Mahlo cardinal: Difference between revisions

If ''X'' is a class of ordinals, them we can form a new class of ordinals ''M''(''X'') consisting of the ordinals α of uncountable cofinality such that α∩''X'' is stationary in α. This operation ''M'' is called the '''Mahlo operation'''. It can be used to define Mahlo cardinals: for example, if ''X'' is the class of regular cardinals, then ''M''(''X'') is the class of weakly Mahlo cardinals. The condition that α has uncountable cofinality ensures that the closed unbounded subsets of α are closed under intersection and so form a filter; in practice the elements of ''X'' often already have uncountable cofinality in which case this condition is redundant. Some authors add the condition that α is in ''X'', which in practice usually makes little difference as it is often automatically satisfied.