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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 11th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexIs this article a duplicate of [[cumulative hierarchy]]? <a href="https://ncatlab.org/nlab/revision/diff/von+Neumann+hierarchy/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/von+Neumann+hierarchy/4">v4</a>, <a href="https://ncatlab.org/nlab/show/von+Neumann+hierarchy">current</a>

   Is this article a duplicate of cumulative hierarchy?

   diff, v4, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 11th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexBumping this thread as it was lost in a bunch of messages about deleted articles. It seems to me that the entire content of this article is contained in the article [[cumulative hierarchy]], except for the name “von Neumann hierarchy”. I suggest deleting it.

   Bumping this thread as it was lost in a bunch of messages about deleted articles.

   It seems to me that the entire content of this article is contained in the article cumulative hierarchy, except for the name “von Neumann hierarchy”.

   I suggest deleting it.

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 12th 2025
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   Author: DavidRoberts
   Format: MarkdownItexI think cumulative hierarchy could be more general, because you have different kinds: $V_\alpha$, $H_\alpha$, $L_\alpha$ (though the last of these of course gets its own special name, and don't give, except under V=L, all sets) and perhaps more. These have abstract characterisation as being a indexing by ORD using a small function (in the sense of AST) plus perhaps other nice properties.

   I think cumulative hierarchy could be more general, because you have different kinds: , , (though the last of these of course gets its own special name, and don’t give, except under V=L, all sets) and perhaps more. These have abstract characterisation as being a indexing by ORD using a small function (in the sense of AST) plus perhaps other nice properties.

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 13th 2025
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   Author: Dmitri Pavlov
   Format: MarkdownItexRe #3: However, the current version of the article talks about “the” cumulative hierarchy, never mentioning any other possibilities other than the von Neumann hierarchy.

   Re #3: However, the current version of the article talks about “the” cumulative hierarchy, never mentioning any other possibilities other than the von Neumann hierarchy.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJun 14th 2025
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   Author: Todd_Trimble
   Format: MarkdownItexI took it upon myself to change one of the "the"s in [[cumulative hierarchy]] to an "a", in the section on algebraic set theory (which I had a hand in writing some years ago). There was another use of "a" earlier in the article. Although I don't have a particular dog in this race, I think I agree with David that "cumulative hierarchy" could be used in a more general way, and maybe [[cumulative hierarchy]] ought to reflect that better. To me, "[von Neumann hierarchy](https://en.wikipedia.org/wiki/Von_Neumann_universe)" has a certain ring to it that is tied to [[pure sets]] (without ur-elements), a specific conception or mental picture of what pure sets actually "are", with realist or Platonist concomitants. The Philosophical Persectives section in the linked Wikipedia article touches on this. With that in mind, maybe [[von Neumann hierarchy]] could be seen as an example of a "concept with an attitude", with certain sociological overtones. (David and Mike Shulman would be more in touch with that sort of thing than I am. At the moment, I'm fairly agnostic.)

   I took it upon myself to change one of the “the”s in cumulative hierarchy to an “a”, in the section on algebraic set theory (which I had a hand in writing some years ago). There was another use of “a” earlier in the article.

   Although I don’t have a particular dog in this race, I think I agree with David that “cumulative hierarchy” could be used in a more general way, and maybe cumulative hierarchy ought to reflect that better. To me, “von Neumann hierarchy” has a certain ring to it that is tied to pure sets (without ur-elements), a specific conception or mental picture of what pure sets actually “are”, with realist or Platonist concomitants. The Philosophical Persectives section in the linked Wikipedia article touches on this. With that in mind, maybe von Neumann hierarchy could be seen as an example of a “concept with an attitude”, with certain sociological overtones.

   (David and Mike Shulman would be more in touch with that sort of thing than I am. At the moment, I’m fairly agnostic.)