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Claim: NGB + $\exists j\colon V\to V$ is equiconsistent with NGB + $\exists j\colon V\to V + \neg RR$.

In order to prove the left to right direction, we will start with a model of NGB + $\exists j\colon V\to V$ and take the symmetric extension that as described in Hamkins and Palumbo's paper. By the results of this paper, in the symmetric extension there is a set that witnesses $\neg RR$. Note that the poset, as well as the groups and the filter, that are used in the symmetric extension are all very low in the cumulative hierarchy (say in $V_{\omega + \omega}$). Therefore, they are fixed by the embedding $j$.

Let $W$ be the symmetric extension and let's extend $j$ to $\tilde{j}\colon W\to W$. We construct $\tilde{j}$ as in the proof of Levy-Solovay's theorem. For a hereditarily symmetric name $\dot{x}$, let's define $\tilde{j}(\dot{x}^G) := j(\dot{x})^G$. Using the fact that $j$ does not move the groups of automorphisms of the forcing poset, as well as the filter $\mathcal{F}$, $j(\dot{x})$ is a hereditarily symmetric name ($sym(j(\dot{x})) = sym(\dot{x})$), and therefore $j(\dot{x})\in W$. $\tilde{j}$ is elementary, by exactly the same argument of Levi-Solovay: using the forcing theorem (every statement in the generic symmetric extension is forces by some condition in the generic filter), and the fact that $j$ doesn't move the forcing or the conditions, $p\Vdash \varphi(\dot{x})$ iff $j(p) = p\Vdash \varphi(j(\dot{x}))$, by the elementarity of $j$ in $V$.

Moreover, the class $\dot{j} = \{(\dot{x}, j(\dot{x})) \mid \dot{x} \text{ is hereditarily symmetric name}\}$ (when writing the pairs as canonical names of pairs) is a hereditarily symmetrical name for a class (it is fixed by itself by any automorphism in the group, and every member of it is a pair of hereditarily symmetric names). So $\tilde{j}$ is a class of $W$.

Claim: NGB + $\exists j\colon V\to V$ is equiconsistent with NGB + $\exists j\colon V\to V + \neg RR$.

In order to prove the left to right direction, we will start with a model of NGB + $\exists j\colon V\to V$ and take the symmetric extension that as described in Hamkins and Palumbo's paper. By the results of this paper, in the symmetric extension there is a set that witnesses $\neg RR$. Note that the poset, as well as the groups and the filter, that are used in the symmetric extension are all very low in the cumulative hierarchy (say in $V_{\omega + \omega}$). Therefore, they are fixed by the embedding $j$.

Let $W$ be the symmetric extension and let's extend $j$ to $\tilde{j}\colon W\to W$. We construct $\tilde{j}$ as in the proof of Levy-Solovay's theorem. For a hereditarily symmetric name $\dot{x}$, let's define $\tilde{j}(\dot{x}^G) := j(\dot{x})^G$.

Since $j$ does not move the forcing notion, $\mathbb{P}$, the groups of automorphisms of the forcing, $\mathcal{G}$, and the filter, $\mathcal{F}$, $j(\dot{x})$ is a hereditarily symmetric name (being hereditarily symmetric is a first order property with parameters $\mathbb{P}, \mathcal{G}, \mathcal{F}$), and therefore $j(\dot{x})\in W$. $\tilde{j}$ is elementary, by exactly the same argument of Levi-Solovay: using the forcing theorem (every statement in the generic symmetric extension is forces by some condition in the generic filter), and the fact that $j$ doesn't move the forcing or the conditions, $p\Vdash \varphi(\dot{x})$ iff $j(p) = p\Vdash \varphi(j(\dot{x}))$, by the elementarity of $j$ in $V$.

Moreover, the class $\dot{j} = \{(\dot{x}, j(\dot{x})) \mid \dot{x} \text{ is hereditarily symmetric name}\}$ (when writing the pairs as canonical names of pairs) is a hereditarily symmetrical name for a class (it is fixed by itself by any automorphism in the group, and every member of it is a pair of hereditarily symmetric names). So $\tilde{j}$ is a class of $W$.

