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Assume that $V = L$ is true in the ground model $M$, and let $G$ be generic for $\mathbb{P} = Fn(I, 2)$, which is just the set of finite functions from $I$ to $2$). Then, ${\rm HOD}^{\mathbb{R}} = L[\mathbb{R}]$ is true in $M[G]$.

I assume that the proof is just a modification of the proof that ${\rm HOD}$ in $M[G]$ is contained in $L[\mathbb{P}]$ in $M$, but I can't figure out how to do it.

As in Kunen, ${\rm HOD}^\mathbb{R}$ are the sets hereditarily definable from ordinal and real parameters, and $L[\mathbb{R}]$ is the constructible hierarchy over the reals, i.e., $L[\mathbb{R}]_0 = \{\mathbb{R}\} \cup {\rm tcl}(\mathbb{R})$, $L[\mathbb{R}]_{\alpha +1} = {\rm Def}(L[\mathbb{R}]_\alpha)$, etc., and where ${\rm tcl}(x)$ is the transitive closure of $x$.

Hopefully the exercise isn't too trivial for mathoverflow.

Assume that $V = L$ is true in the ground model $M$, and let $G$ be generic for $\mathbb{P} = Fn(I, 2)$, which is just the set of finite functions from $I$ to $2$. Then, ${\rm HOD}^{\mathbb{R}} = L[\mathbb{R}]$ is true in $M[G]$.

I assume that the proof is just a modification of the proof that ${\rm HOD}$ in $M[G]$ is contained in $L[\mathbb{P}]$ in $M$, but I can't figure out how to do it.

As in Kunen, ${\rm HOD}^\mathbb{R}$ are the sets hereditarily definable from ordinal and real parameters, and $L[\mathbb{R}]$ is the constructible hierarchy over the reals, i.e., $L[\mathbb{R}]_0 = \{\mathbb{R}\} \cup {\rm tcl}(\mathbb{R})$, $L[\mathbb{R}]_{\alpha +1} = {\rm Def}(L[\mathbb{R}]_\alpha)$, etc., and where ${\rm tcl}(x)$ is the transitive closure of $x$.

Hopefully the exercise isn't too trivial for mathoverflow.

Assume that $V = L$ is true in the ground model $M$, and let $G$ be generic for $\mathbb{P} = Fn(I, 2)$ (which is just the set of finite functions from $I$ to $2$). Then, $HOD^{\mathbb{R}} = L[\mathbb{R}]$ is true in $M[G]$.

I assume that the proof is just a modification of the proof that $HOD$ in $M[G]$ is contained in $L[\mathbb{P}]$ in $M$, but I can't figure out how to do it.

As in Kunen, $HOD^\mathbb{R}$ are the sets hereditarily definable from ordinal and real parameters; and $L[\mathbb{R}]$ is the constructible hierarchy over the reals (i.e. $L[\mathbb{R}]_0 = \{\mathbb{R}\} \cup tcl(\mathbb{R})$, $L[\mathbb{R}]_{\alpha +1} = Def(L[\mathbb{R}]_\alpha)$, etc. (where $tcl(x)$ is the transitive closure of $x$)).

Hopefully the exercise isn't too trivial for mathoverflow.

Assume that $V = L$ is true in the ground model $M$, and let $G$ be generic for $\mathbb{P} = Fn(I, 2)$, which is just the set of finite functions from $I$ to $2$). Then, ${\rm HOD}^{\mathbb{R}} = L[\mathbb{R}]$ is true in $M[G]$.

I assume that the proof is just a modification of the proof that ${\rm HOD}$ in $M[G]$ is contained in $L[\mathbb{P}]$ in $M$, but I can't figure out how to do it.

As in Kunen, ${\rm HOD}^\mathbb{R}$ are the sets hereditarily definable from ordinal and real parameters, and $L[\mathbb{R}]$ is the constructible hierarchy over the reals, i.e., $L[\mathbb{R}]_0 = \{\mathbb{R}\} \cup {\rm tcl}(\mathbb{R})$, $L[\mathbb{R}]_{\alpha +1} = {\rm Def}(L[\mathbb{R}]_\alpha)$, etc., and where ${\rm tcl}(x)$ is the transitive closure of $x$.

Hopefully the exercise isn't too trivial for mathoverflow.

Assume that $V = L$ is true in the ground model $M$, and let $G$ be generic for $\mathbb{P} = Fn(I, 2)$ (which is just the set of finite functions from $I$ to $2$). Then, $HOD^{\mathbb{R}} = L[\mathbb{R}]$ is true in $M[G]$.

I assume that the proof is just a modification of the proof that $HOD$ of $M[G]$ is contained in $L[\mathbb{P}]$ in $M$, but I can't figure out how to do it.

As in Kunen, $HOD^\mathbb{R}$ are the sets hereditarily definable from ordinal and real parameters; and $L[\mathbb{R}]$ is the constructible hierarchy over the reals (i.e. $L[\mathbb{R}]_0 = \{\mathbb{R}\} \cup tcl(\mathbb{R})$, $L[\mathbb{R}]_{\alpha +1} = Def(L[\mathbb{R}]_\alpha)$, etc. (where $tcl(x)$ is the transitive closure of $x$)).

Hopefully the exercise isn't too trivial for mathoverflow.

Assume that $V = L$ is true in the ground model $M$, and let $G$ be generic for $\mathbb{P} = Fn(I, 2)$ (which is just the set of finite functions from $I$ to $2$). Then, $HOD^{\mathbb{R}} = L[\mathbb{R}]$ is true in $M[G]$.

I assume that the proof is just a modification of the proof that $HOD$ in $M[G]$ is contained in $L[\mathbb{P}]$ in $M$, but I can't figure out how to do it.

As in Kunen, $HOD^\mathbb{R}$ are the sets hereditarily definable from ordinal and real parameters; and $L[\mathbb{R}]$ is the constructible hierarchy over the reals (i.e. $L[\mathbb{R}]_0 = \{\mathbb{R}\} \cup tcl(\mathbb{R})$, $L[\mathbb{R}]_{\alpha +1} = Def(L[\mathbb{R}]_\alpha)$, etc. (where $tcl(x)$ is the transitive closure of $x$)).

Hopefully the exercise isn't too trivial for mathoverflow.