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1. HOD


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      Problem 1.2.

      [Poveda]
      1. Is it consistent that every uncountable regular cardinal is supercompact in HOD?

      2. What about \kappa^+ is supercompact in HOD whenever \kappa is singular strong limit of uncountable cofinality.
          (Ben-Neria and Unger) The answer to part 1 is affirmative if supercompact is weakened to measurable “and a bit more”.

      (Shelah) For such a kappa, there exists X\subseteq\kappa such that \mathcal{P}(\kappa)\subseteq\text{HOD}_X. In particular, \kappa^+ cannot be supercompact in HOD.

          Cite this as: AimPL: From ℵ2 to infinity, available at http://aimpl.org/alephtwo.