Assuming 2 ℵ 0 = ℵ 1 and 2 ℵ 1 = ℵ 2 , we build a partial order that forces the existence of a well-order of H ( ω 2 ) lightface definable over ⟨ H ( ω 1 ), ∈⟩ and that preserves cardinal exponentiation and cofinalities.
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Asperó et al. (2012) studied this question.