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Is Zermelo set theory sufficient to prove the existence of the transitive closure of any set $X$?

$TC(X)=\{X, \bigcup X, \bigcup\bigcup X, \ldots \}$

user21820
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Sapiens
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1 Answers1

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No, it does not.

There are two common forms of $\mathsf{Z}$: with or without the axiom of regularity (or foundation). I'll call these "$\mathsf{Z}_+$" and "$\mathsf{Z_-}$" respectively.

Re: the version with regularity, which in my experience is the more common presentation these days (despite the wikipedia page!), see the results of Jensen, Schroder, and Boffa cited in the beginning of Mathias' paper on slim models.

Re: the regularity-free version, Esser and Hinnion showed that even the stronger system $\mathsf{Z_-+AFA+WREP}$ fails to prove the existence of transitive closures, where $\mathsf{AFA}$ is Aczel's antifoundation axiom and $\mathsf{WREP}$ is a weak form of replacement (Theorem $2.12$).

Noah Schweber
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  • But $KP^\omega$ suffices, right? – Sapiens Sep 20 '21 at 02:06
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    @Sapiens Yes, that's right. – Noah Schweber Sep 20 '21 at 02:12
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    I always found Azcel's AFA to be somehow a natural extension of Fnd. It still says that there is a unique solution to each equation; it just says that more equations can be solved, so the universe is still pretty rigid. The big mess starts with "let's have a lot of solutions to the same equations" and you end up with a proper class of atoms or whatever. – Asaf Karagila Sep 20 '21 at 10:12
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    Mathias' paper on slim models cites the results of Jensen, Schroder, and Boffa, but does not elaborate on them. Could someone with access to the papers of Jensen, Schroder, and Boffa summarize the main idea(s) of the results? – Imperishable Night Apr 05 '23 at 17:49
  • @ImperishableNight I couldn't find those papers either. Assume you've probably found something already, but for others looking, see my answer here for a sketch of something that works https://math.stackexchange.com/questions/4859015/is-the-empty-set-always-an-implicit-member-of-all-sets-under-a-pure-set-theory/4859053#4859053 – spaceisdarkgreen Feb 20 '24 at 17:31