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Here is another way to answer, which builds a tight connection between $H_{\omega_1}$ and $H_{\omega_2}$, rather than just the power sets. But meanwhile, the power sets interpret these structures since every set in $H_{\omega_1}$ is coded by real and every set in $H_{\omega_2}$ is coded by a subset of $\omega_1$. In light of that, this theorem can be seen as a strengthening of the phonemenon about which you inquire.

Theorem. (Freire,Hamkins) If ZFC is consistent, then it is consistent with ZFC that there is a membership relation $\bar\in$ definable in $\langle H_{\omega_1},\in\rangle$ such that $\langle H_{\omega_1},\bar\in\rangle\cong\langle H_{\omega_2},\in\rangle$, putting $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ into a bi-interpretation synonymy.

This is theorem 19 of our paper

• Roque Freire, Alfredo; Hamkins, Joel David, Bi-interpretation in weak set theories, Journal of Symbolic Logic, 2020, DOI:10.1017/jsl.2020.72, ZBL07415218.

The proof uses almost disjoint coding, just as in mentioned in Andres's answer, but applying it in a model of Harrington that has an absolutely definable projective well ordering of the reals. Ultimately, a real in $H_{\omega_1}$ codes a subset of $\omega_1$, which codes a relation on $\omega_1$, which codes a set in $H_{\omega_2}$. This is how one interprets $H_{\omega_2}$ in $H_{\omega_1}$. The converse interpretation is simply to cut down to the hereditarily countable sets. This is a bi-interpretation, and one can select representatives using the projective well-order, so it is a synonymy.

Here is another way to answer, which builds a bi-interpretation between $H_{\omega_1}$ and $H_{\omega_2}$, rather than just the power sets, and this is a very nice connection indeed. Meanwhile, the power sets interpret these structures since every set in $H_{\omega_1}$ is coded by real and every set in $H_{\omega_2}$ is coded by a subset of $\omega_1$. In light of that, this theorem can be seen as a strengthening of the phenomenon about which you inquire.

Theorem. (Freire,Hamkins) If ZFC is consistent, then it is consistent with ZFC that there is a membership relation $\bar\in$ definable in $\langle H_{\omega_1},\in\rangle$ such that $\langle H_{\omega_1},\bar\in\rangle\cong\langle H_{\omega_2},\in\rangle$, putting $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ into a bi-interpretation synonymy.

This is theorem 19 of our paper

• Roque Freire, Alfredo; Hamkins, Joel David, Bi-interpretation in weak set theories, Journal of Symbolic Logic, 2020, DOI:10.1017/jsl.2020.72, ZBL07415218.

The proof uses almost disjoint coding, just as in mentioned in Andres's answer, but applying it in a model of Harrington that has an absolutely definable projective well ordering of the reals, mentioned in this MO answer. Ultimately, a real in $H_{\omega_1}$ codes a subset of $\omega_1$, which codes a relation on $\omega_1$, which codes a set in $H_{\omega_2}$. This is how one interprets $H_{\omega_2}$ in $H_{\omega_1}$. The converse interpretation is simply to cut down to the hereditarily countable sets. This is a bi-interpretation, and one can select representatives using the projective well-order, so it is a synonymy.

Here is another way to answer, which builds a tight connection between $H_{\omega_1}$ and $H_{\omega_2}$, rather than just the power sets. But meanwhile, the power sets interpret these structures since every set in $H_{\omega_1}$ is coded by real and every set in $H_{\omega_2}$ is coded by a subset of $\omega_1$. In light of that, this theorem can be seen as a strengthening of the phonemenon about which you inquire.

Theorem. (Freire,Hamkins) If ZFC is consistent, then it is consistent with ZFC that there is a membership relation $\bar\in$ definable in $\langle H_{\omega_1},\in\rangle$ such that $\langle H_{\omega_1},\bar\in\rangle\cong\langle H_{\omega_2},\in\rangle$, putting these structures into bi-interpretation synonymy.

This is theorem 19 of our paper

• Roque Freire, Alfredo; Hamkins, Joel David, Bi-interpretation in weak set theories, Journal of Symbolic Logic, 2020, DOI:10.1017/jsl.2020.72, ZBL07415218.

The proof uses almost disjoint coding, just as in mentioned in Andres's answer, but applying it in a model of Harrington that has an absolutely definable projective well ordering of the reals. Ultimately, a real in $H_{\omega_1}$ codes a subset of $\omega_1$, which codes a relation on $\omega_1$, which codes a set in $H_{\omega_2}$. This is how one interprets $H_{\omega_2}$ in $H_{\omega_1}$. The converse interpretation is simply to cut down to the hereditarily countable sets. This is a bi-interpretation, and one can select representatives using the projective well-order, so it is a synonymy.

