "family of sets" means here: set or class?
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863:{\displaystyle \bigcup _{\alpha \in I}x_{\alpha }}
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645:{\displaystyle \{\{F(\alpha )\}\}_{\alpha \in I}}
474:{\displaystyle \{\{F(\alpha )\}\}_{\alpha \in I}}
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186:. Any U satisfying this definition is either V
808:{\displaystyle \{x_{\alpha }\}_{\alpha \in I}}
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109:and see a list of open tasks.
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500:{\displaystyle \alpha \in I}
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139:on the
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