Mahlo cardinal

An Mahlo cardinal is an ordinal/cardinal number that is strong limit and the inaccessible cardinals. The smallest Mahlo cardinal is sometimes called "the" Mahlo cardinal \(M\).

Breaking down the definition, an inaccessible cardinal \(\alpha\) must be:


 * Uncountable. \(\alpha \geq \omega_1\).
 * Inaccessible. \(\alpha \geq I\).
 * Strong limit. \(\alpha = \beth_\gamma\) for a limit ordinal \(\gamma\).

The Mahlo cardinals are most relevant to googology through ordinal collapsing functions.