Here is another way to answer, which builds a tight connection between $H_{\omega_1}$ and $H_{\omega_2}$, rather than just the power sets. But meanwhile, the power sets interpret these structures since every set in $H_{\omega_1}$ is coded by real and every set in $H_{\omega_2}$ is coded by a subset of $\omega_1$. In light of that, this theorem can be seen as a strengthening of the phonemenon about which you inquire.

Theorem. (Freire,Hamkins) If ZFC is consistent, then it is consistent with ZFC that there is a membership relation $\bar\in$ definable in $\langle H_{\omega_1},\in\rangle$ such that $\langle H_{\omega_1},\bar\in\rangle\cong\langle H_{\omega_2},\in\rangle$, putting $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ into a bi-interpretation synonymy.

This is theorem 19 of our paper

• Roque Freire, Alfredo; Hamkins, Joel David, Bi-interpretation in weak set theories, Journal of Symbolic Logic, 2020, DOI:10.1017/jsl.2020.72, ZBL07415218.

The proof uses almost disjoint coding, just as in mentioned in Andres's answer, but applying it in a model of Harrington that has an absolutely definable projective well ordering of the reals. Ultimately, a real in $H_{\omega_1}$ codes a subset of $\omega_1$, which codes a relation on $\omega_1$, which codes a set in $H_{\omega_2}$. This is how one interprets $H_{\omega_2}$ in $H_{\omega_1}$. The converse interpretation is simply to cut down to the hereditarily countable sets. This is a bi-interpretation, and one can select representatives using the projective well-order, so it is a synonymy.

Here is another way to answer, which builds a tight connection between $H_{\omega_1}$ and $H_{\omega_2}$, rather than just the power sets.

Theorem. (Freire,Hamkins) If ZFC is consistent, then it is consistent with ZFC that there is a membership relation $\bar\in$ definable in $\langle H_{\omega_1},\in\rangle$ such that $\langle H_{\omega_1},\bar\in\rangle\cong\langle H_{\omega_2},\in\rangle$, putting these structures into bi-interpretation synonymy.

This is theorem 19 of our paper

• Roque Freire, Alfredo; Hamkins, Joel David, Bi-interpretation in weak set theories, Journal of Symbolic Logic, 2020, DOI:10.1017/jsl.2020.72, ZBL07415218.

The proof uses almost disjoint coding, just as in mentioned in Andres's answer, but applying it in a model of Harrington that has an absolutely definable projective well ordering of the reals. Ultimately, a real in $H_{\omega_1}$ codes a subset of $\omega_1$, which codes a relation on $\omega_1$, which codes a set in $H_{\omega_2}$. This is how one interprets $H_{\omega_2}$ in $H_{\omega_1}$. The converse interpretation is simply to cut down to the hereditarily countable sets. This is a bi-interpretation, and one can select representatives using the projective well-order, so it is a synonymy.

Here is another way to answer, which builds a tight connection between $H_{\omega_1}$ and $H_{\omega_2}$, rather than just the power sets. But meanwhile, the power sets interpret these structures since every set in $H_{\omega_1}$ is coded by real and every set in $H_{\omega_2}$ is coded by a subset of $\omega_1$. In light of that, this theorem can be seen as a strengthening of the phonemenon about which you inquire.

Theorem. (Freire,Hamkins) If ZFC is consistent, then it is consistent with ZFC that there is a membership relation $\bar\in$ definable in $\langle H_{\omega_1},\in\rangle$ such that $\langle H_{\omega_1},\bar\in\rangle\cong\langle H_{\omega_2},\in\rangle$, putting these structures into bi-interpretation synonymy.

This is theorem 19 of our paper

• Roque Freire, Alfredo; Hamkins, Joel David, Bi-interpretation in weak set theories, Journal of Symbolic Logic, 2020, DOI:10.1017/jsl.2020.72, ZBL07415218.

The proof uses almost disjoint coding, just as in mentioned in Andres's answer, but applying it in a model of Harrington that has an absolutely definable projective well ordering of the reals. Ultimately, a real in $H_{\omega_1}$ codes a subset of $\omega_1$, which codes a relation on $\omega_1$, which codes a set in $H_{\omega_2}$. This is how one interprets $H_{\omega_2}$ in $H_{\omega_1}$. The converse interpretation is simply to cut down to the hereditarily countable sets. This is a bi-interpretation, and one can select representatives using the projective well-order, so it is a synonymy.