I need a global choice function defined by a formula in (a fragment of) ZF. There is no harm in assuming V=L for my purposes. But I wonder if there are any familiar alternative ways to get this? 


The comments make see I also want a weakened part of GCH. Namely the power set of $\aleph\_n,\ n\in\mathbb{N}$ should be $\aleph\_m$ for some $m\in\mathbb{N}$. I see that HOD is known compatible with some extreme failure of CH, but I have not found what. Is it compatible with failure of this weakened part of GCH? 



Are there known ways to posit definable global choice in ZF without positing V=L?

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> Two comments/answers:
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**(1)** By an old theorem of Roguski, for any $\Sigma\_2^{\text{ZFC}}$ sentence $\phi$, the theories $\text{ZFC} + \phi$ and $\text{ZFC + V=HOD} + \phi$ are equiconsistent.


Roguski's result appears in his paper *Extensions of models for ZFC to models for ZF+V=HOD with applications*, in Set theory and hierarchy theory, pp. 241–247. Lecture Notes in Math., Vol. 537, Springer, Berlin, 1976. 


**(2)** Since there is a parenthetical reference to fragments of $\text{ZF}$ in the first line of the question: the formulation and salient consequences of $\text{V=HOD}$ heavily depend on stratification of the universe into rank initial segments of the form $V\_\alpha$, and on the veracity of the Montague-Levy reflection theorem; the latter is equivalent over $\text{ZF}$ without $\{\text{Replacement, Infinity}\}$ to the conjunction of $\text{Replacement}$ and $\text{Infinity}$ (by an old result of Azriel Levy). 


So $\text{V=L}$, rather than $\text{V=HOD}$, is the safe way to arrange global choice, at least for fragments of $\text{ZF}$ that extend Kripke-Platek set theory.


