Timeline for Can we have a spectrum of intermediate choice properties between set choice and global choice?
Current License: CC BY-SA 4.0
13 events
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| Oct 9, 2022 at 19:35 | comment | added | Zuhair Al-Johar | @ElliotGlazer, Thanks! | |
| Oct 9, 2022 at 19:27 | comment | added | Elliot Glazer | @ZuhairAl-Johar Probably a lot, though I haven't thought about this sort of question in a while. You might find this paper relevant since it examines similar questions in the ZF setting: math.bu.edu/people/aki/7.pdf | |
| Oct 9, 2022 at 19:23 | comment | added | Elliot Glazer | In fact, local choice isn't even needed: just over NBG, global choice fails iff $|Ord| < |Ord \cup \bigcup_{\alpha} \{\text{well-orderings of } V_{\alpha} \}| < |V|.$ | |
| Oct 9, 2022 at 19:23 | comment | added | Zuhair Al-Johar | @ElliotGlazer, what's the lower bound on the cardinality of strictly intermediate cardinalities between $|Ord|$ and $|V|$ | |
| Oct 9, 2022 at 19:08 | comment | added | Elliot Glazer | @DmytroTaranovsky The first is inconsistent: under NBG + local choice, if $|Ord| < |V|,$ then $Ord \cup \bigcup_{\alpha} \{\text{well-orderings of } V_{\alpha}\}$ is of strictly intermediate cardinality. | |
| Oct 9, 2022 at 8:42 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Oct 9, 2022 at 8:09 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Oct 9, 2022 at 7:56 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Oct 9, 2022 at 7:30 | comment | added | Zuhair Al-Johar | @DmytroTaranovsky, OK! Nice suggestion. The idea is the effect of those possibilities on choice over classes, I think we can have those versions, but by then can those be considered as various degrees of choice above set choice but below global choice, do they have any known consequences in theorization of set\class theory. | |
| Oct 9, 2022 at 1:05 | comment | added | Dmytro Taranovsky | As it stands, it is not quite clear what you are asking. I suggest rephrasing as follows. Is Morse-Kelley set theory without global choice (but with choice for sets) consistent with $|V|$ being the least cardinal above $|On|$? What about existence of $|V|$-many (or even $2^{|V|}$-many) proper classes, each of different cardinality? Similar question for NBG without global choice. | |
| Oct 8, 2022 at 6:53 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Oct 8, 2022 at 6:29 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Oct 7, 2022 at 20:51 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |