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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