Timeline for Is it consistent with ZFC(A) for the Hartogs number of a proper class to be $\aleph_0$?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| May 8, 2023 at 17:57 | vote | accept | NoLongerBreathedIn | ||
| May 8, 2023 at 4:03 | comment | added | Bokai Yao | @NoLongerBreathedIn BTW, would you consider accepting my answer? | |
| May 8, 2023 at 4:00 | comment | added | Bokai Yao | @NoLongerBreathedIn I recently asked Asaf this question. Asaf told me that using class forcing and symmetric models we can basically make a proper class have any infinite Hartogs number. | |
| May 7, 2023 at 9:00 | comment | added | NoLongerBreathedIn | OK, so that leaves the question of Hartogs numbers of proper classes in ZF unclear. Oh well. | |
| May 6, 2023 at 15:29 | history | edited | Bokai Yao | CC BY-SA 4.0 |
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| May 5, 2023 at 4:38 | comment | added | Bokai Yao | @NoLongerBreathedIn Sorry, my previous comment was not correct and I deleted it. So the situation is this. When a proper class of urelements is allowed, Global Choice is weaker than Global Well-Ordering, which is in turn weaker than Limitation of Size (they are equivalent when there is only a set of urelements). With only a set of urelements and assuming AC for sets , no proper class can have a Hartog number because every ordinal injects into it by a standard ZFC argument as in here math.stackexchange.com/questions/2109874/…. | |
| May 4, 2023 at 23:30 | comment | added | NoLongerBreathedIn | Oh, that's a shame. So there's no model of this in ZF. Well, that answers all my questions on the topic thoroughly: the class of atoms can have any Hartogs number that's an infinite successor cardinal, and with replacement instead of collection it can also be an infinite limit cardinal. I'm glad to see someone else has asked the question. | |
| May 4, 2023 at 19:28 | comment | added | NoLongerBreathedIn | $\aleph_0$ is out because replacement implies collection in the absence of a proper class of urelements. | |
| May 4, 2023 at 19:21 | comment | added | NoLongerBreathedIn | Oh, cool! So global choice is weaker than limitation of size. This raises the question of whether the Hartogs number of a proper class can be $\aleph_1$ even without urelements. | |
| May 4, 2023 at 15:39 | comment | added | Bokai Yao | @JoelDavidHamkins I've edited my answer to address your comment. | |
| May 4, 2023 at 15:38 | history | edited | Bokai Yao | CC BY-SA 4.0 |
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| May 4, 2023 at 11:51 | comment | added | Joel David Hamkins | I think in your dissertation you also have the case of a proper class with Hartog number $\aleph_1$, and indeed this example has the urelement theory with collection and not merely replacement. Isn't it true that one can arrange such a situation with any given uncountable cardinal $\kappa$? (Or at least successor cardinals?) Can we have such classes for more than one $\kappa$ at a time, or perhaps even all $\kappa$? | |
| S May 4, 2023 at 3:55 | review | First answers | |||
| May 4, 2023 at 6:06 | |||||
| S May 4, 2023 at 3:55 | history | answered | Bokai Yao | CC BY-SA 4.0 |