Abstract
Assuming $2^{\aleph_0}=\aleph_1$ and $2^{\aleph_1}=\aleph_2$, we build a partial order that forces the existence of a well—order of $H(\omega_2)$ lightface definable over $\langle H(\omega_2), \in\rangle$ and that preserves cardinal exponentiation and cofinalities.
Citation
David Asperó. Sy-David Friedman. "Definable well-orders of $H(\omega _2)$ and $GCH$." J. Symbolic Logic 77 (4) 1101 - 1121, December 2012. https://doi.org/10.2178/jsl.7704030
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