Can there exist a model V of ZFC in which aleph_omega is the least cardinal kappa such that P^V(kappa)≠P^HOD^V(kappa); that is, P^V(lambda)=P^HOD^V(lambda) for every cardinal lambda<aleph_omega^V, while P^V(aleph_omega^V)≠P^HOD^V(aleph_omega^V)?

Hayut’s HOD power-set disagreement question and optimality of Magidor’s Covering Theorem

Can there exist a model $V$ of $\mathrm{ZFC}$ in which $\aleph_\omega$ is the least cardinal $\kappa$ such that $\mathcal{P}^V(\kappa)\neq\mathcal{P}^{\mathrm{HOD}^V}(\kappa)$ ; that is, $\mathcal{P}^V(\lambda)=\mathcal{P}^{\mathrm{HOD}^V}(\lambda)$ for every cardinal $\lambda<\aleph_\omega^V$ , while $\mathcal{P}^V(\aleph_\omega^V)\neq\mathcal{P}^{\mathrm{HOD}^V}(\aleph_\omega^V)$ ?

References

Primary source

arXiv

Additional references

Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new unrefereed manuscript claims to settle the question conditionally: agreement can occur at the first limit of countable cofinality, certain larger limits are excluded, and an extra assumption in a standard covering theorem is necessary.

Hayut’s question concerns how closely $\mathrm{HOD}$ and $V$ can agree on power sets at singular cardinals, alongside the optimality of Magidor’s Covering Theorem. The latest manuscript claims results addressing all three aspects, but its conclusions remain unrefereed.

New HOD compactness results (date not stated)

The manuscript claims a conditional realization of the relevant disagreement pattern at $\aleph_\omega$ , assuming a measurable cardinal; it rules out first disagreement at singular strong limits of uncountable cofinality; and it shows that the GCH hypothesis in a Magidor covering result is necessary.

Current status (as of August 2026): The manuscript claims a conditional resolution of Hayut’s question and establishes the stated optimality result, but the claims are unverified, and the $\aleph_\omega$ realization depends on a measurable cardinal.

Sources

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