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Yes, it is consistent with AD. In $L(\mathbb{R})$, if AD holds then $\Theta$ injects into $\mathcal{P}(\mathbb{R})$, or equivalently, into $2^{\mathbb{R}}$. The OD sets of reals are Wadge-cofinal and $\Theta$ is regular, so we can define an injection $F : \Theta \to \mathcal{P}(\mathbb{R})$ by induction. Let $F(\alpha)$ be the $<_{OD}$-least OD set of reals with Wadge rank greater than the Wadge ranks of all the sets of reals $F(\gamma)$, $\gamma < \alpha$.

To see that in $L(\mathbb{R})$ the Wadge ranks of the OD sets of reals are cofinal, notice that every set of reals $A$ is OD from some real $y$, say $x \in A \iff \varphi[x,y,\alpha]$, and the OD set $\lbrace (x,y) : \varphi[x,y,\alpha] \rbrace$ has Wadge rank at least that of $A$.

Assuming $AD + V=L(\mathcal{P}(\mathbb{R}))$ the existence of an injection $\Theta \to \mathcal{P}(\mathbb{R})$ is equivalent to the length of the Solovay sequence being a successor ordinal. The meaning of this expression that we will use here is that there is a single set of reals $A$ such that the $OD_A$ sets of reals are Wadge-cofinal in $\mathcal{P}(\mathbb{R})$.

If the Solovay sequence has successor length as witnessed by $A$, then $\Theta$ is regular because any function $g: \mathbb{R} \to \Theta$ is bounded by the Wadge rank of the set of pairs $(x,y)$ such that $y$ is in the $<_{OD_A}$-least $OD_A$ set of reals with Wadge rank at least $g(x)$. So we can just modify the argument given for $L(\mathbb{R})$ by adding $A$ as a parameter.

Conversely, if there is an injection $F:\Theta \to \mathcal{P}(\mathbb{R})$, it must be Wadge-cofinal or else we would get an injection $\Theta \to \mathbb{R}$, which is impossible. If $V = L(\mathcal{P}(\mathbb{R}))$ then the function $F$ is OD from a set $A$ of reals, so its range is a Wadge-cofinal collection of $OD_A$ sets of reals.

Yes, it is consistent with AD. In $L(\mathbb{R})$, if AD holds then $\Theta$ injects into $\mathcal{P}(\mathbb{R})$, or equivalently, into $2^{\mathbb{R}}$. The OD sets of reals are Wadge-cofinal and $\Theta$ is regular, so we can define an injection $F : \Theta \to \mathcal{P}(\mathbb{R})$ by induction. Let $F(\alpha)$ be the $<_{OD}$-least OD set of reals with Wadge rank greater than the Wadge ranks of all the sets of reals $F(\gamma)$, $\gamma < \alpha$.

Going further, we can determine the Hartogs number $\aleph(2^{\mathbb{R}})$ in all models of $AD + V=L(\mathcal{P}(\mathbb{R}))$. The proof splits into two cases corresponding to the length of the Solovay sequence being a successor ordinal or a limit ordinal.

If the length of the Solovay sequence is a successor ordinal, this means that there is a set of reals $A$ such that every set of reals is OD from $A$ and a real. In this case $\aleph(2^{\mathbb{R}})$ has the largest possible value, namely $\Theta(2^{\mathbb{R}})$. Given a surjection $F : \mathcal{P}(\mathbb{R}) \to Z$ for any set $Z$ we can construct an injection $Z \to \mathcal{P}(\mathbb{R})$. Let $G(z)$ be the set of pairs $(x,y)$ of reals such that $y$ is in the least $OD_{A,x}$ set of reals $B$ with $F(B) = z$ (if it exists.)

On the other hand, if the length of the Solovay sequence is a limit ordinal then $\aleph(2^{\mathbb{R}})$ is equal to $\Theta$; that is, to $\Theta(\mathbb{R})$. We have $\aleph(2^{\mathbb{R}}) \ge \Theta$ on general grounds like you said in the question. But any function $F : \Theta \to \mathcal{P}(\mathbb{R})$ is OD from a set $A$ of reals because $V=L(\mathcal{P}(\mathbb{R}))$, so the range of $F$ consists of $OD_A$ sets of reals. The Solovay sequence has limit length, so the range of $F$ cannot be all of $\mathcal{P}(\mathbb{R})$.

Yes, it is consistent with AD. In $L(\mathbb{R})$, if AD holds then $\Theta$ injects into $\mathcal{P}(\mathbb{R})$, or equivalently, into $2^{\mathbb{R}}$. The OD sets of reals are Wadge-cofinal and $\Theta$ is regular, so we can define an injection $F : \Theta \to \mathcal{P}(\mathbb{R})$ by induction. Let $F(\alpha)$ be the $<_{OD}$-least OD set of reals with Wadge rank greater than the Wadge ranks of all the sets of reals $F(\gamma)$, $\gamma < \alpha$.

To see that in $L(\mathbb{R})$ the Wadge ranks of the OD sets of reals are cofinal, notice that every set of reals $A$ is OD from some real $y$, say $x \in A \iff \varphi[x,y,\alpha]$, and the OD set $\lbrace (x,y) : \varphi[x,y,\alpha] \rbrace$ has Wadge rank at least that of $A$.

Assuming $AD + V=L(\mathcal{P}(\mathbb{R}))$, the existence of an injection $\Theta \to \mathcal{P}(\mathbb{R})$ is equivalent to the length of the Solovay sequence being a successor ordinal. The meaning of this expression that we will use here is that there is a set of reals $A$ such that the $OD_A$ sets of reals are Wadge-cofinal in $\mathcal{P}(\mathbb{R})$. This implies that $\Theta$ is regular, so we just modify the argument given for $L(\mathbb{R})$ by adding $A$ as a parameter.

