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We prove that the assertion “There is a singular cardinal κ such that for all x ⊆ κ, HODx does not contain the entire powerset of κ” is equiconsistent with the assertion that there is a cardinal κ such that {o(ν) | ν < κ} is unbounded in κ. Shelah [9] proved that for every singular cardinal κ of uncountable cofinality there exists x ⊆ κ such that P(κ) ⊆ HODx. In work by Cummings, Friedman, Magidor, Rinot, and Sinapova [2] it has been shown that the statement is consistently false for a singular κ of cofinality ω, starting from the large cardinal assumption of an inaccessible cardinal λ and an infinite sequence of < λ-supercompact cardinals below it. In fact they prove the stronger assertion that it is consistent that there is a singular cardinal κ of cofinality ω such that for all x ⊆ κ, κ+ is inaccessible in HODx. Gitik and Merimovich ([6]) and Kafkoulis ([7]) have obtained the same result from a slightly weaker assumption (see Remarks 6,7 in [2]). We isolate the exact strength of the failure of Shelah’s theorem at a singular cardinal of cofinality ω. In particular, we prove the following theorems. Theorem 1. Assuming there is a cardinal κ of cofinality ω such that {o(ν) | ν < κ} is unbounded in κ, there is a forcing extension in which for every subset x of κ, HODx does not contain the powerset of κ. Theorem 2. Assuming that κ is a singular strong limit cardinal of cofinality ω such that {oK(ν) | ν < κ} is bounded in κ, there is x ⊆ κ such that P(κ) ⊆ HODx. We also show that in Theorem 1, κ can be made into אω using standard arguments. Recently, there has been renewed interest in the extent of covering properties for HOD. In particular, we mention papers of Cummings, Friedman and Golshani [3], the first and fourth author [1], and Gitik and Merimovich [6]. The central idea in all of these works is the notion of homogeneity of forcing posets. Theorem 1 is a continuation of this study, but at a considerably lower consistency strength. Date: August 30, 2019 This work is based upon research supported by the National Science Foundation under Grants No. DMS-1363364, DMS-1764029 (Neeman), DMS-1700425 (Unger), and DMS1800613 (Ben-Neria), and the Israel Science Foundation Grant No. 58/14 (Gitik).

On the powersets of singular cardinals in HOD