Claim: NGB + $\exists j\colon V\to V$ is equiconsistent with NGB + $\exists j\colon V\to V + \neg RR$.

In order to prove the left to right direction, we will start with a model of NGB + $\exists j\colon V\to V$ and take the symmetric extension that as described in Hamkins and Palumbo's paper. By the results of this paper, in the symmetric extension there is a set that witnesses $\neg RR$. Note that the poset, as well as the groups and the filter, that are used in the symmetric extension are all very low in the cumulative hierarchy (say in $V_{\omega + \omega}$). Therefore, they are fixed by the embedding $j$.

Let $W$ be the symmetric extension and let's extend $j$ to $\tilde{j}\colon W\to W$. We construct $\tilde{j}$ as in the proof of Levy-Solovay's theorem. For a symmetric name $\dot{x}$, let's define $\tilde{j}(\dot{x}^G) := j(\dot{x})^G$. Using the fact that $j$ does not move the groups of automorphisms of the forcing poset, as well as the filter $\mathcal{F}$, $j(\dot{x})$ is a symmetric name ($fix(j(\dot{x})) = fix(\dot{x})$), and therefore $j(\dot{x})\in W$. $\tilde{j}$ is elementary, by exactly the same argument of Levi-Solovay: using the forcing theorem (every statement in the generic symmetric extension is forces by some condition in the generic filter), and the fact that $j$ doesn't move the forcing or the conditions, $p\Vdash \varphi(\dot{x})$ iff $j(p) = p\Vdash \varphi(j(\dot{x}))$, by the elementarity of $j$ in $V$.

Moreover, the class $\dot{j} = \{(\dot{x}, j(\dot{x})) \mid fix(\dot{x})\in \mathcal{F}\}$ (when writing the pairs as canonical names of pairs) is a symmetrical name for a class (it is fixed by any automorphism in the group). So $\tilde{j}$ is a class of $W$.

Claim: NGB + $\exists j\colon V\to V$ is equiconsistent with NGB + $\exists j\colon V\to V + \neg RR$.

In order to prove the left to right direction, we will start with a model of NGB + $\exists j\colon V\to V$ and take the symmetric extension that as described in Hamkins and Palumbo's paper. By the results of this paper, in the symmetric extension there is a set that witnesses $\neg RR$. Note that the poset, as well as the groups and the filter, that are used in the symmetric extension are all very low in the cumulative hierarchy (say in $V_{\omega + \omega}$). Therefore, they are fixed by the embedding $j$.

Let $W$ be the symmetric extension and let's extend $j$ to $\tilde{j}\colon W\to W$. We construct $\tilde{j}$ as in the proof of Levy-Solovay's theorem. For a hereditarily symmetric name $\dot{x}$, let's define $\tilde{j}(\dot{x}^G) := j(\dot{x})^G$. Using the fact that $j$ does not move the groups of automorphisms of the forcing poset, as well as the filter $\mathcal{F}$, $j(\dot{x})$ is a hereditarily symmetric name ($sym(j(\dot{x})) = sym(\dot{x})$), and therefore $j(\dot{x})\in W$. $\tilde{j}$ is elementary, by exactly the same argument of Levi-Solovay: using the forcing theorem (every statement in the generic symmetric extension is forces by some condition in the generic filter), and the fact that $j$ doesn't move the forcing or the conditions, $p\Vdash \varphi(\dot{x})$ iff $j(p) = p\Vdash \varphi(j(\dot{x}))$, by the elementarity of $j$ in $V$.

Moreover, the class $\dot{j} = \{(\dot{x}, j(\dot{x})) \mid \dot{x} \text{ is hereditarily symmetric name}\}$ (when writing the pairs as canonical names of pairs) is a hereditarily symmetrical name for a class (it is fixed by itself by any automorphism in the group, and every member of it is a pair of hereditarily symmetric names). So $\tilde{j}$ is a class of $W$.

