Mahlo cardinal: Difference between revisions

Ord is Mahlo has interesting consistency strength, as we've mentioned. Recall from [[Reflection principle|here]] that a cardinal \( \kappa \) is sound if \( V_\kappa \) is a full elementary substructure of \( V \). Such cardinals are massive, but their existence is provable in \( \mathrm{ZFC} \), due to the reflection principle. In particular, for all \(n\), we have a club of cardinals which are \(\Sigma_n\)-sound, and thus their intersection is also club. Meanwhile, say a cardinal \( \kappa \) is totally reflecting if it is sound and strongly inaccessible. Such cardinals are hyper-inaccessible and larger than virtually any other large cardinal axiom size-wise other than possibly stationary superhuges or Reinhardt cardinals. However, their consistency strength is not particularly high: it turns out that Ord is Mahlo has the same consistency strength as the existence of a totally reflecting cardinal, which shows that slight modifications of reflection principles can give large consistency strength.