Remarkable cardinal

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Remarkable cardinals were introduced by Schinder in [1] to provide precise consistency strength of the statement that $L(\mathbb R)$ cannot be modified by proper forcing. A cardinal $\kappa$ is remarkable if for each regular $\lambda>\kappa$, there exists a countable transitive $M$ and an elementary embedding $e:M\rightarrow H_\lambda$ with $\kappa\in \text{ran}(e)$ and also a countable transitive $N$ and an elementary embedding $\theta:M\to N$ such that:

  • the critical point of $\theta$ is $e^{-1}(\kappa)$,
  • $\text{Ord}^M$ is a regular cardinal in $N$,
  • $M=H^N_{\text{Ord}^M}$,
  • $\theta(e^{-1}(\kappa))>\text{Ord}^M$.

Remarkable cardinals could be called virtually supercompact, because the following alternative definition is an exact analogue of the definition of supercompact cardinals by Magidor [Mag71]: A cardinal $κ$ is remarkable iff for every $η > κ$, there is $α < κ$ such that in a set-forcing extension there is an elementary embedding $j : V_α → V_η$ with $j(\mathrm{crit}(j)) = κ$.[2]

Remarkable cardinals and the constructible universe

  • Remarkable cardinals are downward absolute to $L$. [1]
  • If $0^\sharp$ exists, then every Silver indiscernible is remarkable in $L$. [1]

Relations with other large cardinals

References

  1. Schindler, Ralf-Dieter. Proper forcing and remarkable cardinals. Bull Symbolic Logic 6(2):176--184, 2000. www   DOI   MR   bibtex
  2. Gitman, Victoria and Shindler, Ralf. Virtual large cardinals. www   bibtex
  3. Gitman, Victoria and Welch, Philip. Ramsey-like cardinals II. J Symbolic Logic 76(2):541--560, 2011. www   arχiv   MR   bibtex
  4. Nielsen, Dan Saattrup and Welch, Philip. Games and Ramsey-like cardinals. , 2018. arχiv   bibtex
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