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I seek a name for an operation I will call for the moment "recursive union". It yields the union of a set of sets, and all the sets from those sets, ..."all the way down".

E.g., applying this operation to a Zermelo ordinal yields the equivalent Von Neumann ordinal.

(I assume for the moment that all the sets involved are well founded, e.g., no infinite descending chains of containment. As of course are all ordinals, being well ordered. But I intend to consider the contrary case, later.)

I am sure I have read something in which this operation was used, although I think in another context, and I can't recall what. And I think another name for the operation was used, but I can't recall that.

Is there an established term that names this operation?

RobPratt
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Karl Botts
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    Are you thinking of the transitive closure? – user14111 Apr 22 '25 at 02:29
  • Sounds like Grothendiek universes, as well. – Malady Apr 22 '25 at 02:37
  • This seems relevant to the example, that is, the part of the question about ordinals: https://math.stackexchange.com/questions/2975907/why-do-we-use-von-neumann-ordinals-and-not-zermelo-ordinals – Karl Botts Apr 22 '25 at 04:46
  • It's not Grothendieck Universe, because that adds powersets and pairs that are not in the recursive union. – Karl Botts Apr 22 '25 at 06:07
  • I think, tentatively, transitive closure is right. That is probably what I was thinking of. I'm pretty sure for ordinals it is exactly right. But I want to think for a couple of days to look for counterexamples, where recursive union and transitive closure are not the same. If I can't think of any, I will come back and accept the transitive closure answer. – Karl Botts Apr 22 '25 at 06:10

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This is the transitive closure. See also this Metamath proof.

user76284
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