This question concerns the possibility of the bi-interpretation synonymy of the structure $\langle H_{\omega_1},\in\rangle$, consisting of the hereditarily countable sets, and the structure $\langle H_{\omega_2},\in\rangle$, consisting of sets of hereditary size at most $\aleph_1$. These are both models of Zermelo-Fraenkel set theory $\text{ZFC}^-$, without the power set axiom. The structure $\langle H_{\omega_1},\in\rangle$ is of course a definable submodel of $\langle H_{\omega_2},\in\rangle$, which provides one direction of interpretation.

Depending on the set-theoretic background, it is also possible that there is a converse interpretation in the other direction. Indeed, in my recent paper with Afredo Roque Freire,

- Joel David Hamkins and Alfredo Roque Freire, Bi-interpretation in weak set theories, blog post, arXiv:2001.05262

we prove that it is relatively consistent with ZFC that the structures $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ are bi-interpretable (see theorem 17). This is true, specifically, in the Solovay-Tennenbaum model, obtained by c.c.c. forcing over $L$ to achieve $\text{MA}+\neg\text{CH}$. What is needed is (i) $H_{\omega_1}$ has a definable almost disjoint $\omega_1$-sequence of reals; and (ii) every subset $A\subseteq\omega_1$ is coded by a real via almost-disjoint coding with respect to that sequence. The basic idea is that objects in $H_{\omega_2}$ are coded by a well-founded relation on $\omega_1$, which is in turn coded by a real, and so in $H_{\omega_1}$ we can define the class $U$ of reals coding a set in this manner and an equivalence relation on those reals $x\equiv y$ for when they code the same set, and a relation $\bar\in$ on those reals, so that $\langle H_{\omega_2},\in\rangle$ is isomorphic to the quotient structure $\langle U,\bar\in\rangle/\equiv$.

The argument seems to use the equivalence relation in a fundamental manner, and the question I have about this here is whether one can omit the need for the equivalence relation. This ultimately amounts to the following, which is question 18 in the paper:

**Question.** Is it relatively consistent with ZFC that there is a
binary relation $\bar\in$ that is definable in $H_{\omega_1}$ such
that $$\langle H_{\omega_1},\bar\in\rangle\cong \langle
H_{\omega_2},\in\rangle?$$

This is what it would mean for these structures to form a bi-interpretation synonymy.

For a positive answer, it would be enough to show the consistency with ZFC of the existence of a definable global well-order in $H_{\omega_1}$, together with the almost-disjoint coding of hypothesis (ii) above. Is that possible?

Apart from $H_{\omega_1}$ and $H_{\omega_2}$ specifically, a related question we have is whether one can prove any instance of interpretation in a model of $\text{ZFC}^-$ that requires the quotient by an equivalence relation.

**Question.** Is there a structure that is interpretable in a model
of $\text{ZFC}^-$, but only by means of a nontrivial equivalence
relation?

This is question 9 in the paper.