Timeline for Large cardinal axioms and Grothendieck universes
Current License: CC BY-SA 4.0
21 events
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| S Mar 5 at 12:48 | history | suggested | C7X | CC BY-SA 4.0 |
κ being limit of inaccessible cardinals alone would not be enough for V_κ |= ZFC, for example if κ were the least limit of inaccessible cardinals, V_κ fails replacement
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| Mar 5 at 10:19 | review | Suggested edits | |||
| S Mar 5 at 12:48 | |||||
| Jun 14, 2023 at 6:49 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
deleted 3 characters in body
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| Nov 23, 2018 at 10:32 | comment | added | Thomas Benjamin | @JoelDavidHamkins: When you write that "superhuge cardinals imply AU directly", do you mean all variants of superhuge (that is, the n-fold variants, if such exist for superhugeness)? Also, what is the definition of "implies AU directly", and does the fact that "none others from Kanamori's chart (except 0=1)" imply AU directly have any consequences for Zermelo-Grothendieck potentialism? | |
| Mar 12, 2014 at 6:39 | comment | added | Trevor Wilson | A minor correction: I believe that projective generic absoluteness is equiconsistent with infinitely many strong cardinals, not unboundedly many. | |
| Feb 4, 2010 at 19:17 | vote | accept | algori | ||
| Jan 24, 2010 at 4:22 | comment | added | Joel David Hamkins | @Mike: Yes, I agree. (Although in set theory we work inside all different kinds of various universes, and have to pay attention to what is absolute and what is not, so I think it would be possible to do it and get used to it.) But for convenience, carry on with AU, in my opinion. I look forward to the future category theoretic arguments that (need to employ) more of the large cardinal hierarchy. For exmple, I could imagine constructions that ask for a stationary tower of universes, which would be something like a Mahlo cardinal. | |
| Jan 24, 2010 at 4:05 | comment | added | Mike Shulman | @F.G.Dorais: Which "Wiki" do you mean? A wiki is a type of web site, not any particular one. | |
| Jan 24, 2010 at 4:02 | comment | added | Mike Shulman | Weakening universes to small models of ZFC (or fragments) is surely sufficient for most theorems of category theory in a logical sense, but with current technology it makes arguments and definitions much more complicated. E.g. if U models ZFC and Set is the category of U-sets, then "Set is complete" can't mean (if it is to be true) "every diagram in Set on a U-small category has a limit" but instead has to mean "every U-small diagram on a U-small category has a limit." I think most users of category theory would find keeping track of such distinctions tedious, at best. | |
| Jan 24, 2010 at 3:53 | comment | added | Joel David Hamkins | Yes, I agree. In fact, I mention AU in my book (on foricng and large cardinals) for precisely this reason. My book is still in progress (and MO gives me yet another excuse not to be done!) | |
| Jan 24, 2010 at 3:46 | comment | added | algori | Thanks, Joel. It's a pity Grothendieck universes are not mentioned explicitly in Kanamori's book (or any other book on set theory that I know), since this is one of the parts of set theory an average mathematician would be interested in. | |
| Jan 24, 2010 at 3:37 | comment | added | Joel David Hamkins | Superhuge and extendible cardinals also imply AU directly, but none others from Kanamori's chart (other than 0=1). | |
| Jan 24, 2010 at 3:31 | comment | added | Joel David Hamkins | Well, the AU itself would be regarded by set theorists as a large cardinal axiom, so that is the weakest one. It's just that whenever a set theorist states a large cardinal axiom, it goes without saying that you could also state as another axiom that there were two of them, or infinitely many or a proper class of them (or a stationary proper class of them). It's somewhat artificial to make the assertion of a proper class of them the fundamental axiom. But if you want axioms from Kanamori's chart, then the only that has a direct implication to AU seems to be Vopenka's principle. | |
| Jan 24, 2010 at 3:20 | comment | added | François G. Dorais | I don't know. Wiki seems to agree with you. They also state all axioms of ZF as "axiom of blah" instead of the "blah axiom." | |
| Jan 24, 2010 at 3:13 | comment | added | algori | Joel, thanks! You say that "few of the large cardinal axioms imply AU directly". But are there any that do? If so, which one is the weakest one? | |
| Jan 24, 2010 at 3:08 | comment | added | Joel David Hamkins | I thought it was usually called the Axiom of Universes. Is this wrong? | |
| Jan 24, 2010 at 3:02 | comment | added | François G. Dorais | Why AU? (French ordering?) | |
| Jan 24, 2010 at 2:52 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
typos
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| Jan 24, 2010 at 2:32 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
Changed GU to AU
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| Jan 24, 2010 at 2:24 | history | edited | Joel David Hamkins | CC BY-SA 2.5 |
added 110 characters in body
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| Jan 24, 2010 at 2:17 | history | answered | Joel David Hamkins | CC BY-SA 2.5 |