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Theorems prove compactness in HOD for strong limit singular cardinals, suggesting impacts on inner models.
We prove two compactness theorems for HOD. First, if κ is a strong limit singular cardinal with uncountable cofinality and for stationarily many δ <κ , (δ ⁺)HOD=δ ⁺ , then (κ ⁺)HOD=κ ⁺ . Second, if κ is a singular cardinal with uncountable cofinality and stationarily many δ <κ are singular in HOD , then κ is singular in HOD . We also discuss the optimality of these results and show that the first theorem does not extend from HOD to other ω -club amenable inner models.
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Goldberg et al. (2025) studied this question.
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