Assume AD+DC. Assume $\kappa$ is an infinite cardinal and $N$ is a (set or class) transitive model of ZFC containing $\kappa$. 


Is it true that for all $\alpha<\kappa$, $N$ thinks that the power set of $\alpha$ has size less than $\kappa$?


This seems like obvious, at least for $\kappa<\Theta$, but I don't find a reference or an idea yet.



Assuming AD, is every infinite cardinal closed under power set in a choice model?
Some partial answers:


(1) Under the (possibly) additional assumption of $\mathsf{AD}^+$ we can prove the weaker conclusion that $\alpha^{+N} < \alpha^+$ for every infinite ordinal $\alpha < \Theta$:


* Assume toward a contradiction that $\alpha^{+N} = \alpha^+$ and use $\mathsf{AC}$ in $N$ to take a sequence of functions $(f\_\xi : \xi < \alpha^+)$ such that each function $f\_\xi$ is a surjection from $\alpha$ onto $\xi$. This shows that $\alpha^+$ is regular in $V$; otherwise we could combine cofinally many of these surjections to get a surjection from $\alpha \times \alpha$ to $\alpha^+$, a contradiction.
* By a theorem of Woodin every uncountable regular cardinal less than $\Theta$ is measurable (as stated by Koellner and Woodin, [Large cardinals from determinacy](http://logic.harvard.edu/koellner/LCFD.pdf), p. 12,) so $\alpha^+$ is measurable. But then we can take an ultrapower of the sequence $(f\_\xi : \xi < \alpha^+)$ to again get a surjection from $\alpha$ to $\alpha^+$, a contradiction.


(2) Under the stronger assumption of $\mathsf{AD} + V = L(\mathbb{R})$ we have $(2^\alpha)^N < \alpha^+$ for every infinite ordinal $\alpha < \Theta$, which implies the desired conclusion. This follows directly from Steel, [An outline of inner model theory](http://math.berkeley.edu/~steel/papers/steel1.pdf), Theorem 8.26, which says that for every infinite cardinal $\alpha < \Theta$, every wellordered family of subsets of $\alpha$ has cardinality at most $\alpha$.
Steel calls this result the "boldface $\mathsf{GCH}$ for $L(\mathbb{R})$" and proves it from $\mathsf{GCH}$ in $\text{HOD}\_x$ where $x$ is a real, which can be understood as a fine-structural model.


Note that the theorem that every uncountable regular cardinal less than $\Theta$ is measurable was first proved under the hypothesis $\mathsf{AD} + V = L(\mathbb{R})$ also using fine structure (see Steel, Theorem 8.27) before being generalized to the $\mathsf{AD}^+$ setting by Woodin. I don't know whether we should expect a similar generalization of the boldface $\mathsf{GCH}$.


(3) For cardinals $\kappa = \alpha^+ > \Theta$, even the weaker statement $\alpha^{+N} < \alpha^+$ can fail. Consider the case that $V = L(\mathcal{P}(\mathbb{R}))$, $\mathsf{AD}^+ + \mathsf{AD}\_\mathbb{R}$ holds, and $N = \text{HOD}$. We can add $\mathcal{P}(\mathbb{R})$ back to $\text{HOD}$ by a Vopěnka-type forcing of cardinality $\Theta$; this argument is due to Woodin and is written up in Section 4.3.4.2 of Nam Trang's [thesis](http://math.cmu.edu/~namtrang/thesis.pdf). This forcing doesn't collapse cardinals above $\Theta$, so $\alpha^{+N} = \alpha^+ = \kappa$. 