Claim: NGB + $\exists j\colon V\to V$ is equiconsistent with NGB + $\exists j\colon V\to V + \neg RR$.

In order to prove the left to right direction, we will start with a model of NGB + $\exists j\colon V\to V$ and take the symmetric extension that as described in Hamkins and Palumbo's paper. By the results of this paper, in the symmetric extension there is a set that witnesses $\neg RR$. Note that the poset, as well as the groups and the filter, that are used in the symmetric extension are all very low in the cumulative hierarchy (say in $V_{\omega + \omega}$). Therefore, they are fixed by the embedding $j$.

Let $W$ be the symmetric extension and let's extend $j$ to $\tilde{j}\colon W\to W$. We construct $\tilde{j}$ as in the proof of Levi-Solovay's theorem. For a symmetric name $\dot{x}$, let's define $\tilde{j}(\dot{x}^G) := j(\dot{x})^G$. Using the fact that $j$ does not move the groups of automorphisms of the forcing poset, as well as the filter $\mathcal{F}$, $j(\dot{x})$ is a symmetric name ($fix(j(\dot{x})) = fix(\dot{x})$), and therefore $j(\dot{x})\in W$. $\tilde{j}$ is elementary, by exactly the same argument of Levi-Solovay: using the forcing theorem (every statement in the generic symmetric extension is forces by some condition in the generic filter), and the fact that $j$ doesn't move the forcing or the conditions, $p\Vdash \varphi(\dot{x})$ iff $j(p) = p\Vdash \varphi(j(\dot{x}))$, by the elementarity of $j$ in $V$.

Moreover, the class $\dot{j} = \{(\dot{x}, j(\dot{x})) \mid fix(\dot{x})\in \mathcal{F}\}$ (when writing the pairs as canonical names of pairs) is a symmetrical name for a class (it is fixed by any automorphism in the group). So $\tilde{j}$ is a class of $W$.

Claim: NGB + $\exists j\colon V\to V$ is equiconsistent with NGB + $\exists j\colon V\to V + \neg RR$.

In order to prove the left to right direction, we will start with a model of NGB + $\exists j\colon V\to V$ and take the symmetric extension that as described in Hamkins and Palumbo's paper. By the results of this paper, in the symmetric extension there is a set that witnesses $\neg RR$. Note that the poset, as well as the groups and the filter, that are used in the symmetric extension are all very low in the cumulative hierarchy (say in $V_{\omega + \omega}$). Therefore, they are fixed by the embedding $j$.

Let $W$ be the symmetric extension and let's extend $j$ to $\tilde{j}\colon W\to W$. We construct $\tilde{j}$ as in the proof of Levy-Solovay's theorem. For a symmetric name $\dot{x}$, let's define $\tilde{j}(\dot{x}^G) := j(\dot{x})^G$. Using the fact that $j$ does not move the groups of automorphisms of the forcing poset, as well as the filter $\mathcal{F}$, $j(\dot{x})$ is a symmetric name ($fix(j(\dot{x})) = fix(\dot{x})$), and therefore $j(\dot{x})\in W$. $\tilde{j}$ is elementary, by exactly the same argument of Levi-Solovay: using the forcing theorem (every statement in the generic symmetric extension is forces by some condition in the generic filter), and the fact that $j$ doesn't move the forcing or the conditions, $p\Vdash \varphi(\dot{x})$ iff $j(p) = p\Vdash \varphi(j(\dot{x}))$, by the elementarity of $j$ in $V$.

Moreover, the class $\dot{j} = \{(\dot{x}, j(\dot{x})) \mid fix(\dot{x})\in \mathcal{F}\}$ (when writing the pairs as canonical names of pairs) is a symmetrical name for a class (it is fixed by any automorphism in the group). So $\tilde{j}$ is a class of $W$.