Working in ZF+AD, let $$\theta\_0(X)=\min\{\alpha\in ON: \not\exists f: X\rightarrow \alpha\mbox{ surjective and OD}\}$$ be the least ordinal onto which $X$ does not surject in an OD way, for $X\subseteq \mathbb{R}$ uncountable and OD. Clearly if $X$ has an OD perfect subset, then $\theta\_0(X)=\theta\_0$ (the first term in the Solovay sequence); howevr, this need not occur, as Andreas Blass' answer to [Ordinal-definable witnesses to the perfect set property?](https://mathoverflow.net/questions/243402/ordinal-definable-witnesses-to-the-perfect-set-property) shows (consider the set of non-OD reals).


My question is: is it consistent with ZF+AD that there is an uncountable OD set of reals $X$ with $\theta\_0(X)<\theta\_0$? If so, what can we say about the possible values of the $\theta\_0$-map? 




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*My motivation is, ultimately, the question [Cardinal characteristics without choice](https://mathoverflow.net/questions/191545/cardinal-characteristics-without-choice); I'm especially interested in cardinal characteristics which have natural "$\kappa$-versions" for arbitrary $\kappa$ (e.g. the idea of the dominating number makes sense on $\kappa^\kappa$, not just $\omega^\omega$), and the possibility of comparing such cardinal characteristics by looking at the Solovay-type sequences their "large" sets induce; but that's jumping the gun by a substantial amount. I mention this here only in case this motivation is helpful re: finding references, etc.*



Comparing the sizes of uncountable sets of reals under AD
*The following is due to John Steel - who is not on mathoverflow - following a suggestion (see [Ordinal-definable witnesses to the perfect set property?](https://mathoverflow.net/questions/243402/ordinal-definable-witnesses-to-the-perfect-set-property)) of Vladimir Kanovei; since none of this is my work, and indeed I don't currently understand the crucial theorem used, I've made this community wiki.* 


*I will leave this up for a while before accepting it, for two reasons: first, this uses enough inner model theory that I am not comfortable claiming correctness, and I want to see if the inner model theorists around here agree with the argument below. Second, it leaves open the question of whether the opposite answer is **consistent**, and I would love information on that, too.*




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Extremely surprisingly, at least to me, the answer to the question is consistently **yes**! We can indeed have "small $\theta\_0$". This raises the question of what the possible values of $\theta\_0(X)$ are, but I think that's a job for a later question.


**CLAIM**. Assume $AD^++V=L(\mathcal{P}(\mathbb{R}))$. Then the set $X$ of reals which are Cohen over HOD satisfies $\theta\_0(X)=\omega\_1$.


To prove this claim, we use the following theorem (John says it follows from <http://www.math.uci.edu/~ntrang/AD+reflection.pdf>, but I don't immediately see how):


**Theorem**. Assume $AD^++V=L(\mathcal{P}(\mathbb{R}))$. Then there are definable maps $A$ and $W$ from reals such that


* $A(x)=\mathbb{P}\_x$ is a forcing notion,
* $W(x)=W\_x$ is a $\mathbb{P}\_x$-name for a symmetric submodel of the generic extension of $HOD[x]$ by $\mathbb{P}\_x$, and
* for every real $x$, $V=W\_x^G$ for some $\mathbb{P}\_x$-generic $G$.




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OK, let's take that for granted for a moment; so what?


Well, let $f:\mathbb{R}\rightarrow ON$ be OD via $\varphi$ and ordinal parameters $\overline{\gamma}$. Then we have $$\mbox{$f(x)=\alpha$ $\quad$ iff $\quad$ $HOD[x]\models$"$\Vdash\_{\mathbb{P}\_x}$"$W\_x\models \varphi(x, \overline{\gamma}, \alpha)."$"}$$ But then the set of possible values for $f(x)$, for $x$ Cohen over $HOD$, is countable in $HOD$! 


Why? Well, if $f(x)=\alpha$ for $x\in X$, then this was forced by some $p\prec x$. But there are only countably many Cohen conditions, so we're done.




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Now, I believe that this generalizes to the following (this is *not* in John's email, so perhaps there is a subtlety of Cohen-ness I'm missing):


**SECOND CLAIM**: Assume $$\mbox{$AD^++V=L(\mathcal{P}(\mathbb{R}))+$"There are uncountably many $\mathbb{Q}$-generic reals over $HOD$"}$$ (for $\mathbb{Q}$ some forcing notion). Then the set $Y\_\mathbb{Q}$ of reals which are $\mathbb{Q}$-generic over $HOD$ has $\theta\_0(Y\_\mathbb{Q})\le \vert\mathbb{Q}\vert^+$.


Based on this, I suspect it is in fact consistent with $AD^++V=L(\mathcal{P}(\mathbb{R}))$ that - for each $\omega<\kappa<\Theta$ - there is an OD set $Z\_\kappa$ such that $\theta\_0(Z\_\kappa)=\kappa$. But I am galactically far from proving anything like this! Heck, I don't even understand everything I've written above. :P


The crucial point - even assuming everything above is correct - is establishing the lower bound. In the case of the original $X$, the lower bound was trivial: any uncountable OD set $A$ surjects onto $\omega$ in an OD way, since we can (OD-ly) find a sequence $I\_n$ ($n\in\omega$) of disjoint rational intervals such that $I\_n\cap A\not=\emptyset$ for all $n$. But this breaks down above $\omega$, so I'm completely lost!


