At [Hamkins - Different set theories are never bi-interpretable](http://jdh.hamkins.org/different-set-theories-are-never-bi-interpretable/), it is mentioned that different set theories extending ZF are never bi-interpretable.


Where different means "not theoretically equivalent", i.e. there must be a theorem that one has and the other doesn't.



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> Question: Would that same result hold for Z, i.e. is it the case that any different [in the same sense mentioned in that article] theories extending Z are never bi-interpretable?
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> More generally: even if the above fails, the question is about whether this result needs the full strength of ZF, and if not then what would be the least fragment of ZF for which this result holds?
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Can two set theories extending Z be different and yet bi-interpretable?
**UPDATE (January 30, 2022):** The *first* question was answered (in the negative) by Hamkins and Freire for $\mathrm{Z}$ (Zermelo set theory) and $\mathrm{ZF}^{-}$ ($\mathrm{ZF}$ without powerset). Their paper "Bi-interpretation in weak set theories" was recently published in the Journal of Symbolic Logic. See [here](https://arxiv.org/abs/2001.05262) for a preprint of their paper, and [here](http://jdh.hamkins.org/bi-interpretation-in-set-theory-oberwolfach-2022/) for recent blog of Hamkins about this topic. The *second* question remains open.



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> What follows is my old answer (June 19, 2018).
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To my knowledge the two questions you are asking are wide open; indeed the second question is one of the two open questions posed at the end of [this 2016 paper](https://arxiv.org/abs/1702.07093) of mine, which gives many examples (and non-examples) of theories $T$ that satisfy the principle "different consistent extensions of $T$ are not bi-interpretable".


The aforementioned paper was published as:


A. Enayat, “Variations on a Visserian theme,” in Liber Amicorum Alberti: a tribute to Albert Visser, Jan van Eijck, Rosalie Iemhoff and Joost J. Joosten (eds.) Pages, 99-110. ISBN, 978-1848902046. College Publications, London, 2016.


