For $\gamma$ an ordinal, let “$H\_\gamma$” be the statement:



> 
> For all ordinals $\alpha$, we have $2^{\aleph\_\alpha} = \aleph\_{\alpha+\gamma}$.
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> 
> 


So clearly $H\_0$ is false, and so is $H\_\omega$; in fact, $H\_\gamma$ implies that $\gamma$ is successor (because otherwise $\operatorname{cf}\gamma = \omega\_\alpha$, say, and $\aleph\_{\alpha+\gamma} = 2^{\aleph\_\alpha} = (2^{\aleph\_\alpha})^{\aleph\_\alpha} = (\aleph\_{\alpha+\gamma})^{\aleph\_\alpha} = (\aleph\_{\alpha+\gamma})^{\operatorname{cf}\aleph\_{\alpha+\gamma}}$ gives a contradiction).


On the other hand, $H\_1$ is precisely the generalized continuum hypothesis (GCH) and it is consistent relative to ZFC.


I understand from [this question](https://mathoverflow.net/questions/79920/failure-of-the-gch) and [this one](https://mathoverflow.net/questions/138308/woodins-unpublished-proof-of-the-global-failure-of-gch) that $H\_2$ is known to be consistent relative to certain large cardinal assumptions, and perhaps even $H\_k$ for any concrete $k < \omega$.


What else, if anything, is known about the consistency of the various $H\_\gamma$? Might we perhaps construct¹ a successor ordinal $\gamma$ for which $H\_\gamma$ is demonstrably false?


1. Admittedly, I don't know how to phrase this question properly, because clearly “the smallest successor ordinal for which $H\_\gamma$ is false” is a definable successor ordinal for which $H\_\gamma$ is provably false, which is clearly not what I'm asking about. But a proof in ZFC that $H\_{\omega+1}$ is false (say) would be a good answer to my question.



What is known about the consistency of $2^{\aleph_\alpha} = \aleph_{\alpha+\gamma}$ for all $\alpha$?
1. By a result of Patai, $\gamma$ should be finite (this is exercise 5.15 in Jech's book).
2. For any finite $n>0, H\_n$ is consistent, see Merimovich's paper [*A power function with a fixed finite gap everywhere*](https://arxiv.org/abs/math/0005179).




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For completeness and since not everyone has Jech's book at hand, here's how the proof of (1) goes (note that Jech uses "$\beta$" in place of "$\gamma$"). Suppose $\gamma\ge \omega$ is such that $2^{\aleph\_\eta}=\aleph\_{\eta+\gamma}$ for all $\eta$. Letting $\alpha$ be minimal such that $\alpha+\gamma>\gamma$ we have $0<\alpha\le\gamma$ and $\alpha$ is a limit. Consider $\kappa=\aleph\_{\alpha\cdot 2}$. This $\kappa$ is obviously singular and by choice of $\alpha$ we have $$2^{\aleph\_{\alpha+\xi}}=\aleph\_{\alpha+\xi+\gamma}=\aleph\_{\alpha+\gamma}$$ for each $\xi<\alpha$. From this we get $$2^\kappa=\aleph\_{\alpha+\gamma}$$ (more generally, if $\lambda$ is singular and $\mu=\max\{2^\theta:\theta<\lambda\}$ exists then $2^\lambda=\mu$). On the other hand, we have $$2^\kappa=\aleph\_{\alpha\cdot 2+\gamma}=\aleph\_{\alpha+(\alpha+\gamma)}>\aleph\_{\alpha+\gamma}$$ again by choice of $\alpha$. So we have a contradiction.


