Talk:Rowbottom cardinal

Equivalent property

On p. 5 of Mitchell's "Inner Models for Large Cardinals" I found that there is a model-theoretic characterization of Rowbottomness of $\kappa$, however I cannot find Mitchell's characterization in either The Higher Infinite or Rowbottom's paper. Theorem 2.1 in Rowbottom's paper does almost give a model-theoretic characterization of $\lambda$-Rowbottom cardinals (written as ${\mathcal {C}}_{\omega }(\kappa ,\lambda ,\kappa ,\lambda )$ in the terminology of the paper), but not quite, as for the theorem to apply to $\lambda$-Rowbottomness, the quintuple $\omega ,\kappa ,\lambda ,\kappa ,\lambda$ would have to be relevant, but $\kappa >\lambda >\lambda \geq \omega$ fails. Is anyone aware of a source for either of these or any similar characterizations in order to add them? C7XWiki (talk) 07:32, 12 May 2024 (UTC)