Conversely, if there is an injection $F:\Theta \to \mathcal{P}(\mathbb{R})$, it must be Wadge-cofinal or else we would get an injection $\Theta \to \mathbb{R}$, which is impossible. Because $V = L(\mathcal{P}(\mathbb{R}))$, the function $F$ is OD from a set $A$ of reals, so its range is a Wadge-cofinal collection of $OD_A$ sets of reals.

Yes, it is consistent with AD. In $L(\mathbb{R})$, if AD holds then $\Theta$ injects into $\mathcal{P}(\mathbb{R})$, or equivalently, into $2^{\mathbb{R}}$. The OD sets of reals are Wadge-cofinal and $\Theta$ is regular, so we can define an injection $F : \Theta \to \mathcal{P}(\mathbb{R})$ by induction. Let $F(\alpha)$ be the $<_{OD}$-least OD set of reals with Wadge rank greater than the Wadge ranks of all the sets of reals $F(\gamma)$, $\gamma < \alpha$.

To see that in $L(\mathbb{R})$ the Wadge ranks of the OD sets of reals are cofinal, notice that every set of reals $A$ is OD from some real $y$, say $x \in A \iff \varphi[x,y,\alpha]$, and the OD set $\lbrace (x,y) : \varphi[x,y,\alpha] \rbrace$ has Wadge rank at least that of $A$.

Assuming $AD + V=L(\mathcal{P}(\mathbb{R}))$ the existence of an injection $\Theta \to \mathcal{P}(\mathbb{R})$ is equivalent to the length of the Solovay sequence being a successor ordinal. The meaning of this expression that we will use here is that there is a single set of reals $A$ such that the $OD_A$ sets of reals are Wadge-cofinal in $\mathcal{P}(\mathbb{R})$.

If the Solovay sequence has successor length as witnessed by $A$, then $\Theta$ is regular because any function $g: \mathbb{R} \to \Theta$ is bounded by the Wadge rank of the set of pairs $(x,y)$ such that $y$ is in the $<_{OD_A}$-least $OD_A$ set of reals with Wadge rank at least $g(x)$. So we can just modify the argument given for $L(\mathbb{R})$ by adding $A$ as a parameter.

Conversely, if there is an injection $F:\Theta \to \mathcal{P}(\mathbb{R})$, it must be Wadge-cofinal or else we would get an injection $\Theta \to \mathbb{R}$, which is impossible. If $V = L(\mathcal{P}(\mathbb{R}))$ then the function $F$ is OD from a set $A$ of reals, so its range is a Wadge-cofinal collection of $OD_A$ sets of reals.

Yes, $\Theta$ injects into $\mathcal{P}(\mathbb{R})$ (equivalently, into $2^{\mathbb{R}})$ in $L(\mathbb{R})$. The OD sets of reals are Wadge-cofinal and $\Theta$ is regular, so we can define an injection $F : \Theta \to \mathcal{P}(\mathbb{R})$ by induction: Let $F(\alpha)$ be the $<_{OD}$-least OD set of reals with Wadge rank greater than the Wadge ranks of all the sets of reals $F(\gamma)$, $\gamma < \alpha$.

To see that the Wadge ranks of the OD sets of reals are cofinal, notice that in $L(\mathbb{R})$ every set of reals $A$ is OD from some real $y$, say $x \in A \iff \varphi[x,y,\alpha]$, and the OD set $\lbrace (x,y) : \varphi[x,y,\alpha] \rbrace$ has Wadge rank at least that of $A$.

A similar argument works in any model of $AD^+ + \neg AD_{\mathbb{R}}$, which implies that there is a set of reals $A$ such that every set of reals is OD from $A$ and a real, and also implies that $\Theta$ is regular.

Yes, it is consistent with AD. In $L(\mathbb{R})$, if AD holds then $\Theta$ injects into $\mathcal{P}(\mathbb{R})$, or equivalently, into $2^{\mathbb{R}}$. The OD sets of reals are Wadge-cofinal and $\Theta$ is regular, so we can define an injection $F : \Theta \to \mathcal{P}(\mathbb{R})$ by induction. Let $F(\alpha)$ be the $<_{OD}$-least OD set of reals with Wadge rank greater than the Wadge ranks of all the sets of reals $F(\gamma)$, $\gamma < \alpha$.

To see that in $L(\mathbb{R})$ the Wadge ranks of the OD sets of reals are cofinal, notice that every set of reals $A$ is OD from some real $y$, say $x \in A \iff \varphi[x,y,\alpha]$, and the OD set $\lbrace (x,y) : \varphi[x,y,\alpha] \rbrace$ has Wadge rank at least that of $A$.

Assuming $AD + V=L(\mathcal{P}(\mathbb{R}))$, the existence of an injection $\Theta \to \mathcal{P}(\mathbb{R})$ is equivalent to the length of the Solovay sequence being a successor ordinal. The meaning of this expression that we will use here is that there is a set of reals $A$ such that the $OD_A$ sets of reals are Wadge-cofinal in $\mathcal{P}(\mathbb{R})$. This implies that $\Theta$ is regular, so we just modify the argument given for $L(\mathbb{R})$ by adding $A$ as a parameter.

Conversely, if there is an injection $F:\Theta \to \mathcal{P}(\mathbb{R})$, it must be Wadge-cofinal or else we would get an injection $\Theta \to \mathbb{R}$, which is impossible. Because $V = L(\mathcal{P}(\mathbb{R}))$, the function $F$ is OD from a set $A$ of reals, so its range is a Wadge-cofinal collection of $OD_A$ sets of reals.