Mahlo cardinal

In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by Paul Mahlo (1911, 1912, 1913). As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by ZFC (assuming ZFC is consistent).

A cardinal number ${\displaystyle \kappa }$ is called strongly Mahlo if ${\displaystyle \kappa }$ is strongly inaccessible and the set ${\displaystyle U=\{\lambda <\kappa \mid \lambda {\text{ is strongly inaccessible}}\}}$ is stationary in κ.

A cardinal ${\displaystyle \kappa }$ is called weakly Mahlo if ${\displaystyle \kappa }$ is weakly inaccessible and the set of weakly inaccessible cardinals less than ${\displaystyle \kappa }$ is stationary in ${\displaystyle \kappa }$.

The term "Mahlo cardinal" now usually means "strongly Mahlo cardinal", though the cardinals originally considered by Mahlo were weakly Mahlo cardinals.