$C^{(n)}$

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The classes $C^{(n)}$ are used by Bagaria to describe fine structure in the large cardinal hierarchy in a uniform way.[1] For $n\in\omega$, $C^{(n)}$ is defined as the class of $\Sigma_n$-correct ordinals, i.e. ordinals $\alpha$ where $V_\alpha\prec_{\Sigma_n}V$.

Applications

  • $C^{(n)}$-measurable
  • $C^{(n)}$-superstrong
  • $C^{(n)}$-extendible
  • $C^{(n)}$-$E_i$, where $E_i$ is a property related to rank-into-ranks