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?

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    $\begingroup$ You can always assume V=HOD (the latter the hereditarily ordinal definable sets). Indeed HOD is the largest class for which there exists a definable bijection with the class of ordinals. See for example, Drake or Jech's book. $\endgroup$ Commented Dec 10, 2014 at 14:27
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    $\begingroup$ To follow up on Philp Welch's comment: see the following posting by Joel Hamkins: mathoverflow.net/questions/180727/… $\endgroup$
    – Ali Enayat
    Commented Dec 10, 2014 at 14:32
  • $\begingroup$ You can force global choice over any model of $ZFC$ without adding any new sets. $\endgroup$ Commented Dec 10, 2014 at 14:40
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    $\begingroup$ @Colin : You can, consistently relative to ZF, have full GCH in HOD if you wish, or else weak failures of the kind ($\aleph_\omega$ is a strong limit) you mention. $\endgroup$ Commented Dec 10, 2014 at 15:03
  • $\begingroup$ @Mohammad: To be accurate, global choice can be forced over models of $\sf NBG+AC$ without adding sets. Doing this over models of $\sf ZFC$ requires us to extend the language by adding a the generic class. Sure, this is not an actual issue, but it's more accurate this way. $\endgroup$
    – Asaf Karagila
    Commented Dec 10, 2014 at 16:15

1 Answer 1


Two comments/answers:

(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.


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