Timeline for Can Z + Ranks + Successor cardinals + Ordinal inaccessibility be equal to ZF?
Current License: CC BY-SA 4.0
18 events
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May 2, 2021 at 19:45 | comment | added | Farmer S | Okay, I see now, I was confused about the definition "transitive set of transitive sets that are $\in$-wellfounded". I see now that under ZF, all transitive sets of transitive sets, are in fact ordinals. So my comment saying "under ZF all sets are $\in$-wellfounded, but not in general $\in$-wellordered", is irrelevant, sorry! | |
May 1, 2021 at 9:06 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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May 1, 2021 at 7:49 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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Apr 30, 2021 at 19:30 | answer | added | Greg Kirmayer | timeline score: 5 | |
Apr 30, 2021 at 5:57 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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Apr 30, 2021 at 0:37 | history | became hot network question | |||
Apr 29, 2021 at 22:09 | vote | accept | Zuhair Al-Johar | ||
Apr 29, 2021 at 22:07 | comment | added | Zuhair Al-Johar | @FarmerS, well here in this theory Foundation is not an axiom, since it is not an axiom of Z. So, we need to define ordinals in a way that makes them built-in $\in$-well founded, in order to get Foundation as a theorem (from the Ranks axiom). | |
Apr 29, 2021 at 22:05 | comment | added | Farmer S | Hmm, under ZF (in particular, Foundation), all sets are $\in$-wellfounded, but not in general $\in$-wellordered. | |
Apr 29, 2021 at 22:02 | answer | added | Farmer S | timeline score: 7 | |
Apr 29, 2021 at 21:59 | comment | added | Zuhair Al-Johar | @FarmerS, I mean a transitive set whose elements are strictly well ordered by $\in$. An this is the usual official definition of von Neumann ordinals, and it is also equivalent to the one I gave in my prior comment. | |
Apr 29, 2021 at 21:57 | comment | added | Farmer S | Do you mean "transitive set whose elements are strictly linearly ordered by $\in$"? | |
Apr 29, 2021 at 21:54 | comment | added | Zuhair Al-Johar | @FarmerS, thanks for spotting the typo. About ordinals, those are the usual von Neumann ordinals, i.e. transitive sets of transitive sets, that are $\in$-well founded. | |
Apr 29, 2021 at 21:51 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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Apr 29, 2021 at 21:37 | comment | added | Farmer S | Can you specify exactly what you mean by ordinal in this theory? And I think there is a typo in the definition of ordinal inaccessibility; is $\lambda=\gamma$? | |
Apr 29, 2021 at 20:30 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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Apr 29, 2021 at 16:56 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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Apr 29, 2021 at 16:33 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |