Does $\mathbf{Z}$ prove existence of transitive closure of every set?

Question

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

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

Answer 1

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 $2.12$).