It is a well-known result by Woodin that the Hartogs number $h(\mathbb{R})$ (more commonly known as $\Theta$) is a Woodin cardinal (in HOD) assuming ZF + AD + DC. This is equivalent to $h(\mathcal{P}(\aleph\_0))$. However, I was wondering the simple question of this very generalization; specifically the following question:


$$\exists\kappa\exists\alpha\in\mathrm{Ord}(\aleph\_\alpha\not\leq^\*|\mathcal{P}(\kappa)|)?$$


If $\kappa$ satisfies the inner quantification of this formula, one could say $\kappa$ satisfies the Hartogs property (or $\mathrm{H}(\kappa)$). 


In, ZF + AD + DC what properties does $\mathrm{H}(\aleph\_1)$ have? What about even larger ordinals? What about in HOD, or even assuming $V=\mathrm{HOD}?$
(The original question was quickly shown true)



Under ZF + DC + AD, is it known what the properties are of the Hartogs number for $\mathcal{P}(\kappa)$ for some $\kappa>\aleph_0$?
This is a bit confused, but I *think* what you're asking is:



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> Let $S(\alpha)$ be the supremum of all the ordinals onto which $\mathcal{P}(\alpha)$ surjects. What can we say about e.g. $S(\omega\_1)$?
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*(And your original question was whether $S(\alpha)$ always exists, which it does.)* **If** this is your question, I think it's a good one (if it's not then you need to clarify what your question is), and the key fact is **Moschovakis' coding lemma** *(see around page 397 in [Kanamori's book](http://www.springer.com/us/book/9783540888666))*:



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> (ZF+AD) Suppose there is a surjection from $\mathcal{P}(\omega)$ to $\alpha$. Then there is a surejction from $\mathcal{P}(\omega)$ to $\mathcal{P}(\alpha)$.
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(There are also non-AD versions of the coding lemma.) This gives us:



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> Corollary (ZF+AD): Suppose $S(\omega)>\alpha$ (so $\mathcal{P}(\omega)$ surjects onto $\alpha$). Then $S(\omega)=S(\alpha)$.
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> *That is: for $\alpha<\Theta$, $S(\alpha)=\Theta$. In particular, $S(\omega\_1)=\Theta$ and $\Theta$ is a limit ordinal, since the powerset of a cardinal always surjects onto that cardinal's successor.*
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Proof: Since $\mathcal{P}(\omega)$ surjects onto $\alpha$, by the coding lemma $\mathcal{P}(\omega)$ surjects onto $\mathcal{P}(\alpha)$. But if $\beta<S(\alpha)$ this means that $\mathcal{P}(\alpha)$ surjects onto $\beta$, so by composing surjections we get that $\mathcal{P}(\omega)$ surjects onto $\beta$, so $\beta<S(\omega)$. (And conversely clearly $S(\omega)\le S(\alpha)$.)




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Having calculated $S(\alpha)$ up to $\Theta$, it's worth saying a bit at this point about $\Theta$ itself. The coding lemma shows that $\Theta$ is a limit, but this isn't the end of the story - lots more can be said. For example, Solovay showed that $\Theta=\omega\_\Theta$ and that $\Theta$ has uncountable cofinality. Additional hypotheses give you further "largeness" properties of $\Theta$, and in particular "$\Theta$ is regular" plays an important role in inner model theory (see e.g. [this paper of Sargsyan](https://arxiv.org/pdf/1206.2712.pdf)).


Note that it is quite possible that $\Theta$ is *not* regular - the regularity of $\Theta$ is a strong assumption, especially in conjunction with stronger determinacy principles (e.g. AD$\_\mathbb{R}$). In fact, $\Theta$ can have cofinality $\omega$. Interestingly, this cannot occur if $V=L(\mathbb{R})$ - Solovay showed that in $ZF+AD+V=L(\mathbb{R})$, $\Theta$ is regular.


