Mahlo cardinal: Difference between revisions

a cardinal κ is called α-Mahlo for some ordinal α if κ is strongly inaccessible and for every ordinal β<α, the set of β-Mahlo cardinals below κ is stationary in κ.<ref>W. Boos, ''[https://www.proquest.com/openview/22276484b3832b4d487425ee882b5b09/1 Nonstandard Large Cardinals]'' (1971).</ref>^{p. 3} However the condition "κ is strongly inaccessible" is sometimes replaced by other conditions, such as "κ is regular" or "κ is weakly inaccessible" or "κ is Mahlo". We can define "hyper-Mahlo", "α-hyper-Mahlo", "hyper-hyper-Mahlo", "weakly α-Mahlo", "weakly hyper-Mahlo", "weakly α-hyper-Mahlo", and so on, by analogy with the definitions for inaccessibles, so for example a cardinal κ is called hyper-Mahlo if it is κ-Mahlo.

A regular uncountable cardinal κ is '''greatly Mahlo''' if and only if it is inaccessible and there is a normal (i.e. nontrivial and closed under [[diagonal intersection]]s) κ-complete [[Filter (mathematics)|filter]] on the power set of κ that is closed under the Mahlo operation, which maps the set of ordinals ''S'' to {α<math>\in</math>''S'': α has uncountable cofinality and S∩α is stationary in α}