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4. Uniform Indiscernibles

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

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      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.