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4. Uniform Indiscernibles

    1.     Let \kappa be an infinite cardinal and put \lambda=\kappa^+. Define u_2(\kappa)=\sup\left\{(\lambda^+)^{L[A]}:A\subseteq\kappa\right\}.

      Add remark

      Problem 4.25.

      [Magidor]
      1. Is u_2(\omega_1)=\omega_3 consistent with (sufficiently) large cardinals?
      2. How to increase u_2(\omega_1) without collapsing \omega_1 or \omega_2?
        1. Remark. The answer is trivially yes if V=L, so the relevant cardinals should be sufficiently large.

          It is known that u_2(\omega)=\omega_2 implies u_2(\omega_1)<\omega_3 under certain large cardinals assumptions (\text{NS}_{\omega_1} is saturated and it has a sharp).
            • Remark. [Larson] u_2(\omega_1)=\omega_3 is impossible if \text{NS}_{\omega_1} is saturated (more generally, if there is a generic embedding with critical point \omega_1 which fixed \omega_3).

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